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American Journal of Physics

Transition Probabilities for the Forced Quantum Oscillator

Leonard M. Scarfone

Published Feb 1, 1964• 11 citations
Publisher

Abstract

Transition probabilities for the forced, undamped quantum harmonic oscillator are obtained by expressing S-matrix elements entirely in terms of Heisenberg states and operators. The application of this method to this elementary-field theoretic model provides a simple example of the contraction technique of modern field theory whereby particles are converted into their corresponding currents. This problem was recently analyzed in the interaction picture from a pedagogical point of view and the treatment presented here is given in the same spirit.

DOI
10.1119/1.1970144

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Transition probabilities for the forced, undamped quantum harmonic oscillator are obtained by expressing S-matrix elements entirely in terms of Heisenberg states and operators. The application of this method to this elementary-field theoretic model provides a simple example of the contraction technique of modern field theory whereby particles are converted into their corresponding currents. This problem was recently analyzed in the interaction picture from a pedagogical point of view and the treatment presented here is given in the same spirit.
The exact solution for a forced, undamped quantum harmonic oscillator is obtained by S-matrix techniques. The force F(t) is assumed to vanish at t = ±∞. The only restriction is that the Fourier transform of F(t) must exist. The transition probabilities are obtained in closed form in terms of Laguerre polynomials. The mean energy transfer to the oscillator is found to be independent of the initial state and is in agreement with the classical result for an oscillator originally at rest. This problem provides a good example of field theoretical procedures in an elementary context. Therefore, all field theoretical concepts are carefully defined and explained as they are introduced in order that the discussion may be self-contained.
A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.
For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.
Quantization of the damped harmonic oscillator is taken as leitmotiv to gently introduce elements of quantum probability theory for physicists. To this end, we take (graduate) students in physics as entry level and explain the physical intuition and motivation behind the, sometimes overwhelming, math machinery of quantum probability theory. The main text starts with the quantization of the (undamped) harmonic oscillator from the Heisenberg and Schroedinger point of view. We show how both treatments are special instances of a quantum probabilistic quantization procedure: the second quantization functor. We then apply the second quantization functor to the damped harmonic oscillator and interpret the quantum dynamics of the position and energy operator as stochastic processes.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
The interaction picture of quantum mechanics is used to calculate the unitary time development operator for a harmonic oscillator subject to an arbitrary time-dependent force. The result is identical to that obtained from the more usual method of the Heisenberg equations of motion, except for a phase factor which the Heisenberg picture method is unable to determine. A simplified derivation of the phase factor is given in the adiabatic limit, as well as a discussion of some possible physical effects which would depend upon the phase factor.