The Wigner-Heisenberg Algebra in Quantum Mechanics
R. de Lima Rodrigues
ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }
InTech eBooks•Apr 3, 2013•1 citation