Quantum Mechanics for Nuclear Structure, Volume 2. Professor Kris Heyde

of suitable magnetic fields it is possible to effect the rotation of the state of a neutron. This has been done using the experimental arrangement shown schematically in figure 1.2. A picture of the silicon crystal, which is the essential component of the interferometer is shown in figure 1.3. The neutron beam is divided and recombined in such a way that one part passes through a magnetic field B which causes the neutron state ket to undergo a phase change. The recombined beam exhibits an interference pattern which can be varied by changing B. Some results are shown in figure 1.4. (Note: by ‘divided’ it is meant that for each individual neutron, it is not certain which path it takes. It is not a situation where some neutrons take one path and the other neutrons take the other path.)

      Figure 1.2. A schematic diagram of the paths of a beam of neutrons through the neutron interferometer shown in figure 1.3. The lattice planes are continuous from slab to slab and the distances a, d₁, and d₂ are machined to optical precision. The phase shift (state ket rotation) is effected in the darkened region using a magnetic field. The distances d₁ and d₂ are typically 3 cm and a is typically 0.5 cm.

      Figure 1.3. The essential component of the neutron interferometer in use at the University of Missouri. It consists of three silicon slabs machined from a single crystal of high-purity silicon to ensure alignment of crystal planes from slab to slab. (Reproduced from [1], with the permission of the American Institute of Physics.)

      Figure 1.4. Observed neutron intensities in counts/2 min in the O beam and H beam, i.e. in counters C₃ and C₂, respectively, in figure 1.2. This is effected by changing the magnetic field action (given in Gauss cm) on the neutron beam by varying the magnet current (given in milliamps). One oscillation corresponds to a rotation of 4π not 2π. (Reprinted from [2], Copyright (1975), with permission from Elsevier.)

      The phase change produced by the magnetic field is eiωT2, where T is the time spent by the neutrons in the magnetic field, ω is the spin-precession frequency,

      ω=2μnBℏ,(1.264)

      μn is the magnetic moment of the neutron, and the magnetic field is assumed to be of uniform constant strength B. The phase change is the standard result for a magnetic field B acting for a time T on a magnetic moment μn, causing the spin to precess. The connection between precession and rotation is seen to follow directly from the Hamiltonian for a neutron in the magnetic field (chosen to be in the z direction)

      Hˆ=ωSˆz,(1.265)

      the time evolution operator for the system

      U(t,0)=exp−iHˆtℏ=exp−iSˆzωtℏ,(1.266)

      and a comparison with the rotation operator about the z-axis

      Dz(ϕ)=exp−iSˆzϕℏ,(1.267)

      i.e.

      ϕ=ωt.(1.268)

      For a monoenergetic beam of neutrons, T is fixed. To produce the results shown in figure 1.4, B is varied (by varying the current to the magnet). The change in B necessary to yield successive maxima is given by

ΔB=ℏ2μnλmd,(1.269)

      where λ is the de Broglie wavelength of the neutrons, m is the neutron mass, and d is the length of the path for which B≠0.

      The extraordinary property of the states of spin-12 particles, that they must be rotated through 4π to ‘bring them back to their unchanged orientation’, does not parallel our experience of rotating everyday objects. Such states are called spinors.

1.17 The Bargmann representation

      The functions

χn(z)≔znn!(1.270)

      provide an orthonormal basis for expanding functions realised on z-space (the complex plane), with scalar products defined in terms of z-space integrals with Bargmann measure³, e−∣z∣2π [3]. This space is called Bargmann space.

      The relevance of these functions to coherent states is implicit in the normalized coherent state form ∣z〉I, i.e. (cf. Volume 1, section 5.5)

      ∣z〉I≔e−∣z∣22∑n=0∞(z*)nn!∣n〉.(1.271)

      Whence, consider

      K≔∫∫dz∣z〉II〈z∣=∫∫dze−∣z∣2∑n(z*)nn!∣n〉∑mzmm!〈m∣;(1.272)

      which, for z=reiϕ, gives

      K=∫0∞rdr∫02πdϕe−r2∑n,mei(m−n)ϕrn+mn!m!∣n〉〈m∣.(1.273)

      Now,

      ∫02πdϕei(m−n)ϕ=2πδmn,(1.274)

      ∴K=∑n∫0∞drr2n+1e−r22πn!∣n〉〈n∣=∑nΓ(n+1)22πn!∣n〉〈n∣=π∑n∣n〉〈n∣=πI.(1.275)

      Thus, the resolution of the identity on Bargmann space is:

      I=∫∫dzπ∣z〉II〈z∣≔∫∫dze−∣z∣2π∣z〉IIII〈z∣,(1.276)

      where ∣z〉II↔χn(z), cf. equation (1.270). Then,

      〈Ψ1∣Ψ2〉=∫∫dze−∣z∣2π〈Ψ1∣z〉IIII〈z∣Ψ2〉=∫∫dze−∣z∣2πΨ1*(z)Ψ2(z)=∫∫dμ(z)Ψ1*(z)Ψ2(z),(1.277)

      where

      Ψ(z)≔II〈z∣Ψ〉,(1.278)

      dμ(z)≔e−∣z∣2πdz.(1.279)

      Bargmann representations of functions are transformed into position representations of functions by the Bargmann transformation,

Ψ(x)=∫∫dμ(z)A(x,z*)Ψ(z),(1.280)

      where

      A(x,z*)≔1π14exp−12x2+2xz*−12(z*)2(1.281)

      is the Bargmann kernel function.

      Comments:

      1 The orthogonality of the χn(z) is evident in a polar coordinate representation which gives (z*)nzm→ei(m−n)ϕ and ∫02πdϕei(m−n)ϕ=2πδmn.

      2 The normalizability of the χn(z) is evident from the Gaussian form of Bargmann measure which ‘quenches’ the scalar products for large ∣z∣. (Indeed, the scalar products involve ‘camouflaged’ Hermite polynomials.)

      3 The functions χn(z) are trivially generalised to tensor product functions,χn1(z1)⊗χn2(z2)⊗⋯which yields functions∑n1,n2,…αn1,n2,…z1n1n!!z2n2n2!⋯(cf. equations (1.147) and (1.176)).

1.17.1 Representation of operators

      Consider the operator O and its representation, O↔Γ(O) in terms of z and ∂∂z, O(z,∂∂z) acting on z-space wave functions, Ψ(z). This is similar to the procedure presented in Volume 1, chapter 8, where, e.g. the operator p_x