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Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

Stephen C. Rand

Print publication date: 2016

Print ISBN-13: 9780198757450

Published to Oxford Scholarship Online: August 2016

DOI: 10.1093/acprof:oso/9780198757450.001.0001

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(p.331) Appendix A Expectation Values

(p.331) Appendix A Expectation Values

Source:
Lectures on Light
Author(s):

Stephen C. Rand

Publisher:
Oxford University Press

In quantum mechanics, operators appear in place of the continuous variables from classical mechanics and their commutation properties must be taken into account when predicting the outcome of measurements. For an operator Oˆ that in general does not commute with wavefunction ψ‎, the average value of Oˆ expected in a series of measurements could in principle be defined in two ways:

(A.1)
Oˆ=Oˆψψd3r,

or

(A.2)
Oˆ=ψOˆψd3r.

We can judge whether Eq. (A.1) or (A.2) is correct by testing the outcome of each expression against Schrödinger’s wave equation (1.3.5). As an example, consider the Hamiltonian operator Hˆ. If we ignore operator commutation rules we can imagine two possible expressions for the first moment, namely

(A.3)
Hˆ=it(ψψ)d3r,

(A.4)
Hˆ=iψtψdx.

The correct expression for Hˆ must however be fully consistent with Eq. (1.3.5). If we multiply through Hˆψ=iψt by ψ and integrate over volume, we find

(A.5)
Hˆ=ψHˆψd3r=iψψtd3r,

in agreement with definition (A.2) and Eq. (1.3.11). If instead we operate with Hˆ on ψψ, we find

(A.6)
Hˆψψd3r=itψψd3r=iψtψ+ψtψd3r.

Evaluation of the last integral for the simplest case in which ψ is an eigenstate with eigenvalue E yields 2E. The outcome 2E clearly contradicts the starting assumption and is therefore inconsistent with Schrödinger’s equation. Only the second definition yields the correct result, uniquely establishing Eq. (A.2) as the definition of expectation value. Overall, this illustrates the importance of commutation properties of operator expressions in quantum mechanics.