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International Journal of Modern Physics B

ENTANGLED STATE REPRESENTATIONS FOR DESCRIBING 2-DIMENSIONAL ELECTRON SYSTEM IN UNIFORM MAGNETIC FIELD

Hong-Yi Fan

Published Aug 30, 2004• 4 citations
Publisher

Abstract

We review how to rely on the quantum entanglement idea of Einstein–Podolsky–Rosen and the developed Dirac's symbolic method to set up two kinds of entangled state representations for describing the motion and states of an electron in uniform magnetic field. The entangled states can be employed for conveniently expressing Landau wave function and Laughlin wave function with a fresh look. We analyze the entanglement involved in electron's coordinates (or momenta) eigenstates, and in the angular momentum-orbit radius entangled state. Various applications of these two representations, such as in developing angular momentum theory, squeezing mechanism, Wigner function and tomography theory for this system are presented. Thus the present review systematically summarizes a distinct approach for tackling this physical system.

DOI
10.1142/s0217979204026196

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We review how to rely on the quantum entanglement idea of Einstein–Podolsky–Rosen and the developed Dirac's symbolic method to set up two kinds of entangled state representations for describing the motion and states of an electron in uniform magnetic field. The entangled states can be employed for conveniently expressing Landau wave function and Laughlin wave function with a fresh look. We analyze the entanglement involved in electron's coordinates (or momenta) eigenstates, and in the angular momentum-orbit radius entangled state. Various applications of these two representations, such as in developing angular momentum theory, squeezing mechanism, Wigner function and tomography theory for this system are presented. Thus the present review systematically summarizes a distinct approach for tackling this physical system.
Based on the entangled state representation for describing electron's coordinates under uniform magnetic field, we establish a one-to-one correspondence between quantum Hamiltonian and its classical Weyl function through the introduction of Wigner operator. We also study some new important properties of Wigner function of electron's quantum states, such as its upper bound, and its relation with electron's wave function. These discussions demonstrate the beauty and elegance of the entangled state representation.
In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.