# This Quantum World/Atoms#What does an atom look like.3F

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

None of these images depicts an atom *as it is*. This is because it is
impossible to even visualize an atom *as it is*. Whereas the best you
can do with the images in the first row is to erase them from your
memory---they represent a way of viewing the atom that is too simplified
for the way we want to start thinking about it---the eight fuzzy images
in the next row deserve scrutiny. Each represents an aspect of a
stationary state of atomic hydrogen. You see neither the nucleus (a
proton) nor the electron. What you see is a fuzzy position. To be
precise, what you see are cloud-like blurs, which are symmetrical about
the vertical and horizontal axes, and which represent the atom\'s
internal relative position---the position of the electron relative to
the proton *or* the position of the proton relative to the electron.

-   What is the *state* of an atom?
-   What is a *stationary* state?
-   What exactly is a *fuzzy* position?
-   How does such a blur represent the atom\'s internal relative
    position?
-   Why can we not describe the atom\'s internal relative position *as
    it is*?

## Quantum states

In quantum mechanics, **states** are
probability algorithms. We use them to calculate the probabilities of
the possible outcomes of
measurements on the
basis of actual measurement outcomes. A quantum state takes as its input

-   one or several measurement outcomes,
-   a measurement M,
-   the time of M,

and it yields as its output the probabilities of the possible outcomes
of M.

A quantum state is called **stationary** if the probabilities it assigns
are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density
function
$\rho(\boldsymbol{r})$. Imagine a small region $R$ like the little box
inside the first blur. And suppose that this is a region of the
(mathematical) space of positions relative to the proton. If you
integrate $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. If it
were inside, the probability of finding it outside would be zero, and if
it were outside, the probability of finding it inside would be zero.
*After* the measurement, on the other hand, the electron is either
inside or outside $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    is neither true nor false; it lacks a (definite) truth
    value.
-   A measurement generally changes the state of the system on which it
    is performed.

As mentioned before, probabilities are assigned not only *to*
measurement outcomes but also *on the basis of* measurement outcomes.
Each density function $\rho_{nlm}$ serves to assign probabilities to the
possible outcomes of a measurement of the electron\'s position relative
to the proton. And in each case the assignment is based on the outcomes
of a simultaneous measurement of three observables: the atom\'s energy
(specified by the value of the principal quantum number $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ is neither 0 nor 1.

What about observables that are generally described as continuous, like
a position?

The description of an observable as \"continuous\" is potentially
misleading. For one thing, we cannot separate an observable and its
possible values from a measurement and its possible outcomes, and a
measurement with an uncountable set of possible outcomes is not even in
principle possible. For another, there is not a single observable called
\"position\". Different partitions of space define different position
measurements with different sets of possible outcomes.

-   Corollary: The possible outcomes of a position measurement (or the
    possible values of a position observable) are defined by a partition
    of space. They make up a finite or countable set of *regions* of
    space. An exact position is therefore neither a possible measurement
    outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron\'s fuzzy
position relative to the proton? Strictly speaking, they graphically
represent probability densities in the mathematical space of exact
relative positions, rather than fuzzy positions. It is these probability
densities that represent fuzzy positions by allowing us to calculate the
probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom\'s internal
relative position *as it is*. To describe a fuzzy observable is to
assign probabilities to the possible outcomes of a measurement. But a
description that rests on the assumption that a measurement is made,
does not describe an observable *as it is* (by itself, *regardless of
measurements*).

```{=html}
<div class="noprint">
```
**NEXT \>**

```{=html}
</div>
```


# This Quantum World/Atoms#Like this?

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

None of these images depicts an atom *as it is*. This is because it is
impossible to even visualize an atom *as it is*. Whereas the best you
can do with the images in the first row is to erase them from your
memory---they represent a way of viewing the atom that is too simplified
for the way we want to start thinking about it---the eight fuzzy images
in the next row deserve scrutiny. Each represents an aspect of a
stationary state of atomic hydrogen. You see neither the nucleus (a
proton) nor the electron. What you see is a fuzzy position. To be
precise, what you see are cloud-like blurs, which are symmetrical about
the vertical and horizontal axes, and which represent the atom\'s
internal relative position---the position of the electron relative to
the proton *or* the position of the proton relative to the electron.

-   What is the *state* of an atom?
-   What is a *stationary* state?
-   What exactly is a *fuzzy* position?
-   How does such a blur represent the atom\'s internal relative
    position?
-   Why can we not describe the atom\'s internal relative position *as
    it is*?

## Quantum states

In quantum mechanics, **states** are
probability algorithms. We use them to calculate the probabilities of
the possible outcomes of
measurements on the
basis of actual measurement outcomes. A quantum state takes as its input

-   one or several measurement outcomes,
-   a measurement M,
-   the time of M,

and it yields as its output the probabilities of the possible outcomes
of M.

A quantum state is called **stationary** if the probabilities it assigns
are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density
function
$\rho(\boldsymbol{r})$. Imagine a small region $R$ like the little box
inside the first blur. And suppose that this is a region of the
(mathematical) space of positions relative to the proton. If you
integrate $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. If it
were inside, the probability of finding it outside would be zero, and if
it were outside, the probability of finding it inside would be zero.
*After* the measurement, on the other hand, the electron is either
inside or outside $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    is neither true nor false; it lacks a (definite) truth
    value.
-   A measurement generally changes the state of the system on which it
    is performed.

As mentioned before, probabilities are assigned not only *to*
measurement outcomes but also *on the basis of* measurement outcomes.
Each density function $\rho_{nlm}$ serves to assign probabilities to the
possible outcomes of a measurement of the electron\'s position relative
to the proton. And in each case the assignment is based on the outcomes
of a simultaneous measurement of three observables: the atom\'s energy
(specified by the value of the principal quantum number $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ is neither 0 nor 1.

What about observables that are generally described as continuous, like
a position?

The description of an observable as \"continuous\" is potentially
misleading. For one thing, we cannot separate an observable and its
possible values from a measurement and its possible outcomes, and a
measurement with an uncountable set of possible outcomes is not even in
principle possible. For another, there is not a single observable called
\"position\". Different partitions of space define different position
measurements with different sets of possible outcomes.

-   Corollary: The possible outcomes of a position measurement (or the
    possible values of a position observable) are defined by a partition
    of space. They make up a finite or countable set of *regions* of
    space. An exact position is therefore neither a possible measurement
    outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron\'s fuzzy
position relative to the proton? Strictly speaking, they graphically
represent probability densities in the mathematical space of exact
relative positions, rather than fuzzy positions. It is these probability
densities that represent fuzzy positions by allowing us to calculate the
probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom\'s internal
relative position *as it is*. To describe a fuzzy observable is to
assign probabilities to the possible outcomes of a measurement. But a
description that rests on the assumption that a measurement is made,
does not describe an observable *as it is* (by itself, *regardless of
measurements*).

```{=html}
<div class="noprint">
```
**NEXT \>**

```{=html}
</div>
```


# This Quantum World/Atoms#Or like this?

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

None of these images depicts an atom *as it is*. This is because it is
impossible to even visualize an atom *as it is*. Whereas the best you
can do with the images in the first row is to erase them from your
memory---they represent a way of viewing the atom that is too simplified
for the way we want to start thinking about it---the eight fuzzy images
in the next row deserve scrutiny. Each represents an aspect of a
stationary state of atomic hydrogen. You see neither the nucleus (a
proton) nor the electron. What you see is a fuzzy position. To be
precise, what you see are cloud-like blurs, which are symmetrical about
the vertical and horizontal axes, and which represent the atom\'s
internal relative position---the position of the electron relative to
the proton *or* the position of the proton relative to the electron.

-   What is the *state* of an atom?
-   What is a *stationary* state?
-   What exactly is a *fuzzy* position?
-   How does such a blur represent the atom\'s internal relative
    position?
-   Why can we not describe the atom\'s internal relative position *as
    it is*?

## Quantum states

In quantum mechanics, **states** are
probability algorithms. We use them to calculate the probabilities of
the possible outcomes of
measurements on the
basis of actual measurement outcomes. A quantum state takes as its input

-   one or several measurement outcomes,
-   a measurement M,
-   the time of M,

and it yields as its output the probabilities of the possible outcomes
of M.

