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Quantum State Time Dependence

Alice Flarend, Robert Hilborn

Published Jul 9, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780197904930.003.0018

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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.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
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.
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This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
This chapter outlines the governing equations in non-relativistic quantum physics. The Schrödinger equation (the Schrödinger picture) looms large. Its intuitive simplicity and elegance render it particularly appealing to aspiring engineers and materials scientists. The other two equivalent pictures presented are the Heisenberg picture and the interaction picture. The Heisenberg picture emphasizes time-dependent operators, and the interaction picture splits the so-called interaction Hamiltonian from the total Hamiltonian of the system considered. Much attention has been paid in this chapter to establish relationships between the pictures, as meaningful as possible at this level of treatment.