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College Physics

A new approach to study the parity operator

LI Ti-jun

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In the quantum mechanics,the parity operator P^ can be expressed by ∫d2zπ|z〉〈-z|、∫dq|q〉〈-q| and ∫dp|p〉〈-p|.By using the Dirac symbol and the completeness of quantum representations,an alternative method is proposed.By this method,these integrations for asymmetric projection operator are realized,and then the explicit form of the asymmetric projection operator is obtained.

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In the quantum mechanics,the parity operator P^ can be expressed by ∫d2zπ|z〉〈-z|、∫dq|q〉〈-q| and ∫dp|p〉〈-p|.By using the Dirac symbol and the completeness of quantum representations,an alternative method is proposed.By this method,these integrations for asymmetric projection operator are realized,and then the explicit form of the asymmetric projection operator is obtained.
Abstract Operators play an important role in the formalism of quantum mechanics. By means of the technique of integration within an ordered product of operators and Dirac notation, we analytically prove that a new kind of asymmetric two-mode projection integral operator and another asymmetric two-mode Fock state projection summation operator are equal to the same Hermitian operator. For such Hermitian operator, we also rigorously demonstrate it corresponds to a parity measurement of a two-mode quantum state after passing through a beam splitter. The method used in this work provides a good exercise in quantum mechanics at the graduate level.
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) }