A quantum state is called **stationary** if the probabilities it assigns
are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density
function
$\rho(\boldsymbol{r})$. Imagine a small region $R$ like the little box
inside the first blur. And suppose that this is a region of the
(mathematical) space of positions relative to the proton. If you
integrate $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. If it
were inside, the probability of finding it outside would be zero, and if
it were outside, the probability of finding it inside would be zero.
*After* the measurement, on the other hand, the electron is either
inside or outside $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    is neither true nor false; it lacks a (definite) truth
    value.
-   A measurement generally changes the state of the system on which it
    is performed.

As mentioned before, probabilities are assigned not only *to*
measurement outcomes but also *on the basis of* measurement outcomes.
Each density function $\rho_{nlm}$ serves to assign probabilities to the
possible outcomes of a measurement of the electron\'s position relative
to the proton. And in each case the assignment is based on the outcomes
of a simultaneous measurement of three observables: the atom\'s energy
(specified by the value of the principal quantum number $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ is neither 0 nor 1.

What about observables that are generally described as continuous, like
a position?

The description of an observable as \"continuous\" is potentially
misleading. For one thing, we cannot separate an observable and its
possible values from a measurement and its possible outcomes, and a
measurement with an uncountable set of possible outcomes is not even in
principle possible. For another, there is not a single observable called
\"position\". Different partitions of space define different position
measurements with different sets of possible outcomes.

-   Corollary: The possible outcomes of a position measurement (or the
    possible values of a position observable) are defined by a partition
    of space. They make up a finite or countable set of *regions* of
    space. An exact position is therefore neither a possible measurement
    outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron\'s fuzzy
position relative to the proton? Strictly speaking, they graphically
represent probability densities in the mathematical space of exact
relative positions, rather than fuzzy positions. It is these probability
densities that represent fuzzy positions by allowing us to calculate the
probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom\'s internal
relative position *as it is*. To describe a fuzzy observable is to
assign probabilities to the possible outcomes of a measurement. But a
description that rests on the assumption that a measurement is made,
does not describe an observable *as it is* (by itself, *regardless of
measurements*).

```{=html}
<div class="noprint">
```
**NEXT \>**

```{=html}
</div>
```


# This Quantum World/Atoms#Quantum states

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

None of these images depicts an atom *as it is*. This is because it is
impossible to even visualize an atom *as it is*. Whereas the best you
can do with the images in the first row is to erase them from your
memory---they represent a way of viewing the atom that is too simplified
for the way we want to start thinking about it---the eight fuzzy images
in the next row deserve scrutiny. Each represents an aspect of a
stationary state of atomic hydrogen. You see neither the nucleus (a
proton) nor the electron. What you see is a fuzzy position. To be
precise, what you see are cloud-like blurs, which are symmetrical about
the vertical and horizontal axes, and which represent the atom\'s
internal relative position---the position of the electron relative to
the proton *or* the position of the proton relative to the electron.

-   What is the *state* of an atom?
-   What is a *stationary* state?
-   What exactly is a *fuzzy* position?
-   How does such a blur represent the atom\'s internal relative
    position?
-   Why can we not describe the atom\'s internal relative position *as
    it is*?

## Quantum states

In quantum mechanics, **states** are
probability algorithms. We use them to calculate the probabilities of
the possible outcomes of
measurements on the
basis of actual measurement outcomes. A quantum state takes as its input

-   one or several measurement outcomes,
-   a measurement M,
-   the time of M,

and it yields as its output the probabilities of the possible outcomes
of M.

A quantum state is called **stationary** if the probabilities it assigns
are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density
function
$\rho(\boldsymbol{r})$. Imagine a small region $R$ like the little box
inside the first blur. And suppose that this is a region of the
(mathematical) space of positions relative to the proton. If you
integrate $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. If it
were inside, the probability of finding it outside would be zero, and if
it were outside, the probability of finding it inside would be zero.
*After* the measurement, on the other hand, the electron is either
inside or outside $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    is neither true nor false; it lacks a (definite) truth
    value.
-   A measurement generally changes the state of the system on which it
    is performed.

As mentioned before, probabilities are assigned not only *to*
measurement outcomes but also *on the basis of* measurement outcomes.
Each density function $\rho_{nlm}$ serves to assign probabilities to the
possible outcomes of a measurement of the electron\'s position relative
to the proton. And in each case the assignment is based on the outcomes
of a simultaneous measurement of three observables: the atom\'s energy
(specified by the value of the principal quantum number $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ is neither 0 nor 1.

What about observables that are generally described as continuous, like
a position?

The description of an observable as \"continuous\" is potentially
misleading. For one thing, we cannot separate an observable and its
possible values from a measurement and its possible outcomes, and a
measurement with an uncountable set of possible outcomes is not even in
principle possible. For another, there is not a single observable called
\"position\". Different partitions of space define different position
measurements with different sets of possible outcomes.

-   Corollary: The possible outcomes of a position measurement (or the
    possible values of a position observable) are defined by a partition
    of space. They make up a finite or countable set of *regions* of
    space. An exact position is therefore neither a possible measurement
    outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron\'s fuzzy
position relative to the proton? Strictly speaking, they graphically
represent probability densities in the mathematical space of exact
relative positions, rather than fuzzy positions. It is these probability
densities that represent fuzzy positions by allowing us to calculate the
probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom\'s internal
relative position *as it is*. To describe a fuzzy observable is to
assign probabilities to the possible outcomes of a measurement. But a
description that rests on the assumption that a measurement is made,
does not describe an observable *as it is* (by itself, *regardless of
measurements*).

```{=html}
<div class="noprint">
```
**NEXT \>**

```{=html}
</div>
```


# This Quantum World/Atoms#Fuzzy observables

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

None of these images depicts an atom *as it is*. This is because it is
impossible to even visualize an atom *as it is*. Whereas the best you
can do with the images in the first row is to erase them from your
memory---they represent a way of viewing the atom that is too simplified
for the way we want to start thinking about it---the eight fuzzy images
in the next row deserve scrutiny. Each represents an aspect of a
stationary state of atomic hydrogen. You see neither the nucleus (a
proton) nor the electron. What you see is a fuzzy position. To be
precise, what you see are cloud-like blurs, which are symmetrical about
the vertical and horizontal axes, and which represent the atom\'s
internal relative position---the position of the electron relative to
the proton *or* the position of the proton relative to the electron.

-   What is the *state* of an atom?
-   What is a *stationary* state?
-   What exactly is a *fuzzy* position?
-   How does such a blur represent the atom\'s internal relative
    position?
-   Why can we not describe the atom\'s internal relative position *as
    it is*?

## Quantum states

In quantum mechanics, **states** are
probability algorithms. We use them to calculate the probabilities of
the possible outcomes of
measurements on the
basis of actual measurement outcomes. A quantum state takes as its input

-   one or several measurement outcomes,
-   a measurement M,
-   the time of M,

and it yields as its output the probabilities of the possible outcomes
of M.

A quantum state is called **stationary** if the probabilities it assigns
are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density
function
$\rho(\boldsymbol{r})$. Imagine a small region $R$ like the little box
inside the first blur. And suppose that this is a region of the
(mathematical) space of positions relative to the proton. If you
integrate $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. If it
were inside, the probability of finding it outside would be zero, and if
it were outside, the probability of finding it inside would be zero.
*After* the measurement, on the other hand, the electron is either
inside or outside $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    is neither true nor false; it lacks a (definite) truth
    value.
-   A measurement generally changes the state of the system on which it
    is performed.

As mentioned before, probabilities are assigned not only *to*
measurement outcomes but also *on the basis of* measurement outcomes.
Each density function $\rho_{nlm}$ serves to assign probabilities to the
possible outcomes of a measurement of the electron\'s position relative
to the proton. And in each case the assignment is based on the outcomes
of a simultaneous measurement of three observables: the atom\'s energy
(specified by the value of the principal quantum number $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ is neither 0 nor 1.

What about observables that are generally described as continuous, like
a position?

The description of an observable as \"continuous\" is potentially
misleading. For one thing, we cannot separate an observable and its
possible values from a measurement and its possible outcomes, and a
measurement with an uncountable set of possible outcomes is not even in
principle possible. For another, there is not a single observable called
\"position\". Different partitions of space define different position
measurements with different sets of possible outcomes.

-   Corollary: The possible outcomes of a position measurement (or the
    possible values of a position observable) are defined by a partition
    of space. They make up a finite or countable set of *regions* of
    space. An exact position is therefore neither a possible measurement
    outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron\'s fuzzy
position relative to the proton? Strictly speaking, they graphically
represent probability densities in the mathematical space of exact
relative positions, rather than fuzzy positions. It is these probability
densities that represent fuzzy positions by allowing us to calculate the
probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom\'s internal
relative position *as it is*. To describe a fuzzy observable is to
assign probabilities to the possible outcomes of a measurement. But a
description that rests on the assumption that a measurement is made,
does not describe an observable *as it is* (by itself, *regardless of
measurements*).

