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Modern Physics Letters B

NEW REPRESENTATION FOR DESCRIBING LOWERING AND ASCENDING OF AN ELECTRON'S ANGULAR MOMENTUM IN A UNIFORM MAGNETIC FIELD

Hong-Yi Fan, H. ZOU, Yue Fan

Published Apr 10, 1999• 5 citations
Publisher

Abstract

We construct a new quantum mechanical representation |l,r> for describing an electron's angular momentum in a uniform magnetic field, where l is the eigenvalue of the azimuthal angular momentum Lz and r is the orbit radius of the electron's motion. The ascending and lowering operators for |l,r>→|l± 1,r> are also obtained. Throughout our discussion the <λ| representation is fully employed (see Hong-Yi Fan and Yong Ren, Mod. Phys. Lett.B10, 523 (1996).

DOI
10.1142/s0217984999000336

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We construct a new quantum mechanical representation |l,r> for describing an electron's angular momentum in a uniform magnetic field, where l is the eigenvalue of the azimuthal angular momentum Lz and r is the orbit radius of the electron's motion. The ascending and lowering operators for |l,r>→|l± 1,r> are also obtained. Throughout our discussion the <λ| representation is fully employed (see Hong-Yi Fan and Yong Ren, Mod. Phys. Lett.B10, 523 (1996).
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.
We use the complete and orthonormal ⟨λ| representation (Phys. Lett. A126 (1987) 150), constructed in terms of guiding centres [ K ± = M Ω/2 1/2 ( x 0 ∓i y 0 )] and kinetic momenta Π ± describing an electron in a uniform magnetic field, to identify that the Laughlin state vector is ∏ i < j N ( K + i - K + j ) m ||0⟩, where ||0⟩ is annihilated by (Π - , K - ). Following this, we further employ the ⟨λ| representation to derive the angular momentum of the Laughlin ground state and excited state in a direct and convenient way. This approach to studying some properties of the Laughlin state seems to be new.