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Journal of Mathematical Physics

Phase-space representation of quantum state vectors

Masashi Ban

Published Apr 1, 1998• 20 citations
Publisher

Abstract

Phase-space representation of quantum state vectors is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained. Phase-space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative-position states and the relative-momentum states. The absolute-square of the phase-space function represents the probability distribution of the phase-space variables. Time-evolution of a quantum system is investigated in terms of the phase-space functions. The relations to the phase-space representations formulated by the other methods are obtained.

DOI
10.1063/1.532262

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Phase-space representation of quantum state vectors is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained. Phase-space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative-position states and the relative-momentum states. The absolute-square of the phase-space function represents the probability distribution of the phase-space variables. Time-evolution of a quantum system is investigated in terms of the phase-space functions. The relations to the phase-space representations formulated by the other methods are obtained.
A relative-state formulation of quantum systems is presented in terms of relative-coordinate states, relative-number states, and relative-energy states. The relative-coordinate states are used to describe quantum systems in position and momentum representations. The probability distribution is calculated in terms of the relative-coordinate states and is shown to be equivalent to the functional definition of the quantum probability in phase space. It is shown that a quantum-mechanical phase operator can be constructed in terms of the relative-number states without the well-known difficulties. The results are compared with those obtained by the Pegg-Barnett phase-operator formalism and the relations to various other phase-operator methods are also discussed. The energy-measurement and energy-probability distributions are discussed in terms of the relative-energy states. Furthermore, a relative-state formulation is developed in the Liouville space. A phase representation in the Liouville space is introduced to investigate the time evolution of quantum coherence. In the Liouville space a time operator is defined as a canonical conjugate of the time-evolution generator, but not the Hamiltonian energy operator. The relation to the internal time presented by Prigogine and Misra is discussed.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.
A relative number state (RNS) representation of a system composed of two distinguishable subsystems is proposed. A phase variable as a quantum mechanical operator conjugate to a relative number operator is defined based on the RNS representation space. The phase operator is expressed as a unitary exponential operator. The properties of the relative number operator, the phase operator, and their eigenstates are investigated in detail. The phase variable has a maximum uncertainty in any stationary state. Also, a time operator as a dynamical variable can be defined in the RNS representation space. The RNS representation is closely related to the Liouville space formulation and to thermofield dynamics. The RNS representation is shown to be a suitable method for investigating the Josephson junction with ultrasmall capacitance. A basic formulation of number-phase quantization in the Josephson junction is given in terms of the RNS representation.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.