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Q. M. Pictures; Heisenberg Equation; Linear Response; Superoperators and Non-Markovian Equations

Norman J. M. Horing

Published Jul 27, 2017• 0 citations

Abstract

Abstract Three fundamental and equivalent mathematical frameworks (“pictures”) in which quantum theory can be lodged are exhibited and their relations and relative advantages/disadvantages are discussed: (1) The Schrödinger picture considers the dynamical development of the overall system state vector as a function of time relative to a fixed complete set of time-independent basis eigenstates; (2) The Heisenberg picture (convenient for the use of Green’s functions) embeds the dynamical development of the system in a time-dependent counter-rotation of the complete set of basis eigenstates relative to the fixed, time-independent overall system state, so that the relation of the latter fixed system state to the counter-rotating basis eigenstates is identically the same in the Heisenberg picture as it is in the Schrödinger picture; (3) the Interaction Picture addresses the situation in which a Hamiltonian, H=H0+H1, involves a part H0 whose equations are relatively easy to solve and a more complicated part, H1, treated perturbatively. The Heisenberg equation of motion for operators is discussed, and is applied to annihilation and creation operators. The S-matrix, density matrix and von Neumann equation, along with superoperators and non-Markovian kinetic equations are also addressed (e.g. the intracollisional field effect).

DOI: 10.1093/oso/9780198791942.003.0003

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Abstract Three fundamental and equivalent mathematical frameworks (“pictures”) in which quantum theory can be lodged are exhibited and their relations and relative advantages/disadvantages are discussed: (1) The Schrödinger picture considers the dynamical development of the overall system state vector as a function of time relative to a fixed complete set of time-independent basis eigenstates; (2) The Heisenberg picture (convenient for the use of Green’s functions) embeds the dynamical development of the system in a time-dependent counter-rotation of the complete set of basis eigenstates relative to the fixed, time-independent overall system state, so that the relation of the latter fixed system state to the counter-rotating basis eigenstates is identically the same in the Heisenberg picture as it is in the Schrödinger picture; (3) the Interaction Picture addresses the situation in which a Hamiltonian, H=H0+H1, involves a part H0 whose equations are relatively easy to solve and a more complicated part, H1, treated perturbatively. The Heisenberg equation of motion for operators is discussed, and is applied to annihilation and creation operators. The S-matrix, density matrix and von Neumann equation, along with superoperators and non-Markovian kinetic equations are also addressed (e.g. the intracollisional field effect).

In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. The description of quantum dynamics where the wave function is time-dependent and the operators describing the observables are time-independent is known as the Schrodinger picture. The use of total time derivative of operator that depends on the time evolution of the wave function as well as on any intrinsic time-dependence in the operators simplifies the development of quantum mechanics and allows the development to more closely follow the corresponding development of classical mechanics. The difference between the Schrodinger and Heisenberg pictures is best understood as the difference in the basic set of observables.

Quantum mechanics takes various forms. For atomic and nuclear physics, the Schrödinger picture is convenient. For quantum field theory and for correspondence with classical mechanics through Poisson brackets, the Heisenberg picture is useful. The interaction picture is useful in calculations. The Wigner phase-space is convenient for studying the uncertainty relations and their symmetries. The density matrix and probability distribution are necessary for extracting measurable dynamic quantities and are essential for ensemble averages. Mathematical instruments, such as the Lie algebra of the rotation group, the creation and annihilation operators of the Fock space and the Lie algebra governing Dirac’s two oscillators are shown to be derivable from Heisenberg’s commutation relations.

Abstract Using the second quantization procedure, we have so far only treated energy eigenstates with a trivial time dependence e−iωt, instant processes at a single time, and systems where interactions are approximated by time-independent mean-field theory. But how does one then treat the general case of time dependence in second quantization? That question will be addressed in this chapter, where the formal theory of time evolution is discussed using three representations, or “pictures:” the Schrödinger picture, the Heisenberg picture, and the interaction picture.

In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrödinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states. Quantum dynamics Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrödinger and Heisenberg pictures. The Schrödinger picture Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.

This chapter briefly explains the two elementary points concerning quantum mechanics: the mean value rule and the Heisenberg picture. The Schrodinger and Heisenberg pictures are two different but equivalent sets of rules. In other words, when applied to a given physical situation they lead to the same observable predictions. In some respects the Heisenberg picture has definite technical advantages. It is particularly convenient, for instance, in quantum field theory. For describing pure cases the use of wave functions in place of kets is sometimes convenient. The mean value of an observable A and the probability that a measurement of A yields eigenvalue a k can be expressed by means of wave functions. The state vector of a system that does not undergo a measurement does not change with time, obviously this state vector is suddenly modified—also in the Heisenberg picture—when a measurement takes place.

The description of the time evolution of systems is technically known as the "Schrodinger picture." An important alternative description is the "Heisenberg picture," which indeed was the one Heisenberg used when he first formulated quantum mechanics. The Schrodinger and Heisenberg pictures represent two different but equivalent sets of rules. In other words, when applied to a given physical situation they lead to the same observable predictions. In some respects, the Heisenberg picture presents definite technical advantages. It is particularly convenient, for instance, in quantum field theory. The chapter discusses work consistently with the equivalent Schrodinger set of rules. For this reason, it describes briefly the Heisenberg formulation. The chapter shows that the expressions used in the Schrodinger picture can be carried over to the Heisenberg picture. It implies, however, that the kets associated with the possible values that can be found upon measurement of an observable are themselves time-dependent kets.