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Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

Stephen Rand

Print publication date: 2010

Print ISBN-13: 9780199574872

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199574872.001.0001

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Appendix B The Heisenberg Uncertainty Principle

Appendix B The Heisenberg Uncertainty Principle

Source:
Lectures on Light
Publisher:
Oxford University Press

Appendix B

The Heisenberg Uncertainty Principle

To derive the Heisenberg uncertainty principle analytically for specific operators such as x and p x, the following argument can be used. The squared standard deviations of position and momentum observables are

< Δ x 2 > = ψ ( x x < x > ) 2 ψ d V ,
(B.1)
< Δ p x 2 > = ψ ( i ħ x < p x > ) 2 ψ d V .
(B.2)

Taking the average values to be 0, (<x> = <p x> = 0), we find

< Δ p x 2 > < Δ x 2 > = ψ ( ħ 2 2 x 2 ) ψ d r ψ x 2 ψ d V .
(B.3)

Integration by parts permits us to write

ψ 2 ψ x 2 d V = ψ x ψ x d V ,
(B.4)
< Δ p x 2 > < Δ x 2 > = ħ 2 ψ x ψ x d V ψ x 2 ψ d V .
(B.5)

Now, using the Schwartz inequality, we obtain

f f d V g g d V [ 1 2 ( f g d V + g f d V ) ] 2 .
(B.6)

Setting f ≡ ∂ψ/∂x and gxψ in this expression, one finds

< Δ p x 2 > < Δ x 2 > ħ 2 4 ( ψ x x ψ d V + x ψ ψ x d V ) 2 ħ 2 4 ( x x ( ψ ψ ) d V ) 2 = ħ 2 4 ħ 2 4 .
(B.7)

(p.265) The final integral above is again evaluated by parts and is equal to -1. Defining the uncertainties in momentum and position as Δ p x < Δ p x 2 > 1 / 2 and Δx ≡ <Δ x 2>1/2, we obtain

Δ p x Δ x ħ 2 .
(B.8)

The operators x and p x provide one example of two Hermitian operators  and with a commutator that is Hermitian and nonzero. Note that this results in an uncertainty principle between  and . Next, this result is extended to the general case.

Consider two Hermitian operators  and with a Hermitian commutator

[ A ^ , B ^ ] = i C ^ .
(B.9)

It is shown below that the standard deviations ΔA and ΔB have the product

Δ A Δ B 1 2 | < C > | .
(B.10)

The standard deviations (uncertainties) in  and are defined as

Δ A A ^ < A > ,
(B.11)
Δ B B ^ < B > ,
(B.12)

hence observed values of their squares are

< ( Δ A ) 2 > = < ψ Δ A | Δ A ψ > = Δ A ψ 2
(B.13)

and

< ( Δ B ) 2 > = < ψ Δ B | Δ B ψ > = Δ B ψ 2 .
(B.14)

The Schwartz inequality is equivalent to the inequality expressing the fact that the inner product of two vector operators, namely <ψΔAǀΔBψ>, is less than the product of their lengths, given by ǀΔAψǁǀΔBψǁ. That is,

| | Δ A ψ | | 2 | | Δ B ψ | | 2 | < ψ Δ A | Δ B ψ > | 2 ,
(B.15)

or

( Δ A ) 2 ( Δ B ) 2 | < ψ Δ A | Δ B ψ > | 2 = | < ψ | Δ A Δ B | ψ > | 2 .
(B.16)

The last step above is due to the Hermiticity of ΔA. The product ΔAΔB on the right hand side of this result can be reexpressed in terms of the commutator of  and , because any operator can be written as a linear combination of two Hermitian (p.266) operators:

Δ A Δ B = 1 2 ( Δ A Δ B + Δ B Δ A ) + 1 2 [ Δ A , Δ B ] = 1 2 ( Δ A Δ B + Δ B Δ A ) + 1 2 [ A , B ] = D ^ + i 2 C ^ .
(B.17)

Hence,

( Δ A ) 2 ( Δ B ) 2 | < ψ | D ^ + i 2 C ^ | ψ > | 2 = | < D > + i 2 < C > | 2 .
(B.18)

Because and Ĉ are Hermitian, their expectation values are real. Consequently, the squared expression on the right above is simply

( Δ A ) 2 ( Δ B ) 2 | < D > | 2 + 1 4 | < C > | 2 1 4 | < C > | 2 .
(B.19)

The uncertainties in A and B therefore satisfy the general relation

( Δ A ) ( Δ B ) 1 2 | < C > | .
(B.20)