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# This Quantum World/Serious illnesses

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## Planck

Quantum mechanics began as a desperate measure to get around some
spectacular failures of what subsequently came to be known as classical
physics.

In 1900 Max Planck discovered a law that
perfectly describes the spectrum of a glowing hot object. Planck\'s
radiation formula
turned out to be irreconcilable with the physics of his time. (If
classical physics were right, you would be blinded by ultraviolet light
if you looked at the burner of a stove, aka the UV
catastrophe.) At first, it was just a fit
to the data, \"a fortuitous guess at an interpolation formula\" as
Planck himself called it. Only weeks later did it turn out to imply the
quantization of energy for the emission of electromagnetic
radiation: the energy $~E~$ of
a quantum of radiation is proportional to the frequency $\nu$ of the
radiation, the constant of proportionality being Planck\'s
constant $~h~$

$$~E = h\nu~$$. We can of course use the angular
frequency $~\omega=2\pi\nu~$ instead of
$\nu$. Introducing the reduced Planck constant $~\hbar=h/2\pi~$, we then
have

$$E = \hbar\omega$$. This theory is valid at all temperatures and
helpful in explaining radiation by black bodies.

## Rutherford

In 1911 Ernest Rutherford proposed a
model of the atom based on experiments
by Geiger and Marsden. Geiger
and Marsden had directed a beam of alpha
particles at a thin gold foil. Most of the
particles passed the foil more or less as expected, but about one in
8000 bounced back as if it had encountered a much heavier object. In
Rutherford\'s own words this was as incredible as if you fired a 15 inch
cannon ball at a piece of tissue paper and it came back and hit you.
After analysing the data collected by Geiger and Marsden, Rutherford
concluded that the diameter of the atomic nucleus (which contains over
99.9% of the atom\'s mass) was less than 0.01% of the diameter of the
entire atom. He suggested that the atom is spherical in shape and the
atomic electrons orbit the nucleus much like planets orbit a star. He
calculated mass of electron as 1/7000th part of the mass of an alpha
particle. Rutherford\'s atomic model is also called the Nuclear model.

The problem of having electrons orbit the nucleus the same way that a
planet orbits a star is that classical electromagnetic theory demands
that an orbiting electron will radiate away its energy and spiral into
the nucleus in about 0.5×10^-10^ seconds. This was the worst
quantitative failure in the history of physics, under-predicting the
lifetime of hydrogen by at least forty orders of magnitude! (This figure
is based on the experimentally established lower bound on the proton\'s
lifetime.)

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```
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# This Quantum World/Serious illnesses#Planck

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## Planck

Quantum mechanics began as a desperate measure to get around some
spectacular failures of what subsequently came to be known as classical
physics.

In 1900 Max Planck discovered a law that
perfectly describes the spectrum of a glowing hot object. Planck\'s
radiation formula
turned out to be irreconcilable with the physics of his time. (If
classical physics were right, you would be blinded by ultraviolet light
if you looked at the burner of a stove, aka the UV
catastrophe.) At first, it was just a fit
to the data, \"a fortuitous guess at an interpolation formula\" as
Planck himself called it. Only weeks later did it turn out to imply the
quantization of energy for the emission of electromagnetic
radiation: the energy $~E~$ of
a quantum of radiation is proportional to the frequency $\nu$ of the
radiation, the constant of proportionality being Planck\'s
constant $~h~$

$$~E = h\nu~$$. We can of course use the angular
frequency $~\omega=2\pi\nu~$ instead of
$\nu$. Introducing the reduced Planck constant $~\hbar=h/2\pi~$, we then
have

$$E = \hbar\omega$$. This theory is valid at all temperatures and
helpful in explaining radiation by black bodies.

## Rutherford

In 1911 Ernest Rutherford proposed a
model of the atom based on experiments
by Geiger and Marsden. Geiger
and Marsden had directed a beam of alpha
particles at a thin gold foil. Most of the
particles passed the foil more or less as expected, but about one in
8000 bounced back as if it had encountered a much heavier object. In
Rutherford\'s own words this was as incredible as if you fired a 15 inch
cannon ball at a piece of tissue paper and it came back and hit you.
After analysing the data collected by Geiger and Marsden, Rutherford
concluded that the diameter of the atomic nucleus (which contains over
99.9% of the atom\'s mass) was less than 0.01% of the diameter of the
entire atom. He suggested that the atom is spherical in shape and the
atomic electrons orbit the nucleus much like planets orbit a star. He
calculated mass of electron as 1/7000th part of the mass of an alpha
particle. Rutherford\'s atomic model is also called the Nuclear model.

The problem of having electrons orbit the nucleus the same way that a
planet orbits a star is that classical electromagnetic theory demands
that an orbiting electron will radiate away its energy and spiral into
the nucleus in about 0.5×10^-10^ seconds. This was the worst
quantitative failure in the history of physics, under-predicting the
lifetime of hydrogen by at least forty orders of magnitude! (This figure
is based on the experimentally established lower bound on the proton\'s
lifetime.)

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```
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# This Quantum World/Serious illnesses#Rutherford

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```
**\< PREVIOUS**

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## Planck

Quantum mechanics began as a desperate measure to get around some
spectacular failures of what subsequently came to be known as classical
physics.

In 1900 Max Planck discovered a law that
perfectly describes the spectrum of a glowing hot object. Planck\'s
radiation formula
turned out to be irreconcilable with the physics of his time. (If
classical physics were right, you would be blinded by ultraviolet light
if you looked at the burner of a stove, aka the UV
catastrophe.) At first, it was just a fit
to the data, \"a fortuitous guess at an interpolation formula\" as
Planck himself called it. Only weeks later did it turn out to imply the
quantization of energy for the emission of electromagnetic
radiation: the energy $~E~$ of
a quantum of radiation is proportional to the frequency $\nu$ of the
radiation, the constant of proportionality being Planck\'s
constant $~h~$

$$~E = h\nu~$$. We can of course use the angular
frequency $~\omega=2\pi\nu~$ instead of
$\nu$. Introducing the reduced Planck constant $~\hbar=h/2\pi~$, we then
have

$$E = \hbar\omega$$. This theory is valid at all temperatures and
helpful in explaining radiation by black bodies.

## Rutherford

In 1911 Ernest Rutherford proposed a
model of the atom based on experiments
by Geiger and Marsden. Geiger
and Marsden had directed a beam of alpha
particles at a thin gold foil. Most of the
particles passed the foil more or less as expected, but about one in
8000 bounced back as if it had encountered a much heavier object. In
Rutherford\'s own words this was as incredible as if you fired a 15 inch
cannon ball at a piece of tissue paper and it came back and hit you.
After analysing the data collected by Geiger and Marsden, Rutherford
concluded that the diameter of the atomic nucleus (which contains over
99.9% of the atom\'s mass) was less than 0.01% of the diameter of the
entire atom. He suggested that the atom is spherical in shape and the
atomic electrons orbit the nucleus much like planets orbit a star. He
calculated mass of electron as 1/7000th part of the mass of an alpha
particle. Rutherford\'s atomic model is also called the Nuclear model.

The problem of having electrons orbit the nucleus the same way that a
planet orbits a star is that classical electromagnetic theory demands
that an orbiting electron will radiate away its energy and spiral into
the nucleus in about 0.5×10^-10^ seconds. This was the worst
quantitative failure in the history of physics, under-predicting the
lifetime of hydrogen by at least forty orders of magnitude! (This figure
is based on the experimentally established lower bound on the proton\'s
lifetime.)

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```
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```


# This Quantum World/Feynman route

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# The Feynman route to Schrödinger

The probabilities of the possible outcomes of measurements performed at
a time $t_2$ are determined by the Schrödinger wave function
$\psi(\mathbf{r},t_2)$. The wave function $\psi(\mathbf{r},t_2)$ is
determined via the Schrödinger
equation by
$\psi(\mathbf{r},t_1).$ What determines $\psi(\mathbf{r},t_1)$ ? Why,
the outcome of a measurement performed at $t_1$ --- what else? Actual
measurement outcomes determine the probabilities of possible measurement
outcomes.

## Two rules

In this chapter we develop the quantum-mechanical probability algorithm
from two fundamental rules. To begin with, two definitions:

-   **Alternatives** are possible sequences of measurement outcomes.
-   With each alternative is associated a complex
    number called **amplitude**.

