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arXiv:0808.2149 · The Journal of Chemical Physics

Explicit state vector for Torres-Vega–Frederick phase space representation and its statistical behavior

Liyun Hu, Hong-Yi Fan, Hai-liang Lu

Published Feb 5, 2008• 8 citations
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Abstract

We find the explicit state vector for Torres-Vega-Frederick phase space representation [Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 (1993)], denoted by Gamma. This set of states make up a complete and nonorthogonal representation. The Weyl ordered form of Gamma Gamma [see text for the sign] is derived, which can clearly exhibit the statistical behavior of marginal distribution of Gamma Gamma [see text for the sign]. The minimum uncertainty relation for mid R:Gamma is demonstrated, which shows it being a coherent squeezed state.

DOI
10.1063/1.2827480

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We find the explicit state vector for Torres-Vega-Frederick phase space representation [Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 (1993)], denoted by Gamma. This set of states make up a complete and nonorthogonal representation. The Weyl ordered form of Gamma Gamma [see text for the sign] is derived, which can clearly exhibit the statistical behavior of marginal distribution of Gamma Gamma [see text for the sign]. The minimum uncertainty relation for mid R:Gamma is demonstrated, which shows it being a coherent squeezed state.
We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.
In this article we argue that the state-vector phase-space representation recently proposed by Torres-Vega and co-workers [introduced in J. Chem. Phys. 98, 3103 (1993)] coincides with the totality of coherent-state representations for the Heisenberg-Weyl group. This fact leads to ambiguities when one wants to solve the stationary Schrödinger equation in phase space and we devise two schemes for the removal of these ambiguities. The physical interpretation of the phase-space wave functions is discussed and a procedure for computing expectation values as integrals over phase space is presented. Our formal points are illustrated by two examples.
The theory of the quantum mechanical representation in phase space recently proposed by Torres-Vega and Frederick is rigorously verified to be extended to the bound systems whose potential energy functions may contain terms of negative powers of coordinate and the quantum Liouville equation in the theory is also generalized to the same systems. It is suggested that the kernels of projection transformation take the forms e i pq/ 2 ħ and e -(i pq/ 2 ħ ) instead of {1/√(4π ħ )}e i pq/ 2 ħ and {1/√(4π ħ )}e -(i pq/ 2 ħ ) in order to keep the consistency for projections of the wave functions in all quantum problems. The validity of quantum average values and the virial theorem in the phase space representation are discussed in detail.
Theories of Torres-Vega and Fredrick (1993 J. Chem. Phys. 98 3103), Harriman (1994 J. Chem. Phys. 100 3651) and of Ban (1998 J. Math. Phys. 39 1744), in which phase space points ( p , q ) are used as configurational variables to formulate quantum mechanics are considered from the standpoint of a class of quantization schemes associating phase space functions with operators. The connection between these schemes and the theories given in Torres-Vega and Fredrick (1993 J. Chem. Phys. 98 3103), Harriman (1994 J. Chem. Phys. 100 3651), Dirac (1930 The Principles of Quantum Mechanics (Oxford: Oxford University Press)), Møller, Jørgensen and Torres-Vega (1997 J. Chem. Phys. 106 7228), Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), Li, Wei and Lü (2004 Phys. Rev. A 70 022105), Ban (1998 J. Math. Phys. 39 1744) is made by means of augmented wavefunctions ψ (λ) σ ( p , q ; t ), where λ = 0 corresponds to the ordering of Wigner and Weyl. For that case we use these functions to define a family of positive operator-valued measures for the phase angle of an harmonic oscillator.
For the first time we introduce the Husimi operator Delta_h(gamma,varepsilon;kappa) for studying Husimi distribution in phase space(gamma,varepsilon) for electron's states in uniform magnetic field, where kappa is the Gaussian spatial width parameter. Using the Wigner operator in the entangled state |lambda> representation [Hong-Yi Fan, Phys. Lett. A 301 (2002)153; A 126 (1987) 145) we find that Delta_h(gamma,varepsilon;kappa) is just a pure squeezed coherent state density operator |gamma,varepsilon>_kappa kappa
Using the time-dependent annihilation and creation operators, the invariant operators, for a free mass and an oscillator, we find the coherent-squeezed state representation of a travelling general Gaussian wave packet with initial expectation values, x0x_0x0​ and p0p_0p0​, of the position and momentum and variances, Δx0Δx_0Δx0​ and Δp0Δp_0Δp0​. The initial general Gaussian wave packet takes, up to a normalization factor, the form eip0x/ℏe−(1∓iδ)(x−x0)2/4(Δx0)2e^{i p_0 x/\hbar} e^{- (1 \mp i δ) (x - x_0)^2 / 4 (Δx_0)^2}eip0​x/ℏe−(1∓iδ)(x−x0​)2/4(Δx0​)2, where δ=(2Δx0Δp0/ℏ)2−1δ= \sqrt{(2Δx_0 Δp_0/\hbar)^2 -1}δ=(2Δx0​Δp0​/ℏ)2−1​ denotes a measure of deviation from the minimum uncertainty or the initial position-momentum correlation δ=2Δ(xp)0/ℏδ= 2Δ(xp)_0 / \hbarδ=2Δ(xp)0​/ℏ. The travelling Gaussian wave packet takes, up to a time-dependent phase and normalization factor, the form eipcx/ℏe−(1−2iΔ(xp)t/ℏ)(x−xc)2/4(Δxt)2e^{i p_c x/\hbar} e^{- (1 - 2 i Δ(xp)_t/\hbar) (x - x_c)^2 / 4 (Δx_t)^2}eipc​x/ℏe−(1−2iΔ(xp)t​/ℏ)(x−xc​)2/4(Δxt​)2 and the centroid follows the the classical trajectory with xc(t)x_c(t)xc​(t) and pc(t)p_c(t)pc​(t). The position variance is found to have additionally a linearly time-dependent term proportional to δδδ with both positive and negative signs.