Symmetries

Abstract

Symmetries and symmetry transformations in quantum systems are considered. Rotation operators, Rotation Matrices and general rotations expressed in terms of the Euler angles are analyzed. Tensor operators are introduced and the Wigner-Eckart theorem is established. Spatial and temporal translations are considered. Spatial Reflection (Parity) and Time-Reversal are studied as examples of Discrete Transformations. Dynamical Symmetries are discussed and the example of the Runge-Lenz vector in the Hydrogen atom is analyzed.

Problems and Exercises

19.1

If $\psi (\mathbf{{ r}},t)$ is the wave function of a particle, show that the wave function $\psi ^*(\mathbf{{r}}, -t)$ describes a particle having the opposite momentum.

19.2

A particle of spin $s=1$ is described by the Hamiltonian

$$\hat{H}\,=\,a\hat{S}_z^2\,+\,b\left( \hat{S}_x^2-\hat{S}_y^2\right) \,,$$

with $a,\,b$ real parameters. Is the energy invariant under time reversal? How do the energy eigenstates change under a time reversal?

19.3

Consider the three-dimensional isotropic harmonic oscillator. Prove that the angular momentum $\mathbf{{ L}}$ and the tensor operators $\hat{Q}_{ij}\,=\,m\omega \left( x_ix_j-\frac{1}{3}\delta _{ij}r^2\right) \,+\,\frac{1}{m\omega }\left( \hat{p}_i\hat{p}_j-\frac{1}{3}\delta _{ij}\hat{p}^2\right)$ satisfy the relation

$$\hat{L}^2\,+\,\frac{1}{2}Tr\left( \hat{Q}^2\right) \,=\,-3\hbar ^2\,+\,\frac{4\hat{H}^2}{3\omega ^2}\,.$$

19.4

Show that, if the Hamiltonian of a system is time reversal invariant, we can always choose the wave function to be real. How is this compatible with the fact that at some particular time the wave function of a free particle can be a plane wave $e^{i\mathbf{{ k}}\cdot \mathbf{{ r}}}$?

19.5

A system consists of two particles of the same mass interacting through a potential $V(|\mathbf{{ r}}_1-\mathbf{{ r}}_2|)$. Discuss the rotational properties of the system and the associated conserved quantities.

19.6

Let $\alpha ,\,\beta ,\,\gamma$ be the Euler angles. If $U\,=\,e^{-i\alpha X_1}\,e^{-i\beta X_2}\,e^{-i\gamma X_3}$ is to represent a rotation, find the commutation relations that have to be satisfied by the $X_1,\,X_2,\,X_3$. Relate them to angular momentum.

19.7

Consider the Galilean Transformations $\hat{G}\,=\,e^{\frac{i}{\hbar }{} \mathbf{{V}}\cdot \mathbf{{ K}}}$, where their generator is $\mathbf{{ K}}=m\mathbf{{ r}}-t\mathbf{{ p}}$. Show that $\mathbf{{ K}}$ is a vector operator.

19.8

Consider a sequence of Euler rotations represented by

$$D^{(1/2)}(\alpha ,\,\beta ,\,\gamma )\,=\,e^{-i\sigma _3\alpha /2}\,e^{-i\sigma _2\beta /2}\,e^{-i\sigma _3\gamma /2}\,.$$

Show that this is equivalent to a single rotation around an axis by an angle $\theta$. Find $\theta$.

19.9

Consider the simple harmonic oscillator. Show that spatial reflection (parity) can be represented by the operator $-ie^{i\frac{\pi \hat{H}}{\hbar \omega }}$, where $\hat{H}$ is the Hamiltonian.

19.10

For the parity operator $\hat{P}_{\hat{n}}\,=\,\hat{P}\,e^{-\frac{i}{\hbar }\pi \hat{n}\cdot \mathbf{{ J}}}$ prove the following:

$$ \begin{array}{l} \hat{P}_{\hat{n}}{} \mathbf{{r}}\hat{P}_{\hat{n}}\,=\,\mathbf{{ r}}-2\hat{n}\left( \hat{n}\cdot \mathbf{{ r}}\right) \\ \,\\ \hat{P}_{\hat{n}}{} \mathbf{{p}}\hat{P}_{\hat{n}}\,=\,\mathbf{{ p}}-2\hat{n}\left( \hat{n}\cdot \mathbf{{ p}}\right) \\ \,\\ \hat{P}_{\hat{n}}{} \mathbf{{J}}\hat{P}_{\hat{n}}\,=\,-\mathbf{{ J}}+\,2\hat{n}\left( \hat{n}\cdot \mathbf{{ J}}\right) \end{array} $$