General Methods of Quantum Mechanics

Abstract

The preceding chapters have provided the introductory information about Quantum Mechanics. Here the general principles of the theory are illustrated, and the methods worked out for the Hamiltonian operator are extended to the operators associated to dynamical variables different from energy. The important concept of separable operator is introduced, and the property of some operators to commute with each other is related to the mutual compatibility of measurements of the corresponding dynamical variables. Then, the concepts of expectation value and uncertainty are introduced, and the Heisenberg uncertainty principle is worked out. This leads in turn to the analysis of the time derivative of the expectation values, showing that the latter fulfill relations identical to those of Classical Mechanics. The form of the minimum-uncertainty wave packet is worked out in the complements.

Problems

• 10.1 Starting from the wave packet (9.5) describing a free particle, determine the time evolution of its position without resorting to the approximation used in Sect. 9.6.

• 10.2 Using the results of Prob. 10.1, determine the time evolution of the standard deviation of position.

• 10.3 Starting from the wave packet (9.5) describing a free particle, determine the time evolution of its momentum without resorting to the approximation used in Sect. 9.6.

• 10.4 Using the results of Prob. 10.3, determine the time evolution of the standard deviation of momentum.

• 10.5 Consider a one-dimensional wave function that at some instant of time is given by a minimum-uncertainty packet (10.35) whose polar form is

  $$\alpha = \frac{1}{ \sqrt[4]{2\,\pi} \, \sqrt{\sigma} } \, {\rm exp} \left[ { - \frac{ (x-x_0)^2} { 4\, \sigma^2} } \right], \qquad \beta = k_0 \, x.$$

  (10.36)

The wave packet is normalized to 1. Using the concepts introduced in Sect. 9.7.3, find the “convective” and “thermal” parts of the expectation value of the kinetic energy.