uncertainty principle

uncertainty principle

We will find the

Fourier transform

F

⁢

(

ω

)

:=

1

2

⁢

π

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

e

-

i

⁢

ω

⁢

t

⁢

𝑑

t

(1)

of the Gaussian bell-shaped

function

f

⁢

(

t

)

=

C

⁢

e

-

a

⁢

t

2

(2)

where

C

and

a

are positive

constants

.

We get first

F

⁢

(

ω

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

C

⁢

e

-

a

⁢

t

2

⁢

e

-

i

⁢

ω

⁢

t

⁢

𝑑

t

=

C

2

⁢

π

⁢

∫

-

∞

∞

e

-

a

⁢

t

2

-

i

⁢

ω

⁢

t

⁢

𝑑

t

.

Completing the square in

-

a

⁢

t

2

-

i

⁢

ω

⁢

t

=

-

a

⁢

(

t

2

+

i

⁢

ω

⁢

t

a

)

=

-

a

⁢

(

t

+

i

⁢

ω

2

⁢

a

)

2

-

ω

2

4

⁢

a

and substituting

a

⁢

(

t

+

i

⁢

ω

2

⁢

a

)

:=

z

,  we may write

F

⁢

(

ω

)

=

C

2

⁢

π

⁢

∫

-

∞

∞

e

-

a

⁢

(

t

+

i

⁢

ω

2

⁢

a

)

2

⁢

e

-

ω

2

4

⁢

a

⁢

𝑑

t

=

C

2

⁢

π

⁢

a

⁢

e

-

ω

2

4

⁢

a

⁢

∫

l

e

-

z

2

⁢

𝑑

z

,

(3)

where

l

is a line of the

complex plane

parallel to the

real axis

and passing through the point

z

=

i

⁢

ω

2

⁢

a

.  Now we can show that the

integral

I

y

:=

∫

l

e

-

z

2

⁢

𝑑

z

=

∫

-

∞

∞

e

-

(

x

+

i

⁢

y

)

2

⁢

𝑑

x

does not depend on

y

at all.  In fact, we have

∂

⁡

I

y

∂

⁡

y

=

∫

-

∞

∞

∂

∂

⁡

y

⁢

e

-

(

x

+

i

⁢

y

)

2

⁢

𝑑

x

=

-

2

⁢

i

⁢

∫

-

∞

∞

e

-

(

x

+

i

⁢

y

)

2

⁢

(

x

+

i

⁢

y

)

⁢

𝑑

x

=

i

⁢

/

x

=

-

∞

∞

⁡

e

-

(

x

+

i

⁢

y

)

2

=

i

⁢

/

x

=

-

∞

∞

⁡

e

-

x

2

+

y

2

⁢

e

-

2

⁢

i

⁢

x

⁢

y

=

0

.

Hence we may evaluate

I

y

as

I

y

=

I

0

=

∫

-

∞

∞

e

-

x

2

⁢

𝑑

x

=

π

(see the

area under Gaussian curve

).  Putting this value to (3) yields the result

F

⁢

(

ω

)

=

C

2

⁢

a

⁢

e

-

ω

2

4

⁢

a

.

(4)

Thus, we have gotten another Gaussian bell-shaped function (4) corresponding to the given Gaussian bell-shaped function (2).

Interpretation

.

One can take for the

breadth

of the bell the portion of the abscissa axis, outside which the ordinate drops under the

maximum

value divided by

e

, for example.  Then, for the bell (2) one writes

C

⁢

e

-

a

⁢

t

2

=

C

⁢

e

-

1

,

whence

t

=

1

a

giving, by evenness (

http://planetmath.org/EvenFunction

) of the function, the breadth

Δ

⁢

t

=

2

a

.  Similarly, the breadth of the bell (4) is

Δ

⁢

ω

=

4

⁢

a

.  We see that the

product

Δ

⁢

t

⋅

Δ

⁢

ω

=

8

(5)

has a constant value.  One can show that any other shape of the graphs of

f

and

F

produces a

relation

to (5).  The breadths are thus inversely proportional (

http://planetmath.org/

Proportion

).

If

t

is the time and

f

is the on a system of oscillators with their natural frequencies, then in the

formula

f

⁢

(

t

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

F

⁢

(

ω

)

⁢

e

i

⁢

ω

⁢

t

⁢

𝑑

ω

of the inverse Fourier transform,

F

⁢

(

ω

)

means the amplitude of the oscillator with angular frequency

ω

.  We can infer from (5) that the more localised (

Δ

⁢

t

small) the external force is in time, the more spread (

Δ

⁢

ω

great) is its spectrum of frequencies, i.e. the

greater

is the amount of the oscillators the force has excited with roughly the same amplitude.  If one,

conversely

, wants to better the selectivity, i.e. to compress the spectrum narrower, then one has to spread out the external action in time.  The impossibility to simultaneously localise the action in time and enhance the selectivity of the action is one of the manifestations of the quantum-mechanical

uncertainty principle

, which has a fundamental role in modern physics.

References

1

Я. Б. Зельдович &  А. Д. Мышкис:

Элементы  прикладной  математики

.  Издательство
‘‘Наука’’.  Москва (1976).

2

Ya. B. Zel’dovich and A. D. Myshkis: ‘‘

Elements of applied mathematics

’’.
Nauka (Science) Publishers, Moscow (1976).

Title

uncertainty principle

Canonical name

UncertaintyPrinciple

Date of creation

2013-03-22 18:38:25

Last modified on

2013-03-22 18:38:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Example

Classification

msc 42A38

Related topic

UncertaintyTheorem

Related topic

SubstitutionNotation

Related topic

GraphOfEquationXyConstant

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