Suppose that you want to calculate the probability of a possible outcome
of a measurement given the actual outcome of an earlier measurement.
Here is what you have to do:

-   Choose any sequence of measurements that may be made in the
    meantime.
-   Assign an amplitude to each alternative.
-   Apply either of the following rules:

:   \
    **Rule A**: If the intermediate measurements are made (or if it is
    possible to infer from other measurements what their outcomes would
    have been if they had been made), first square the absolute values
    of the amplitudes of the alternatives and then add the results.

```{=html}
<!-- -->
```

:   **Rule B**: If the intermediate measurements are not made (and if it
    is not possible to infer from other measurements what their outcomes
    would have been), first add the amplitudes of the alternatives and
    then square the absolute value of the result.

\
In subsequent sections we will explore the consequences of these rules
for a variety of setups, and we will think about their origin --- their
*raison d\'être*. Here we shall use Rule B to determine the
interpretation of $\overline{\psi}(k)$ given Born\'s probabilistic
interpretation of $\psi(x)$.

In the so-called \"continuum normalization\", the unphysical limit of a
particle with a sharp momentum $\hbar k'$ is associated with the wave
function

$$\psi_{k'}(x,t)=\frac1{\sqrt{2\pi}}\int\delta(k-k')\,e^{i[kx-\omega(k)t]}dk=
\frac1{\sqrt{2\pi}}\,e^{i[k'x-\omega(k')t]}.$$ Hence we may write
$\psi(x,t) = \int\overline{\psi}(k)\,\psi_{k}(x,t)\,dk.$

$\overline{\psi}(k)$ is the amplitude for the outcome $\hbar k$ of an
infinitely precise momentum measurement. $\psi_{k}(x,t)$ is the
amplitude for the outcome $x$ of an infinitely precise position
measurement performed (at time t) subsequent to an infinitely precise
momentum measurement with outcome $\hbar k.$ And $\psi(x,t)$ is the
amplitude for obtaining $x$ by an infinitely precise position
measurement performed at time $t.$

The preceding equation therefore tells us that the *amplitude* for
finding $x$ at $t$ is the product of

1.  the *amplitude* for the outcome $\hbar k$ and
2.  the *amplitude* for the outcome $x$ (at time $t$) subsequent to a
    momentum measurement with outcome $\hbar k,$

summed over all values of $k.$

Under the conditions stipulated by Rule A, we would have instead that
the *probability* for finding $x$ at $t$ is the product of

1.  the *probability* for the outcome $\hbar k$ and
2.  the *probability* for the outcome $x$ (at time $t$) subsequent to a
    momentum measurement with outcome $\hbar k,$

summed over all values of $k.$

The latter is what we expect on the basis of standard probability
theory. But if this holds under the conditions stipulated by Rule A,
then the same holds with \"amplitude\" substituted from \"probability\"
under the conditions stipulated by Rule B. Hence, given that
$\psi_{k}(x,t)$ and $\psi(x,t)$ are amplitudes for obtaining the outcome
$x$ in an infinitely precise position measurement, $\overline{\psi}(k)$
is the amplitude for obtaining the outcome $\hbar k$ in an infinitely
precise momentum measurement.

Notes:

1.  Since Rule B stipulates that the momentum measurement is not
    actually made, we need not worry about the impossibility of making
    an infinitely precise momentum measurement.
2.  If we refer to $|\psi(x)|^2$ as \"the probability of obtaining the
    outcome $x,$\" what we mean is that $|\psi(x)|^2$ *integrated* over
    any interval or subset of the real
    line is the probability of finding
    our particle in this interval or subset.

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# This Quantum World/Feynman route#Two rules

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# The Feynman route to Schrödinger

The probabilities of the possible outcomes of measurements performed at
a time $t_2$ are determined by the Schrödinger wave function
$\psi(\mathbf{r},t_2)$. The wave function $\psi(\mathbf{r},t_2)$ is
determined via the Schrödinger
equation by
$\psi(\mathbf{r},t_1).$ What determines $\psi(\mathbf{r},t_1)$ ? Why,
the outcome of a measurement performed at $t_1$ --- what else? Actual
measurement outcomes determine the probabilities of possible measurement
outcomes.

## Two rules

In this chapter we develop the quantum-mechanical probability algorithm
from two fundamental rules. To begin with, two definitions:

-   **Alternatives** are possible sequences of measurement outcomes.
-   With each alternative is associated a complex
    number called **amplitude**.

Suppose that you want to calculate the probability of a possible outcome
of a measurement given the actual outcome of an earlier measurement.
Here is what you have to do:

-   Choose any sequence of measurements that may be made in the
    meantime.
-   Assign an amplitude to each alternative.
-   Apply either of the following rules:

:   \
    **Rule A**: If the intermediate measurements are made (or if it is
    possible to infer from other measurements what their outcomes would
    have been if they had been made), first square the absolute values
    of the amplitudes of the alternatives and then add the results.

```{=html}
<!-- -->
```

:   **Rule B**: If the intermediate measurements are not made (and if it
    is not possible to infer from other measurements what their outcomes
    would have been), first add the amplitudes of the alternatives and
    then square the absolute value of the result.

\
In subsequent sections we will explore the consequences of these rules
for a variety of setups, and we will think about their origin --- their
*raison d\'être*. Here we shall use Rule B to determine the
interpretation of $\overline{\psi}(k)$ given Born\'s probabilistic
interpretation of $\psi(x)$.

In the so-called \"continuum normalization\", the unphysical limit of a
particle with a sharp momentum $\hbar k'$ is associated with the wave
function

$$\psi_{k'}(x,t)=\frac1{\sqrt{2\pi}}\int\delta(k-k')\,e^{i[kx-\omega(k)t]}dk=
\frac1{\sqrt{2\pi}}\,e^{i[k'x-\omega(k')t]}.$$ Hence we may write
$\psi(x,t) = \int\overline{\psi}(k)\,\psi_{k}(x,t)\,dk.$

$\overline{\psi}(k)$ is the amplitude for the outcome $\hbar k$ of an
infinitely precise momentum measurement. $\psi_{k}(x,t)$ is the
amplitude for the outcome $x$ of an infinitely precise position
measurement performed (at time t) subsequent to an infinitely precise
momentum measurement with outcome $\hbar k.$ And $\psi(x,t)$ is the
amplitude for obtaining $x$ by an infinitely precise position
measurement performed at time $t.$

The preceding equation therefore tells us that the *amplitude* for
finding $x$ at $t$ is the product of

1.  the *amplitude* for the outcome $\hbar k$ and
2.  the *amplitude* for the outcome $x$ (at time $t$) subsequent to a
    momentum measurement with outcome $\hbar k,$

summed over all values of $k.$

Under the conditions stipulated by Rule A, we would have instead that
the *probability* for finding $x$ at $t$ is the product of

1.  the *probability* for the outcome $\hbar k$ and
2.  the *probability* for the outcome $x$ (at time $t$) subsequent to a
    momentum measurement with outcome $\hbar k,$

summed over all values of $k.$

The latter is what we expect on the basis of standard probability
theory. But if this holds under the conditions stipulated by Rule A,
then the same holds with \"amplitude\" substituted from \"probability\"
under the conditions stipulated by Rule B. Hence, given that
$\psi_{k}(x,t)$ and $\psi(x,t)$ are amplitudes for obtaining the outcome
$x$ in an infinitely precise position measurement, $\overline{\psi}(k)$
is the amplitude for obtaining the outcome $\hbar k$ in an infinitely
precise momentum measurement.

Notes:

1.  Since Rule B stipulates that the momentum measurement is not
    actually made, we need not worry about the impossibility of making
    an infinitely precise momentum measurement.
2.  If we refer to $|\psi(x)|^2$ as \"the probability of obtaining the
    outcome $x,$\" what we mean is that $|\psi(x)|^2$ *integrated* over
    any interval or subset of the real
    line is the probability of finding
    our particle in this interval or subset.

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# This Quantum World/Implications and applications

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# The Schrödinger equation: implications and applications

In this chapter we take a look at some of the implications of the
Schrödinger equation

:   \
    `<math>`{=html}

i\\hbar\\,\\frac{\\partial\\psi}{\\partial t} = \\frac{1}{2m}
\\left(\\frac\\hbar i \\frac{\\partial}{\\partial\\mathbf{r}} -
\\mathbf{A}\\right)\^2\\psi + V\\psi. `</math>`{=html}\

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# This Quantum World/Bell

## Bell\'s theorem: the simplest version

Quantum mechanics permits us to create the following scenario.

