10 Momentum and the Free Particle

We’ll now turn to the problem that conventional quantum mechanics courses generally begin with: that of the quantum system describing a free particle moving in physical space ${\mathbf{R}}^3$ . This is something quite diferent from the classical mechanical description of a free particle, which will be reviewed in chapter 14. $\mathrm{A}$ common way of motivating this is to begin with the 1924 suggestion by de Broglie that, just as photons may behave like either particles or waves, the same should be true for matter particles. Photons were known to carry an energy given by $E=\hslash \omega$ , where $\omega$ is the angular frequency of the wave. De Broglie’s proposal was that matter particles with momentum $\mathbf{p}=\hslash \mathbf{k}$ can also behave like a wave, with dependence on the spatial position $\mathbf{q}$ given by

$$e^{i\mathbf{k}\cdot \mathbf{q}}$$

This proposal was realized in Schr¨odinger’s early 1926 discovery of a version of quantum mechanics in which the state space is

$$\mathcal{H}=L^2({\mathbf{R}}^3)$$

which is the space of square-integrable complex-valued functions on ${\mathbf{R}}^3$ , called “wavefunctions”. The operator

$$\mathbf{P}=-i\hslash \nabla$$

will have eigenvalues $\hslash \mathbf{k}$ , the de Broglie momentum, so it can be identified as the momentum operator.

In this chapter our discussion will emphasize the central role of the momentum operator. This operator will have the same relationship to spatial translations as the Hamiltonian operator does to time translations. In both cases, the operators are the Lie algebra representation operators corresponding to a unitary representation on the quantum state space $\mathcal{H}$ of groups of translations (translations in the three space and one time directions respectively).

One way to motivate the quantum theory of a free particle is that, whatever it is, it should have analogous behavior to that of the classical case under translations in space and time. In chapter 14 we will see that in the Hamiltonian form of classical mechanics, the components of the momentum vector give a basis of the Lie algebra of the spatial translation group ${\mathbf{R}}^3$ , the energy a basis of the Lie algebra of the time translation group $R$. Invoking the classical relationship between energy and momentum

$$E=\frac{|\mathbf{p}{|}^2}{2m}$$

used in non-relativistic mechanics relates the Hamiltonian and momentum operators by

$$H=\frac{|\mathbf{P}{|}^2}{2m}$$

On wavefunctions, for this choice of H the abstract Schr¨odinger equation 1.1 becomes the partial diferential equation

$$i\hslash \frac{\partial }{\partial t}\psi (\mathbf{q},t)=\frac{-{\hslash }^2}{2m}{\nabla }^2\psi (\mathbf{q},t)$$

for the wavefunction of a free particle.

中文目录

Chapter contents

• 10.1 The group R and its representations
• 10.2 Translations in time and space
  • 10.2.1 Energy and the group R of time translations
  • 10.2.2 Momentum and the group ${\mathbf{R}}^3$ of space translations
• 10.3 The energy-momentum relation and the Schr¨odinger equation for a free particle
• 10.4 For further reading

Previous chapter: 9 Tensor Products, Entanglement, and Addition of Spin

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来源版本：2025-10-20

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