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Communications in Theoretical Physics

Landau Wavefunction Immediately Derived by Virtue of the 〈λ| Representation

Fan Hong‐Yi, Lin Jing-Xian, Fan Yue

Published Feb 15, 2001• 2 citations
Publisher

Abstract

We show that by virtue of the 〈λ| representation (Hong-Yi Fan, Phys. Lett. A 126 (1987) 150) the Landau wavefunctions of an electron in a uniform magnetic field can be immediately derived without solving the corresponding Schrödinger equation. It turns out that the differential operation form of the electron's dynamic Hamiltonian, adopted in standard quantum mechanics textbooks, is actually expressed in 〈λ| representation.

DOI
10.1088/0253-6102/35/2/153

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We show that by virtue of the 〈λ| representation (Hong-Yi Fan, Phys. Lett. A 126 (1987) 150) the Landau wavefunctions of an electron in a uniform magnetic field can be immediately derived without solving the corresponding Schrödinger equation. It turns out that the differential operation form of the electron's dynamic Hamiltonian, adopted in standard quantum mechanics textbooks, is actually expressed in 〈λ| representation.
By using the technique of integration within an ordered product (IWOP) of operators, we derive the normal-product form of the gauge-invariant Wigner operator of an electron in a uniform magnetic field and its expression in the newly constructed |λ> representation. The virtue of working in the |λ> representation lies in the fact that it is expressed in terms of the ladder operators Π±, K±(see Eq. (10)), thus the magnetic vector potential can naturally be included. On these bases we can easily obtain Wigner functions of some important electron states.
It is shown that the problem of a free eletron in a uniform magnetic field provides an interesting example for the use of recently developed canonical transformation methods in quantum mechanics. The Schrödinger equation is immediately solved by a unitary transformation corresponding to a finite linear symplectic transformation in phase space. The natural invariance and dynamical groups involve affine and not only linear symplectic transformations, and the Weyl group turns out to be an appropriate invariance group to account completely for the infinite degeneracies.