-   Pairs of particles are launched in opposite directions.
-   Each particle is subjected to one of three possible measurements
    (**1**, **2**, or **3**).
-   Each time the two measurements are chosen at random.
-   Each measurement has two possible results, indicated by a red or
    green light.

Here is what we find:

-   If both particles are subjected to the same measurement, identical
    results are never obtained.
-   The two sequences of recorded outcomes are completely random. In
    particular, half of the time both lights are the same color.

\
471px

\
If this doesn\'t bother you, then please explain how it is that the
colors differ whenever identical measurements are performed!

The obvious explanation would be that each particle arrives with an
\"instruction set\" --- some property that pre-determines the outcome of
every possible measurement. Let\'s see what this entails.

Each particle arrives with one of the following 2^3^ = 8 instruction
sets:

:

    :   **RRR**,**RRG**,**RGR**,**GRR**,**RGG**,**GRG**,**GGR**, or
        **GGG**.

(If a particle arrives with, say, **RGG**, then the apparatus flashes
red if it is set to **1** and green if it is set to **2** or **3**.) In
order to explain why the outcomes differ whenever both particles are
subjected to the same measurement, we have to assume that particles
launched together arrive with opposite instruction sets. If one carries
the instruction (or arrives with the property denoted by) **RRG**, then
the other carries the instruction **GGR**.

Suppose that the instruction sets are **RRG** and **GGR**. In this case
we observe different colors with the following five of the 3^2^ = 9
possible combinations of apparatus settings:

:

    :   **1---1**,**2---2**,**3---3**,**1---2**, and **2---1**,

and we observe equal colors with the following four:

:

    :   **1---3**,**2---3**,**3---1**, and **3---2**.

Because the settings are chosen at random, this particular pair of
instruction sets thus results in different colors 5/9 of the time. The
same is true for the other pairs of instruction sets *except* the pair
**RRR**, **GGG**. If the two particles carry these respective
instruction sets, we see different colors *every* time. It follows that
we see different colors *at least* 5/9 of the time.

But different colors are observed half of the time! In reality the
probability of observing different colors is 1/2. Conclusion: the
statistical predictions of quantum mechanics cannot be explained with
the help of instruction sets. In other words, these measurements do not
reveal *pre-existent* properties. They *create* the properties the
possession of which they indicate.

Then how is it that the colors differ whenever identical measurements
are made? How does one apparatus \"know\" which measurement is performed
and which outcome is obtained *by the other apparatus*?

Whenever the joint probability **p(A,B)** of the respective outcomes
**A** and **B** of two measurements does not equal the product
**p(A) p(B)** of the individual probabilities, the outcomes --- or their
probabilities --- are said to be *correlated*. With equal apparatus
settings we have **p(R,R) = p(G,G) = 0**, and this obviously differs
from the products **p(R) p(R)** and **p(G) p(G)**, which equal
$\textstyle\frac12\times\frac12=\frac14.$ What kind of mechanism is
responsible for the correlations between the measurement outcomes?

:   *You understand this as much as anybody else!*

The conclusion that we see different colors at least 5/9 of the time is
*Bell\'s theorem* (or *Bell\'s inequality*) for this particular setup.
The fact that the universe violates the logic of Bell\'s Theorem is
evidence that particles do not carry instruction sets embedded within
them and instead have instantaneous knowledge of other particles at a
great distance. Here is a comment by a distinguished Princeton physicist
as quoted by David Mermin[^1]

:

    :   Anybody who\'s not bothered by Bell\'s theorem has to have rocks
        in his head.

And here is why Einstein wasn\'t happy with quantum mechanics:

:

    :   I cannot seriously believe in it because it cannot be reconciled
        with the idea that physics should represent a reality in time
        and space, free from spooky actions at a distance.[^2]

Sadly, Einstein (1879 - 1955) did not know Bell\'s theorem of 1964. We
know now that

:

    :   there must be a mechanism whereby the setting of one measurement
        device can influence the reading of another instrument, however
        remote.[^3]

:   *Spooky actions at a distance are here to stay!*

\
\-\-\--

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<references/>
```

------------------------------------------------------------------------

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[^1]: N. David Mermin, \"Is the Moon there when nobody looks? Reality
    and the quantum theory,\" *Physics Today*, April 1985. The version
    of Bell\'s theorem discussed in this section first appeared in this
    article.

[^2]: Albert Einstein, *The Born-Einstein Letters*, with comments by Max
    Born (New York: Walker, 1971).

[^3]: John S. Bell, \"On the Einstein Podolsky Rosen paradox,\"
    *Physics* 1, pp. 195-200, 1964.


# This Quantum World/Game

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## A quantum game

Here are the rules:[^1]

-   Two teams play against each other: Andy, Bob, and Charles (the
    \"players\") versus the \"interrogators\".
-   Each player is asked either \"What is the value of **X**?\" or
    \"What is the value of **Y**?\"
-   Only two answers are allowed: +1 or −1.
-   Either each player is asked the **X** question, or one player is
    asked the **X** question and the two other players are asked the
    **Y** question.
-   The players win if the product of their answers is −1 in case only
    **X** questions are asked, and if the product of their answers is +1
    in case **Y** questions are asked. Otherwise they lose.
-   The players are not allowed to communicate with each other once the
    questions are asked. Before that, they are permitted to work out a
    strategy.

Is there a failsafe strategy? Can they make sure that they will win?
Stop to ponder the question.

Let us try pre-agreed answers, which we will call **X~A~**, **X~B~**,
**X~C~** and **Y~A~**, **Y~B~**, **Y~C~**. The winning combinations
satisfy the following equations:

:   \
    `<math>`{=html}

X_AY_BY_C=1,\\quad Y_AX_BY_C=1,\\quad Y_AY_BX_C=1,\\quad
X_AX_BX_C=-1.`</math>`{=html}

\
Consider the first three equations. The product of their right-hand
sides equals +1. The product of their left-hand sides equals
**X~A~X~B~X~C~**, implying that **X~A~X~B~X~C~ = 1**. (Remember that the
possible values are ±1.) But if **X~A~X~B~X~C~ = 1**, then the fourth
equation **X~A~X~B~X~C~ = −1** obviously cannot be satisfied.

:   The bottom line: There is no failsafe strategy with pre-agreed
    answers.

\
\-\-\--

```{=html}
<references/>
```

------------------------------------------------------------------------

```{=html}
<div class="noprint">
```
**NEXT \>**

```{=html}
</div>
```

[^1]: Lev Vaidman, \"Variations on the theme of the
    Greenberger-Horne-Zeilinger proof,\" *Foundations of Physics* 29,
    pp. 615-30, 1999.


# This Quantum World/GHZ

```{=html}
<div class="noprint">
```
**\< PREVIOUS**

```{=html}
</div>
```
## The experiment of Greenberger, Horne, and Zeilinger

And yet there is a failsafe strategy.[^1]

Here goes:

-   Andy, Bob, and Charles prepare three particles (for instance,
    electrons) in a particular way. As a result, they are able to
    predict the probabilities of the possible outcomes of any spin
    measurement to which the three particles may subsequently be
    subjected. In principle these probabilities do not depend on how far
    the particles are apart.
-   Each player takes one particle with him.
-   Whoever is asked the **X** question measures the *x* component of
    the spin of his particle and answers with his outcome, and whoever
    is asked the **Y** question measures the *y* component of the spin
    of his particle and answers likewise. (All you need to know at this
    point about the spin of a particle is that its component with
    respect to any one axis can be measured, and that for the type of
    particle used by the players there are two possible outcomes, namely
    +1 and −1.

Proceeding in this way, the team of players is sure to win every time.

Is it possible for the *x* and *y* components of the spins of the three
particles to be in possession of values before their values are actually
measured?

Suppose that the *y* components of the three spins have been measured.
The three equations

:   \
    `<math>`{=html}

X_AY_BY_C=1,\\quad Y_AX_BY_C=1,\\quad Y_AY_BX_C=1`</math>`{=html}

\
of the previous section tell us
what we would have found if the *x* component of any one of the three
particles had been measured instead of the *y* component. If we assume
that the *x* components are in possession of values even though they are
*not* measured, then their values can be inferred from the measured
values of the three *y* components.

Try to fill in the following table in such a way that

-   each cell contains either +1 or −1,
-   the product of the three X values equals −1, and
-   the product of every pair of Y values equals the remaining X value.

Can it be done?

      A   B   C
  --- --- --- ---
  X
  Y

The answer is negative, for the same reason that the four equations

:   \
    `<math>`{=html}

X_AY_BY_C=1,\\quad Y_AX_BY_C=1,\\quad Y_AY_BX_C=1,\\quad
X_AX_BX_C=-1`</math>`{=html}

\
cannot all be satisfied. Just as there can be no strategy with
pre-agreed answers, there can be no pre-existent values. We seem to have
no choice but to conclude that these spin components are in possession
of values *only if* (and only when) they are actually measured.

Any two outcomes suffice to predict a third outcome. If two
*x* components are measured, the third *x* component can be predicted,
if two *y* components are measured, the *x* component of the third spin
can be predicted, and if one *x* and one *y* component are measurement,
the *y* component of the third spin can be predicted. How can we
understand this given that

-   the values of the spin components are created as and when they are
    measured,
-   the relative times of the measurements are irrelevant,
-   in principle the three particles can be millions of miles apart.

How does the third spin \"know\" which components of the other spins are
measured and which outcomes are obtained? What mechanism correlates the
outcomes?

:   *You understand this as much as anybody else!*

\
\-\-\--

