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 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.
In this chapter, the authors describe the Heisenberg picture of quantum mechanics by formally accomplishing a time-dependent transformation of state vectors and operators of the Schrodinger picture. Any new picture of quantum mechanics will be acceptable, if the Schrodinger state vectors and operators are subjected to a transformation, which does not alter: the spectrum of the eigenvalues of the operators and the scalar product of an arbitrary state vector with the eigenstate vectors. These two conditions are easily fulfilled by means of the unitary transformations. The authors show that a unitary operator represents an isomorph mapping of a certain unitary vector space onto itself. Since the physical quantities measured in the experiment are, in fact, a measure of the scalar product, as implied by the definition of the probability W through the scalar product in, it is clear that the predictions based upon the new, Heisenberg picture must be identical to those obtained in the Schrodinger picture.
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.
This chapter explores the Schrödinger and Heisenberg pictures in quantum mechanics, emphasizing their mathematical equivalence and physical consistency. The two formulations are shown to be connected by a unitary transformation. The Heisenberg equation of motion is derived, and the time evolution of operators—particularly those not explicitly dependent on time—is expressed in terms of their initial values. It is demonstrated that scalar products, expectation values, matrix elements, and commutation relations remain invariant under the change of picture. Drawing a close analogy with classical mechanics, Dirac's insight is presented: the quantum mechanical counterpart of the classical Poisson bracket is the operator commutator divided by iℏ . The chapter concludes by deriving the Lorentz force law within the quantum framework using commutator algebra. Metadata/Keywords: Schrödinger picture, Heisenberg picture, unitary transformation, Heisenberg equation of motion, Poisson bracket, Lorentz force, time evolution, expectation values
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.
In general, the expectation values of observable quantities depend on time. In quantum mechanics, these values are given by the expression: < X >t=< A(t)|X(t)|A(t) >. (10.1) There is an intrinsic ambiguity in determining the time dependence on the various elements (bra, ket and operator) which comprise the right hand side of equation (10.1) because we can transfer this dependence from one element to another, while keeping unchanged the expectation value < X >t, which is all we can measure on the system. The ambiguity gives rise to different descriptions of the motion, connected to unitary time-dependent transformations, thus equivalent to each other. In the following sections, we describe the Schro¨dinger and Heisenberg representations. Subsequently we will introduce a third description of the motion or interactions: the Dirac representation, also called the interaction representation, which is particularly useful in the case of weakly interacting systems.
In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.