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Journal of Mathematical Physics

von Neumann lattices and magnetic orbitals in finite phase space

J. Žák

Published Jul 1, 1989• 14 citations
Publisher

Abstract

It is proven that in the finite phase space of dimensionality M, the number of independent von Neumann states is M−r, where r is the number of distinct zeros of the kq function. When applied to magnetic orbitals, this leads to a linear dependence between them for each zero of the corresponding kq function. Different interesting identities follow from the linear dependence of von Neumann states.

DOI
10.1063/1.528292

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It is proven that in the finite phase space of dimensionality M, the number of independent von Neumann states is M−r, where r is the number of distinct zeros of the kq function. When applied to magnetic orbitals, this leads to a linear dependence between them for each zero of the corresponding kq function. Different interesting identities follow from the linear dependence of von Neumann states.
A rational von Neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers. Zeros of harmonic oscillator functions in the kq representation on such lattices are found. It is shown that the number of zeros of the kq function determines the number of states by which a set on a von Neumann lattice is overcomplete. Interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation.
A formula is derived connecting any wave function 〈k,q‖f〉 in the kq-representation with the corresponding amplitudes of the state ‖f〉 on von Neumann lattices of states. The formula is used for establishing a possible interpretation for these amplitudes, for obtaining linear relationships between them, and for finding sum rules for the squares of their absolute values, and other related sum rules. It can also be used for establishing completeness criteria for the lattices of states and for defining a modified Hilbert space in which they become strictly complete. Particular attention is given to the coherent state lattice, but the discussion is extended to von Neumann lattices generated from an arbitrary state. Lattices generated from harmonic oscillator states are studied explicitly, and shown incidentally to lead to a wealth of summation expressions for Laguerre polynomials.
A general set of states on a lattice in the phase plane is considered. The discrete set of von Neumann coherent states for a harmonic oscillator is a particular case of the above set. The kq\mathrm{kq}kq representation is used in an elementary proof of completeness and orthogonality of states on a discrete phase plane lattice.
A general formula for the number of linearly independent states from (2 s + 1)N-dimensional Hilbert space for a linear chain of N spins s corresponding to a wave number k has been derived. As shown, this density is inhomogeneous within the Brillouin zone, and its shape depends on the decomposition of N into prime factors. It proves that non all elementary excitations in a Heisenberg magnet can be adequately described in terms of quasiparticles, and in particular, they not necessarily exhibit a band structure.
We consider solutions of the non-linear von Neumann equation involving Jacobi's elliptic functions sn, cn, and dn, and 3 linearly independent operators. In two cases one can construct a state-dependent Hamiltonian which is such that the corresponding non-linear von Neumann equation is solved by the given density operator. We prove that in a certain context these two cases are the only possibilities to obtain special solutions of this kind. Well-known solutions of the reduced Maxwell-Bloch equations produce examples of each of the two cases. Also known solutions of the non-linear von Neumann equation in dimension 3 are reproduced by the present approach.
The transition from classical physics to quantum mechanics has been mysterious. Here, we derive the space-independent von Neumann equation for electron spin mathematically from the classical Bloch or Majorana--Bloch equation, which is also derived. Subsequently, the space-independent Schrödinger--Pauli equation is derived in both the quantum mechanical and recently developed co-quantum dynamic frameworks. An additional appendix incorporates the CQD signum branching function into the nonlinear von Neumann induction equation and develops its explicit double-commutator and state-dependent effective-Hamiltonian forms.