```{=html}
<references/>
```

------------------------------------------------------------------------

[^1]: D. M. Greenberger, M. A. Horne, and A. Zeilinger, \"Going beyond
    Bell\'s theorem,\" in *Bell\'s theorem, Quantum Theory, and
    Conception of the Universe*, edited by M. Kafatos (Dordrecht: Kluwer
    Academic, 1989), pp. 69-72.


# Managing Groups and Teams/Introduction

## Foreword

It is often remarked that groups are everywhere, whether in our social
lives, our work lives, or even our families. In each of these
situations, sets of individuals decide to work collectively to achieve
particular goals.

However, although groups are everywhere and we participate in them
constantly, we do not understand them very well. Many of us can tell
stories of groups that seemed perfect for a given task, but which
failed. And we all have reasons (or excuses) that explain such failures.

But our experiences in groups suffer precisely because we are with them.

The study of groups as a phenomenon that is unique and different from
other social phenomena is very active, reflecting both the importance it
has and how much we still don\'t know about groups. `<small>`{=html}S M
Rizwsan Clinical Psychologist ANF`</small>`{=html}

## About this Book

!Hub and Spoke{width="200"}
In this book, we take a challenge-based approach to dealing with groups.
Many other books provide conceptual and descriptive treatments of groups
and teams. Here we will take a prescriptive perspective, one that
focuses on the \"how to\" of managing a group or a team. This
prescriptive perspective, however, will be rooted in social science.

## About Wikibooks and Wikimedia

-   Wikibooks, a
    Wikipedia article about Wikibooks and its
    history.
-   Wikimedia Foundation, a Wikipedia article
    about the non-profit parent organization of Wikibooks.
-   Frequently asked questions about
    Wikibooks.


# Managing Groups and Teams/Creating and Maintaining Team Cohesion

## Team Cohesion Defined

One definition of cohesion is "a group property with individual
manifestations of feelings of belongingness or attraction to the group"
(Lieberman et al., 1973: 337). It is generally accepted that group
cohesion and performance are associated. "However, the issue of a
cause/effect relationship between group cohesion and performance is not
completely resolved. Generally, there tend to be more studies supporting
a positive relationship between group cohesion and performance."
 With that in
mind the following article is an effort to enhance group/team cohesion
and as a result help improve group/team performance.

## The Question

What is team cohesiveness and why does it matter to an organization to
have cohesiveness within its teams?

## Team Composition

### How to promote team cohesion when selecting and identifying diversity within teams

In their journal article Beyond Relational Demography: Time and the
Effects of Surface- and Deep-Level Diversity on Work Group Cohesion,
David A. Harrison, Kenneth H. Price, and Myrtle P. Bell discuss the
composition of teams and its effect on cohesiveness. They describe two
different categories of diversity, namely surface level and deeper
level.

### Surface-Level Diversity:

Surface level attributes are "immutable \[and\] almost immediately
observable."  Such
attributes include age, sex, and race/ethnicity. In general, the
findings have been fairly inconsistent within and across studies as to
how diversity in these areas affect team cohesion.

### Deep-Level Diversity:

Deep-level diversity includes differences among members' attitudes,
beliefs, and values. These attributes are less apparent than
surface-level differences and are "learned through extended,
individualized interaction and information gathering."
 They are communicated
differences which are shared through both verbal and nonverbal behavior.
There has been less research done in this area with regards to teams in
workplace settings, though a number of social psychological studies have
been conducted. The findings consistently suggest that "attitudinal
similarity \[is\] associated with higher group cohesiveness."
 Diversity also
improves communication, reduces personal conflict, attracts friendships,
and gives more satisfaction to group members.

### Summary

Overall, the school of thought that is most widely accepted, in regards
to team cohesion, is that "surface-level differences are less important
and deep-level differences are more important for groups that had
interacted more often"
. Harrison, Price, and
Bell's study concluded that while homogeneous groups interacted and
performed more effectively than heterogeneous groups in the beginning,
with time and information, the diverse groups' performance and processes
improved more rapidly and "had grown more effective in identifying
problems and generating solutions"
. Overall cohesiveness
was strengthened in such cases. Hence, for optimum results, teams ought
to include deep-level diversity as part of the process for achieving
cohesiveness.

## Internal Environment Factors Needed in Team Cohesion

Internally there are several factors that must be present for cohesion
to exist within a team. First good and appropriate communication is
essential to creating and maintaining cohesion. Communication leads to
the second factor, unity of purpose. For a team to work as a cohesive
team they must share a common goal and to collectively work towards that
goal. And finally, the team must have a high level of commitment
understanding that what they do together as a team is better than what
they do on their own.

### Communication

In the article "Building Team Cohesion: Becoming "We" Instead of "Me"
the authors stress the importance of not losing the "human moment" which
they define as "not to lose the powerful impact of face-to-face,
immediate interaction in real time and space." Furthermore, the authors
add the following:

:   "It is communication in the "human moment" that most powerfully
    creates team synergy -- the energy that truly makes "the whole
    greater than the sum of its parts." It is communication in the
    "human moment" that also most powerfully creates team cohesion -- a
    strong sense of loyalty and commitment to the team vision as one's
    own."
:   "Providing communication opportunities in real time and space for
    forensics team members is necessary to build team cohesion. Whether
    a room or lounge where team members can congregate between classes
    and the end of the day, practice space for formal and informal
    coaching sessions, travel time in cars and vans, or social time to
    enjoy pizza and a movie, both quantity and quality of communication
    are necessary to build a cohesive team climate of openness and
    trust...According to Bormann(1990), highly cohesive groups interact
    in an open climate where individuals are free to ask questions and
    disagree with one another; even the ability to work through
    inevitable team conflict in such a constructive climate will only
    serve to strengthen team cohesion."

In order to build cohesion within any team whether it be a sports team
or work team communication is an essential ingredient. Providing
opportunities for the team members to interact socially is necessary to
help build trust. In addition, a safe environment in which the team can
deal with conflict is critical to team cohesion.

### Unity of Purpose or a Common Goal

A critical factor that must be present for groups or teams to experience
cohesion is to have a common goal. In SELF-MANAGING WORK TEAMS:An
Empirical Study of Group Cohesiveness in "Natural Work Groups" at a
Harley-Davidson Motor Company Plant, the authors state: "that highly
cohesive groups tend to perform better because they have high commitment
to attaining group goals (e.g., Stogdill, 1972), and because the members
are more sensitive to others in the group, they are more willing to
assist each other (e.g., Schachter, Ellertson, McBride,&Gregory, 1951)."

Additional support to the importance of a common goal in building and
maintaining a common goal is found in "Building Team Cohesion: Becoming
"We" Instead of "Me" where the author relates the following:

:   "Since cohesion is believed to be one of the distinguishing
    characteristics of a high-performance team, what is this powerful
    team quality and how is it cre-ated? According to Bollen and Hoyle
    (1979), cohesion is the degree of attraction members feel toward one
    another and the team; \"it is a feeling of deep loyalty, of esprit
    de corps, the degree to which each individual has made the team\'s
    goal his or her own, a sense of belonging, and a feeling of morale\"
    (as cited in Beebe & Masterson, 2000, p. 122). Though cohesion is
    rooted in the feelings team mem-bers have for one another as well as
    a common goal, creating, shaping, and strengthening those feelings
    relies on the use of effective communication. Communication scholars
    have long agreed that group or team cohesion is as much about the
    relationships created as the task at hand, and success in both
    fos-ters the development of team cohesion. (Bormann, 1990).

Without a purpose or a common goal a team will eventually splinter into
separate individuals working towards their own personal agendas and not
together toward a team goal. It is important for team members to see
themselves as a part of the group working towards a goal for
cohesiveness to exist.

### Commitment

Teams that are not committed to each other or a common goal do not
experience cohesion and are much more likely to leave the team or even
the organization. In the article \"Commitment and the Control of
Organizational Behavior and Belief\" the author states the following:

:   \"Commitment also derives from the relation of an employee\'s job to
    those of other in the organization. Some jobs are rather isolated
    and can be done independently of other jobs in the organization. It
    has been found that jobs which are not integrated with the work
    activities of others tend to be associated with less favorable
