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Physical review. B, Solid state

Landau Diamagnetism from the Coherent States of an Electron in a Uniform Magnetic Field

Albert Feldman, Arnold H. Kahn

Published Jun 15, 1970• 134 citations
Publisher

Abstract

A complete set of coherent-state wave packets has been constructed for an electron in a uniform magnetic field. These states are nonspreading packets of minimum uncertainty that follow the classical motion. Use was made of the ladder operators that generate all the eigen-states of the Hamiltonian from any one energy eigenstate. The coherent states are the eigenstates of the two ladder operators that annihilate the zero-angular-momentum ground state. We have calculated the partition function, exploiting advantages of the coherent-state basis. The Landau diamagnetism and the de Haas-van Alphen oscillations are contained in the coherent-state framework.

DOI
10.1103/physrevb.1.4584

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•Jan 1, 1995•1 citation
A complete set of coherent-state wave packets has been constructed for an electron in a uniform magnetic field. These states are nonspreading packets of minimum uncertainty that follow the classical motion. Use was made of the ladder operators that generate all the eigen-states of the Hamiltonian from any one energy eigenstate. The coherent states are the eigenstates of the two ladder operators that annihilate the zero-angular-momentum ground state. We have calculated the partition function, exploiting advantages of the coherent-state basis. The Landau diamagnetism and the de Haas-van Alphen oscillations are contained in the coherent-state framework.
Based on the newly constructed state | l , r ⟩⟩ [Fan et al. , Chin. Phys. Lett. 16(1999)706], where l is the angular momentum quantum number and r denotes the electron's orbit radius in a uniform magnetic field, we propose a new angular momentum-phase coherent state by introducing a new operator A . A and A † are annihilation and creation operators in the | l , r ⟩⟩ space, respectively. The coherent state is A 's eigenket and possesses non-orthonormal and overcomplete properties. It is constructed on the certain superposition of zero-angular momentum states along the radius direction.
Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.
This chapter shows how to construct path integral representation using a closure relation based on the eigenstates of the creation or the annihilation operators. It examines how coherent states are used to obtain a path integral representation of the partition function as well as calculating directly the trace defining the partition function. The chapter explores quantum field theory using the Feynman functional and diagrammatic techniques as foundations to apply Quantum Field Theory to a broad range of topics in physics. The minimum uncertainty wave packets for the harmonic oscillator were published as coherent states by Schrodinger. Coherent states have played an important role in many branches of physics and, in particular, quantum field theory and quantum optics. The most important property of coherent states is their overcompleteness in the Fock space, which implies that any vector of the Fock space can be expanded in terms of coherent states.
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the