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Cambridge University Press eBooks

The Schrödinger and Heisenberg Representations: Time-Evolution Operator

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter introduces the Hamiltonian operator in the language of second quantization, which is associated with the many-particle system to be considered throughout, including its time-dependent part. An expression is derived for the corresponding time-evolution operator, which depends only on the Hamiltonian and not on the initial preparation of the system before the time-dependent part begins to act. The connection between the Schrödinger and Heisenberg representations is discussed.

DOI
10.1017/9781009411509.003

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This chapter introduces the Hamiltonian operator in the language of second quantization, which is associated with the many-particle system to be considered throughout, including its time-dependent part. An expression is derived for the corresponding time-evolution operator, which depends only on the Hamiltonian and not on the initial preparation of the system before the time-dependent part begins to act. The connection between the Schrödinger and Heisenberg representations is discussed.
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.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
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.
Abstract All quantum processes involve time. For an algorithm to work, something must change over time. This chapter shows how that is implemented and how time dependence can be calculated. The chapter shows how a linear transformation that connects a state at one time to another state at a short time later is equivalent to a derivative with respect to time. To implement that idea, two separate methods are employed. The first is the perhaps familiar calculus-based Schrödinger differential equation where the Hamiltonian operator determines the time dependence of the states of the quantum system. The second method makes use of the Heisenberg picture where the time dependence resides in the operators.