    attitudes. (Sheperd, 1973). Gow, Clarkand dossett (1974), for
    instance find that telephone operators who quit tend to be those who
    are not integrated into the work group. Work integration can affect
    commitment by the fact that integrated jobs are likely to be
    associated with salient demands from others in the organization. If
    a person has a job which affects the work of others in the
    organization, it is likely that those other will communicate their
    expectations for performance of that job. Such expectations can be
    committing in that the other people implicitly or explicitly hold
    the person accountable for what he does. Earlier we mentioned that
    when individuals did not know what was expected of them they tended
    to be less committed to the organization. One reason an individual
    will not know what is expected is because no one is telling him. In
    general, we would expect that anything which contributes to creating
    definite expectations for a person\'s behavior would enhance his
    felt responsibility, and hence commitment.\"

We learn from the above author that for commitment to exist we employees
need to know what is expected of them and then to know they will be held
accountable either by a manager or other co-workers. Once commitment is
present team members are more likely to stay and work towards the team
goal.

## Role of Management in Team Cohesion

The roles that management has in a team that they oversee are extremely
important. But it is also important for the management to understand the
boundaries of what their roles and responsibilities are and what the
roles and responsibilities of the team itself are. The manager is often
placed in the management position because of their people and technical
skills and experience. A team often benefits from the manager's
abilities, skills, attitudes, insights and ideas. But neither the
management nor the team should ever forget that it is the team's
responsibility to perform the actual work. So what role should
management play in a team that they oversee? How best can they serve the
team to ensure they are successful? A critical role that management can
and should have is to facilitate and encourage team cohesion.

### Establish the Team Vision/Goal

The first step in creating team cohesion and where management should be
involved is in the establishment of the team vision and/or goal.
Management must set a clear vision to which the team can jointly work
towards together. As Tommy Lasorda, former manager of the LA Dodgers,
stated, "My responsibility is to get my 25 guys playing for the name on
the front of their shirt and not the one on the
back." Management must "establish a common
goal for \[the\] team -- an underlying target that will bind \[them\]
together..." The goal must be as clear as
possible for each member of the team. "Goal clarity is critical for team
members to have confidence in their direction and to be committed to
make it happen." A clearly defined goal
articulated to the team in such a way that they all understand will
inspire the team and commit them to the cause.

Once the goal has been clearly defined and clearly articulated,
management must keep the vision and goal alive. Obstacles, tension, and
crises may arise that can distract or discourage away from the common
goal. The management must "continually reinforce and renew the team
goal."

Being that managements "primary responsibility is to ensure that the
team reaches its goal," management must
also facilitate a working environment, set clear expectations and
responsibilities, and lastly, let the team do their job.

### Facilitate a Working Environment

Once the team vision and goal has been established, the most important
contribution management can make "is to ensure a climate that enables
team members to speak up and address the real issues preventing the goal
from being achieved." Such a climate
includes creating an environment of trust, communication and openness
with each other. As Frank Lafasto describes in his book, openness and
supportiveness are "the ability to raise and resolve the real issues
standing the way of a team accomplishing its goal. And to do so in a way
that brings out the best thinking and attitude of everyone involved.
It's too hard for team members to contribute, much less explore the
possibilities, when it is not safe for them to say what's on their
minds. They must be able to speak honestly. They must be able to deal
openly with real obstacles, problems, and opportunities in a way that
promotes listening, understanding of differing perspectives, and
constructively working towards a solution."
The environment and climate in which the team works and operates must be
facilitated by the management to ensure that trust is established,
collective collaboration is demanded, and openness is welcome.

### Set Clear Expectations and Responsibilities

Management responsibility is also to set clear expectations and
responsibilities of the team and individual team members. Patrick
Lencioni describes in his book "The Five Dysfunctions of a Team" that a
team where there is ambiguity about the direction and priorities fails
to commit. Whereas when the expectations, direction and priorities are
clear the team is more likely to commit to the cause and each
other. Management must establish clear
expectations so there is no ambiguity or question of what is expected of
the team, whether it is the timeline, product, requirements, etc.

Also, management must set clear responsibilities. "There are few
behaviors that build confidence as well as personalized expression of
belief in an individual. One of the most direct signals of such belief
is trusting someone with important and meaningful
responsibility." Clear and meaningful
responsibility that allows the team members to stretch enhances their
trust and confidence. And, as Jack Welch, the CEO of General Electric,
put it, "giving people self-confidence is by far the most important
thing I can do. Because then they will
act."

### Training and Staffing

According to Chansler, Swamidass, & Cammann to get a task completed, "a
work team must have the resources to do the job. Specifically, the team
needs trained, competent team members. Training is a planned effort by a
firm to help employees learn job-related competencies (Noe, 1999).
Training is used by companies to gain a competitive advantage over
rivals in their respective industries. A company must provide adequate
resources to an empowered team to staff and train its members
adequately." It is the responsibility of Management to provide such
training. Chansler, Swamidass, & Cammann also suggest management should
provide its workers with both "hard" and "soft" skills. "Hard-skills
training helps them do their jobs properly so that the plant can produce
a quality product cost-effectively. Soft-skills training, on the other
hand, teaches the workers to get along better as part of a functioning
team; this type of skills training improves interpersonal dynamics and
relationships. To effectively and efficiently manufacture quality
product, both types of training are needed."
 It is
therefore the responsibility of management to make sure that group/ team
members have the hard and soft skills to perform tasks and maintain
cohesion.

### Get Out of Their Way

And lastly, the manager's role is to get out of the team's way. Once the
team knows what they are working towards, tasks have been clearly
defined and delegated, expectations are clearly set and they have the
means to build relationships of trust and have open communication, the
manager needs to step back and let the team work. The last thing the
team needs, not only to reach their goal, but also to build strong
cohesion is, as Dr. Travis Bradberry described, a seagull manager; one
that swoops in when problems arise "squawking and dumpling advice, only
to take off and let others clean up the
mess." Management needs to let the
members in the team be smart and informed about key issues and facts
related to their tasks and goal. Then management must trust team members
by providing sufficient autonomy, which will in turn build confidence.

### Summary

Ultimately, the goal and role of management should be to add value to
the team's effort. This can be done by defining a clear vision and goal,
facilitate a working environment, set clear expectations and
responsibilities, and provide the team enough autonomy where they can
work and do their jobs with full commitment and confidence.

## Examples of Team Cohesion: The Good

A good example of team Cohesion is that of the Harley Davidson Motor
Company (HDMC) and its group structure. The well known turnaround of
HDMC occurred in the 1980s when it changed from a "command-and-control"
culture to that of self-managing work teams (SMWT). This change allowed
assembly employees to make important decisions in their work teams
. With group work as
the foundation of HDMC's manufacturing cohesion among group members was
essential.

At its Kansas City Plant HDMC natural work groups (NWG) were organized
to make decisions (and build motorcycles). The plant's employees are
made up of local union members. "This partnership allows the shifting of
the decision-making and financial responsibilities for the operation of
the plant to the assembly floor employees"
.

The structure of the plant divides workers into NWGs. Each NWG is either
assigned to one of four process operations groups (POG) (the Assembly
POG, the Fabrication POG, the Paint POG, or a POG dedicated to future
programs) or provides "computer, human resources, materials, and so
forth, support for the operations NWGs (denoted as RG or Resource
Groups). Each of the NWGs is represented by NWG-elected (on a rotating
basis) members. The highest level of the circular organization is the
lone plant leadership group (PLG), which is co-chaired by the plant
manager and two local union presidents"
.

Within this group structure HDMC provides for widespread access to
information. "All financial and operations information is available to
