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State after Measurement

K. Kong Wan

Published Jul 9, 2019• 0 citations
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Abstract

The disturbance during a measurement on a classical system can be made arbitrarily small. One may then assume that a measurement does not change the state, e.g., the radar used by police to measure the speed of a car has a negligible effect on the speed of the car. Classical dynamics also applies to the measuring process as well. For quantum systems Postulate 26.1(OV) tells us that a measurement of a discrete observable A will yield an eigenvalue of its corresponding operator  . Let the initial state vector be ϕ → i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6175.tif"/> at time t = 0 when the measurement begins and let the final state vector be ϕ → f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6176.tif"/> at time t = Δ t when the measurement ends. A natural question to ask is Can disturbance during a measurement on the system be made arbitrarily small so that the state can remain the same, and if not, does the initial state evolve into the final state in a unitary manner describable in terms of Postulate 29.1.2(TESP) ?

DOI
10.1201/b22504-30

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The disturbance during a measurement on a classical system can be made arbitrarily small. One may then assume that a measurement does not change the state, e.g., the radar used by police to measure the speed of a car has a negligible effect on the speed of the car. Classical dynamics also applies to the measuring process as well. For quantum systems Postulate 26.1(OV) tells us that a measurement of a discrete observable A will yield an eigenvalue of its corresponding operator  . Let the initial state vector be ϕ → i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6175.tif"/> at time t = 0 when the measurement begins and let the final state vector be ϕ → f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6176.tif"/> at time t = Δ t when the measurement ends. A natural question to ask is Can disturbance during a measurement on the system be made arbitrarily small so that the state can remain the same, and if not, does the initial state evolve into the final state in a unitary manner describable in terms of Postulate 29.1.2(TESP) ?
The representation of physical observables by Hermitian linear operators was postulated assuming that an expectation value, defined by Eqs.(2.11) and (2.12) , is associated with any observable A measured in a quantum state Ψ of a dynamical system. In view of measurement uncertainties, defined by the dispersion: ( Δ A ) 2 = 〈 A 2 〉 − 〈 A 〉 2 = 〈 ( A − 〈 A 〉 ) 2 〉 = 〈 ( δ A ) 2 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_1_B.tif"/> the expectation value 〈 A 〉 is taken in a state Ψ if and only if (Δ Α ) 2 vanishes. It is convenient to introduce the deviation operator: 3.1 δ  =  − 〈 A 〉 Î https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_2_B.tif"/> such that the condition of a vanishing dispersion for the observable A in a given state Ψ reads: 〈 Ψ |( δ  ) 2 Ψ 〉 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_3_B.tif"/> Because δ is a Hermitian operator, we can also write: 〈 δ Â Ψ | δ Â Ψ 〉 = | δ Â Ψ | 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_4_B.tif"/> which means that the norm of δ Â Ψ is zero and this implies that we have a null function δ Â Ψ = 0. In other words: 3.2 Â Ψ = 〈 A 〉 Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_5_B.tif"/> It follows that the state function Ψ where a definite expectation value 〈 A 〉 of the observable A is observed, should be a solution of Eq.(3.2) rather than an arbitrary 112 function. The action of the operator  on Ψ has to be simply that of rescaling. All the state functions which are solutions of Eq.(3.2) are called eigenfunctions of  , and the constant factor 〈 A 〉 is said to be the corresponding eigenvalue. Hence we have shown that if ((δ  ) 2 ) = 0, the state function should be an eigenfunction of  , which means that a physical observable only takes definite values in states represented by eigenfunctions of the associated quantum operator. Conversely, given a quantum operator  one can systematically determine all its eigenvalues and eigenfunctions, which are solutions of the so-called eigenvalue equation: 3.3 Â Ψ = λ Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_6_B.tif"/> It is visible that the eigenvalue equation fixes the eigenfunctions only up to an overall scale factor, which may always be chosen such that the eigenfunctions be considered normalized. Then, in view of Eq.(2.11) , Eq.(3.3) gives: 〈  〉 = ∫ − ∞ ∞ Ψ * Â Ψ d x = λ ∫ − ∞ ∞ Ψ * Ψ d x = λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_7_B.tif"/> If we further apply the operator  to Eq.(3.3) , we obtain  2 Ψ = λ 2 Ψ and this leads to: 〈  2 〉 = ∫ − ∞ ∞ Ψ *  2 Ψ d x = λ 2 ∫ − ∞ ∞ Ψ * Ψ d x = λ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_8_B.tif"/> It follows that: ( Δ A ) 2 = 〈  2 〉 − 〈  〉 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_9_B.tif"/> and this shows that the dispersion vanishes if a physical observable is measured in states represented by eigenfunctions of the associated operator.
In this chapter, the simplest quantum measurement situation, i.e., where the state is an eigenvector of the observable to be measured, is considered. The states are therefore non-superimposed and measurement Axiom 3.2 of Section 13.7 in the previous chapter is applicable. We consider states in https://www.w3.org/1998/Math/MathML" display="inline"> ℂ 2 , ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math14_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> ℂ 2 ⊗ ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math14_2.tif "/> . The generalisation to https://www.w3.org/1998/Math/MathML" display="inline"> ⊗ n ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math14_3.tif "/> is then clear.
Postulate 26.1(OV) An observable A of a quantum system is describable by a selfadjoint operator A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq5203.tif"/> in the state space H → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq5204.tif"/> of the system. The measurable values of the observable are given by the elements of the spectrum s p ( A ^ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq5205.tif"/> of the operator .