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CRC Press eBooks

Vortex Lines

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

In the preceding chapters, we have discussed at length the longitudinal oscillations of a superfluid Bose liquid. We now focus our attention on “steady” superfluid flow, in which the superfluid velocity, v s , satisfies the equation d i υ v s = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429495717/2cc6e3e4-f702-48c8-b407-59783ef3d6a6/content/eq332_v2.tif"/>

DOI
10.1201/9780429495717-15

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Commutation Rules and Uncertainty Relations

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In this chapter, we develop results that will prove to be very important later. First, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0001.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0002.tif"/> are compatible , that is, if their Hermitian operator representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0003.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0004.tif"/> commute, then these representatives have a complete orthonormal set of common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0005.tif"/> which are the solutions of the simultaneous eigenvalue problem A ^ Ψ a n , b m ( r ) = a n Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0006.tif"/> and B ^ Ψ a n , b m ( r ) = b n Ψ a n , b m ( r ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0007.tif"/> Consequently, from the discussion at the end of Section 4.5 , when the system is in a quantum state specified by one of the common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0008.tif"/> , both A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0009.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0010.tif"/> have definite values, a n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0011.tif"/> for A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0012.tif"/> and b m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0013.tif"/> for B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0014.tif"/> In contrast, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0015.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0016.tif"/> are incompatible , that is, if their representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0279.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0017.tif"/> do not commute, then these representatives have few if any common eigenfunctions. As a consequence rarely if ever can one find quantum states for which two incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0018.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0019.tif"/> both have definite values rather than spreads of values. Therefore, two incompatible observables cannot (with rare exceptions) be simultaneously precisely measured. Rather, in analogy to the Heisenberg uncertainty relation Δ x Δ p x ≥ ℏ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0020.tif"/> which exists between the incompatible position and momentum observables, a Heisenberg uncertainty relation exists between any pair of incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0021.tif"/> and B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0022.tif"/>

CRC Press eBooks•May 31, 2021•0 citations
In the preceding chapters, we have discussed at length the longitudinal oscillations of a superfluid Bose liquid. We now focus our attention on “steady” superfluid flow, in which the superfluid velocity, v s , satisfies the equation d i υ v s = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429495717/2cc6e3e4-f702-48c8-b407-59783ef3d6a6/content/eq332_v2.tif"/>
The eigenstates of a gas of non-interacting bosons are symmetrized combinations of N different single particle states. As for the case of the corresponding fermion system, the eigenstates may be characterized by a distribution function, n p , equal to 1 if the state p is occupied, zero otherwise. The system energy is ( 2.1 ) E = ∑ n n p p 2 2 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429495717/2cc6e3e4-f702-48c8-b407-59783ef3d6a6/content/eq14_v2.tif"/>
In this chapter, we develop results that will prove to be very important later. First, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0001.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0002.tif"/> are compatible , that is, if their Hermitian operator representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0003.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0004.tif"/> commute, then these representatives have a complete orthonormal set of common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0005.tif"/> which are the solutions of the simultaneous eigenvalue problem A ^ Ψ a n , b m ( r ) = a n Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0006.tif"/> and B ^ Ψ a n , b m ( r ) = b n Ψ a n , b m ( r ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0007.tif"/> Consequently, from the discussion at the end of Section 4.5 , when the system is in a quantum state specified by one of the common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0008.tif"/> , both A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0009.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0010.tif"/> have definite values, a n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0011.tif"/> for A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0012.tif"/> and b m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0013.tif"/> for B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0014.tif"/> In contrast, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0015.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0016.tif"/> are incompatible , that is, if their representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0279.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0017.tif"/> do not commute, then these representatives have few if any common eigenfunctions. As a consequence rarely if ever can one find quantum states for which two incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0018.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0019.tif"/> both have definite values rather than spreads of values. Therefore, two incompatible observables cannot (with rare exceptions) be simultaneously precisely measured. Rather, in analogy to the Heisenberg uncertainty relation Δ x Δ p x ≥ ℏ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0020.tif"/> which exists between the incompatible position and momentum observables, a Heisenberg uncertainty relation exists between any pair of incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0021.tif"/> and B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0022.tif"/>