all team members, which allows them to monitor budgets and production
quotas" . This
access to information facilitates open communication which in turn leads
to greater team cohesion. Cohesion is also furthered by the autonomy of
workers within the group. "Each NWG is empowered to make decisions with
regard to any aspect of the assembly process as long as it does not
cross over its boundary and impede another NWG"
. With freedom to
make any necessary decisions and freedom from continuous managerial
intervention NWGs are free to bend and move as needed in response to any
given situation.

Interestingly in this structure there are no formal team leaders. "NWGs
are collectively led by the members of the group. Traditional leadership
duties such as scheduling, safety monitoring, budget balancing, and so
forth, are rotated among the NWG members on a regular basis (usually
monthly). The NWG controls its own budget, sick pay, overtime, and
consumable production materials. Individual performance measures are not
maintained. The NWG performance is measured on achievement of plant
goals and on the goals that they set for themselves"
. This sharing of
responsibilities fosters cohesion by aligning the goals of the group,
goals each member is included in creating.

## Examples of Team Cohesion: The Bad

The 2010 film "The Social Network" is based on the events and
circumstances that lead to the creation and founding of the social
networking website "Facebook." Founder Mark Zuckerberg and his friend,
co-founder Eduardo Saverin agree to launch the site and split up
ownership of the new company equitably. In the process of developing the
company, other individuals and interests come into play that are
detrimental to the team cohesion developed by Mark and Eduardo
eventually leading to multi-million dollar lawsuits and the end of the
original founding team.

Several factors that lead to the failure of team cohesion:

:\* Team members were unable to work together cooperatively

:\* Team goals were not shared by everyone on the team

:\* Team members felt that they were not recognized for individual
contributions to accomplish team goals

:\* Selfish interests were able to infiltrate the team cohesion

The fact that team members were unable to work cooperatively together is
likely the single biggest factor in the failure of the original
"Facebook" leadership team. In the movie, to help advance the growth of
the company, Mark brought in a third partner, Sean Parker, the
co-founder of the famous music sharing sight "Napster." Mark was
instantly drawn to Sean's charismatic personality and vision for
"Facebook." At the same time, Eduardo was highly skeptical of Sean and
his business history. Immediately Mark began to lean toward the ideas
that Sean had developed for "Facebook" and eventually gave Sean a small
ownership stake in the company as well as a management position. Upon
learning this, Eduardo was very upset that Mark would go ahead and make
the decision to include Sean without consulting him first.

Mark and Eduardo both had visions of keeping this site exclusive for the
elite college institutions around the country and gradually introducing
it to other colleges. When Sean was brought into the company he
presented Mark with a business plan to expand "Facebook" beyond the
college scene and introduce it to the general public. At the same time
he was trying to convince Mark that he needed to relocate the business
to Palo Alto, CA from Boston, MA. Eduardo was never consulted on these
propositions that were made to Mark. Eduardo felt like Sean was trying
to push him out of the company and influence many of the decisions made
by Mark. As the company grew and others were able to influence decision
making, the team goals had clearly changed and not everyone shared the
same vision.

When "Facebook" was originally started Eduardo was designated as the CFO
of the company. In this responsibility he put up the initial seed money
to get it off the ground. He was in charge of all finances and bank
accounts for the company. While Mark was moving the company headquarters
to Palo Alto, Eduardo was spending time in New York working on securing
advertising contracts with prominent advertising firms. When Eduardo
goes to visit the team in Palo Alto he begins to tell Mark all about the
progress he has made with the advertisers but instead he is told all
about the work that Sean and Mark had accomplished and is essentially
told that his time and work in New York will not be needed. Eduardo felt
like his contributions to the company and goals were not being
recognized. This drives Eduardo further and further from the team.

Throughout the life of the original leadership team there were many
occasions where selfish interests were able to infiltrate team cohesion.
Sean was the worst offender of this. Sean was one of the founders of
"Napster." "Napster" was eventually forced to shut down and was facing
many lawsuits from the record industry. Sean saw an opportunity to work
with Mark and Eduardo on "Facebook." Sean could see the potential that
this venture had and also that he could influence the socially
introverted Mark by filling him with visions of big pay days and a life
style full of privilege. At times he appeared to try and relive his days
of "Napster" and treated "Facebook" like it was his own company and he
was trying to accomplish the goals there that weren't achievable at
"Napster." After a party to celebrate the 1 millionth member of
"Facebook," Sean was arrested with several other "Facebook" interns for
possession of cocaine and was eventually dismissed from the company.
Through these actions, Sean clearly was acting in his own self interest
and did not take into account what the effects would be on the group or
company. In many ways the selfish actions of Sean drove a wedge between
Mark and Eduardo that eventually lead to lawsuits and the end of the
original leadership team.

## Conclusion

### Ways to Increase Team Cohesion

Each group environment is different and will present different
challenges. In order to create a cohesive team unit it is important for
team members to be aware of this and work towards it. In Joseph Powell
Stokes's research, he found that "risk taking that occurs in a group,
attraction to individual members of the group, and the instrumental
value of a group are all related to the cohesion of the group". He
proposes that "increasing risk taking, intermember attraction, and the
instrumental value of a personal change group might lead to increased
cohesion, which in turn might lead to increase benefits for group
participants."

As such, groups should attempt to foster an "atmosphere of tolerance and
acceptance" so they can assure openness and honesty and hence, increase
their risk taking and intermember attraction. They can "\[reward\]
members who make risky self-disclosures or give honest feedback to other
group members". They should make sure group members know that they are
expected to "like each other" and can help members "differentiate
between not liking other members' behaviors and not liking the other
members themselves". Group leaders ought to act as examples and make
sure that the group composition and expectations of the group members
are in line with risk-taking and intermember attraction. "Leaders can
maximize the instrumental value of a group for its members by having the
group focus explicitly on its goals and by helping redirect the group
when members' needs are not being met".

### Potential problems

One possible caveat of cohesion is that when there is too much cohesion,
groups are prone to groupthink. "Groupthink is a tendency by groups to
engage in a concurrence seeking manner. Groupthink occurs when group
members give priority to sustaining concordance and internal harmony
above critical examination of the issues under consideration".
 It is important for all group members
to be conscious of this pitfall and to take precautions to prevent such
behavior. See Ways to Prevent
Groupthink.

## References

Chansler, P. A., Swamidass, P. M., & Cammann, C. (2003). Self-Managing
Work Teams : An Empirical Study of Group Cohesiveness in *Natural Work
Groups* at a Harley-Davidson Motor Company Plant. Retrieved November 25,
2010, from Sage Journals Online:
<http://sgr.sagepub.com/content/34/1/101>

Harrison, David A.; Price, Kenneth H.; Bell, Myrtle P. "Beyond
Relational Demography: Time and the Effects of Surface- and Deep-Level
Diversity on Work Group Cohesion", The Academy of Management Journal,
Vol. 41, No. 1 (Feb., 1998), pp. 96-107

Milliken, F. J., & Martins, L. L. 1996. Searching for common threads:
Understanding the multiple effects of diversity in organizational
groups. Academy of Management Review, 21: 402-433

Terborg, J. R., Castore, C., & DeNinno, J. A. 1976. A longitudinal field
investigation of the impact of group composition on group performance
and cohesion. Journal of Personality and Social Psychology, 34: 782-790.

Friedley, Sheryl A. and Bruce B. Manchester. 2005. Building Team
Cohesion: Becoming "We" Instead of "Me". George Mason University.

SELF-MANAGING WORK TEAMS: An Empirical Study of Group Cohesiveness in
"Natural Work Groups" at a Harley-Davidson Motor Company Plant. SMALL
GROUP RESEARCH, Vol. 34 No. 1, February 2003 101-120

Salancik, Gerald R. Organizational Socialization and Commitment:
Commitment and the Control of Organizational Behavior and Belief. pp.
284-290

LaFasto, F., & Larson, C. (2001). *When Teams Work Best.* Thousand Oaks:
Sage Publications

Lencioni, P. (2002). *The Five Dysfunctions of a Team.* San Francisco:
Jossey-Bass.

Bradberry, T. (2008). *Squaqk.* New York: HarperCollins Publishers.

*The Social Network.* (2010, 11 21). Retrieved 11 21, 2010, from
Wikipedia: <http://en.wikipedia.org/wiki/The_Social_Network>

Stokes, Joseph Powell. Components of Group Cohesion : Intermember
Attraction, Instrumental Value, and Risk Taking. Small Group Research
1983 14: 163

*Managing Groups and Teams/Groupthink.* (2010, March 23). Retrieved 11
15, 2010, from Wikibooks:
<http://en.wikibooks.org/wiki/Managing_Groups_and_Teams/Groupthink>