The Uncertainty Principle

Consider a large number N of

identical boxes with identical

particles all described by the
same statefunction

Ψ

(x , y , z ) :

Consider the observable A represented by the operator ˆ

A

Let [ ˆ

A , ˆ

H ]

≠

0

Thus the system described by

Ψ

do not

have sharp value for A.

The average (expectation ) value is defined by :

< A >=

Ψ

*

∫

ˆ

A

Ψ

d

τ

Appendix B

<A>

A

1

A

2

An

The Uncertainty Principle

The measurement of A on each of the n identical
systems will give a different outcome A

i

We define the variance as :

1

n

i

∑

A

i

− <

A

>

(

)

2

=

σ

A

2

=

(

∆

A)

2

σ

A

2

= Ψ

*

( ˆ

A

∫

− <

A

>

)

2

Ψ

d

τ

= Ψ

*

( ˆ

A

2

∫

−

2

<

A

>

ˆ

A

+ <

A

>

2

)

Ψ

d

τ

= Ψ

*

ˆ

A

2

∫

Ψ

d

τ −

2

<

A

> Ψ

*

∫

ˆ

A

Ψ

d

τ+ <

A

>

2

Ψ

*

∫

Ψ

d

τ

= Ψ

*

ˆ

A

2

∫

Ψ

d

τ

− <

A

>

2

=

<

A

2

> − <

A

>

2

Appendix B

The Uncertainty Principle

We define :

∆

A =

σ

A

2

as the standard deviation

We shall later show that two for two observables A and B

∆

A

∆

B =

1

2i

Ψ

*

∫

[ ˆ

A , ˆ

B ]

Ψ

d

τ

Consider as an example ˆ

x and ˆ

p

x

[ ˆ

x , ˆ

p

x

]

=

i

h

Since :

∆

x

∆

p

x

=

1

2i

Ψ

*

∫

[ˆ

x , ˆ

p

x

]

Ψ

d

τ

=

1

2

h

We can not simultaniously obtain sharp values
for x and p

x

Appendix B