Postulates of Quantum Mechanics

**Adapted from *Elements of Quantum Theory* by Frank J. Bockhoff**

## **Postulate 1: Quantum States and Wavefunctions**

The state of a single particle is fully described by a wavefunction

Ψ
(
x
,
y
,
z
,
t
)

{\displaystyle \Psi (x,y,z,t) }

. For stationary energy states, the wavefunction can be expressed as the product of a time-independent amplitude function

ψ
(
x
,
y
,
z
)

{\displaystyle \psi (x,y,z) }

 and a time-dependent function

f
(
t
)

{\displaystyle f(t) }

:

Ψ
(
x
,
y
,
z
,
t
# )
ψ
(
x
,
y
,
z
,
t
)
f
(
t
)
.

{\displaystyle \Psi (x,y,z,t) = \psi (x,y,z,t) f(t). }

Both functions

Ψ
(
x
,
y
,
z
,
t
)

{\displaystyle \Psi (x,y,z,t) }

 and

ψ
(
x
,
y
,
z
)

{\displaystyle \psi (x,y,z) }

 must be single-valued, continuous, and finite for all values of their coordinates, and must be smoothly varying within the boundaries at which they go to zero.

## **Postulate 2: Observables and Hermitian Operators**

(1) The possible state functions

Ψ
(
x
,
y
,
z
,
t
)

{\displaystyle \Psi (x,y,z,t) }

 for a single particle are given by the solution of the time-independent Schrödinger wave equation:

−

ℏ

2

2
m

∇

2

Ψ
V
(
x
,
y
,
z
,
t
)
# Ψ

−
ℏ

i

∂
Ψ

∂
t

.

{\displaystyle \frac{-{\hbar}^{2}}{2m}{\nabla}^{2} \Psi + V(x,y,z,t)\Psi = \frac{-\hbar}{i}\frac{\partial \Psi}{\partial t}. }

(2) In the special case of conservative systems, where

V

{\displaystyle V}

 is not dependent on time, and

Ψ
(
x
,
y
,
z
,
t
# )
ψ
(
x
,
y
,
z
)

e

−
i
E
t

/

ℏ

{\displaystyle \Psi (x,y,z,t) = \psi (x,y,z) {e}^{-iEt/\hbar} }

, the possible time-independent amplitude functions

ψ
(
x
,
y
,
z
)

{\displaystyle \psi (x,y,z) }

 for a single particle are given by the solution of the time-independent Schrödinger wave equation:

−

ℏ

2

2
m

∇

2

ψ
V
(
x
,
y
,
z
)
# ψ
E
ψ
.

{\displaystyle \frac{-{\hbar}^{2}}{2m}{\nabla}^{2} \psi + V(x,y,z)\psi = E \psi. }

## **Postulate 3: Linear Operators and Eigenvalues**

(1) For every dynamical variable there is a corresponding linear operator

A
^

{\displaystyle \hat{A}}

. If the dynamical variable is capable of exact experimental determination, its only possible exact values

A

{\displaystyle A}

 are those given by the eigenvalues of the equation

A
^

# Ψ
A
Ψ
.

{\displaystyle \hat{A} \Psi = A \Psi. }

(2) In the special case of conservative systems and for time-independent operators, the only possible exact values of the dynamical variable are also given by

A
^

# ψ
A
ψ
.

{\displaystyle \hat{A} \psi = A \psi. }

## **Postulate 4: Linear Operators and Eigenvalues**

(1) The term

Ψ

2

d
τ

{\displaystyle {|\Psi|}^{2} d\tau }

 (also written as

Ψ

∗

Ψ
d
τ

{\displaystyle {\Psi}^{\*}\Psi d\tau }

) is the time-dependent probability that a single particle exists at a given time

t

{\displaystyle t}

 in the space element

d
τ

{\displaystyle d\tau }

.

(2) For the special case of a conservative system, in which the single particle is restricted to an energy eigenstate, the time-independent probability that the particle exists in the space element

d
τ

{\displaystyle d\tau }

 is

ψ

2

d
τ

{\displaystyle {|\psi|}^{2} d\tau }

 (also written as

ψ

∗

ψ
d
τ

{\displaystyle {\psi}^{\*}\psi d\tau }

).

## **Postulate 5: Expectation of Observables**

(1) The expectation (mean value)

A
¯

{\displaystyle \bar{A} }

 of a series of measurements of an observable

A

{\displaystyle A}

 made over a large number of particles each in the state represented by

Ψ

{\displaystyle \Psi}

 is

A
^

=

∫

−
∞

∞

Ψ

∗

A
^

Ψ
d
τ

∫

−
∞

∞

Ψ

∗

Ψ
d
τ

.

{\displaystyle \hat{A} = \frac{\int\_{-\infty}^{\infty} {\Psi}^{\*}\hat{A}\Psi d\tau}{\int\_{-\infty}^{\infty} {\Psi}^{\*}\Psi d\tau}. }

where

A
^

{\displaystyle \hat{A}}

 is the linear operator for the observable.

(2) For the special case of conservative systems and time-independent operators and where the wave function are normalized,

A
¯

=

∫

−
∞

∞

ψ

∗

A
^

ψ
d
τ
.

{\displaystyle \bar{A} = \int\_{-\infty}^{\infty} {\psi}^{\*}\hat{A}\psi d\tau. }