Abstract A central problem in quantum mechanics is the calculation of the overlap, that is, the scalar product between two quantum states. In the semiclassical limit (Bohr's correspondence principle) we visualize this quantity as the area of overlap between two bands in phase space. In the case of more than one overlap the contributing amplitudes have to be combined with a phase difference again determined by an area in phase space. In this sense the familiar double‐slit interference experiment is generalized to an interference in phase space. We derive this concept by the WKB approximation, illustrate it by the example of Franck‐Condon transitions in diatomic molecules, and compare it with and contrast it to Wigner's concept of pseudo‐probabilities in phase space.
Semiclassical angular momentum matrices are calculated using a new contour-integral formula for matrix-elements. WKB spherical harmonic functions are found to be exaclty orthonormal with the contour-integral inner-product. Matrix-elements obtained from these wave-functions are accurate to about 2 % and the matrices obey the commutation relations expected of quantum angular momentum operators. The semiclassical wave-functions are related to a superposition of allowed classical orbits and this illustrates the connection to the Feynman pathsummation formulation of quantum mechanics for electrons in a spherically symmetric potential.
We apply geometric phase ideas to coherent states to shed light on interference phenomenon in the phase space description of continuous variable Cartesian quantum systems. The motivating idea is Pancharatnam's concept of two Hilbert space vectors being ’’in phase”. This leads to special sums and integrals of vectors which are locally in-phase superpositions. Geometric phase considerations naturally lead to the emergence of phase space areas. Applied to the overcomplete family of coherent states, we are led to preferred in-phase integral representations for various states of physical significance, such as the position, momentum and Fock states as well as the squeezed vacuum state. Interestingly, the Husimi-Kano Q function is maximized along the line of such superpositions. We also get a fresh perspective on the Bohr-Sommerfeld quantization condition. Finally, we use our exact integral representations to obtain asymptotic expansions for state overlaps and matrix elements, leading to phase space area considerations similar to the ones noted earlier in the seminal works of Schleich, Wheeler and collaborators, but now from a geometric phase perspective.
Quantum states of a spin
21 (a qubit) are parametrized by the space
CP1∼S2, the Bloch sphere. A spin
j for a generic
j (a
2j+1-state system) is represented instead by a point of a larger space,
CP2j. Here we study the state of a single angular momentum/spin in the limit,
j→∞. The special class of states $ | j, {\mathbf n}\rangle \in {\mathbf {CP}}^{2j} $, with spin oriented towards definite spatial directions
n∈S2, i.e., $({\mathbf J}\cdot {\mathbf n} ) \, | j, {\mathbf n}\rangle = j\, |j, {\mathbf n}\rangle $, are found to behave as classical angular momenta,
jn, in this limit. Vice versa, general spin states in
CP2j do not become classical, even at large
j. We discuss these questions, by analysing the Stern-Gerlach processes, the angular-momentum composition rule, and the rotation matrix. Our observations help to clarify better how classical mechanics emerges from quantum mechanics in this context (e.g., with unique trajectories for a particle carrying a large spin), and to make the widespread idea that large spins somehow become classical, more precise.