5.5: Exercises

Exercises

Exercise $\PageIndex{1}$

In Section 5.3, we derived the vector potential operator, in an infinite volume, to be

$$\hat{\mathbf{A}}(\mathbf{r},t) = \int d^3k \sum_{\lambda} \sqrt{\frac{\hbar}{16\pi^3\epsilon_0\omega_{\mathbf{k}}}}\, \Big(\hat{a}_{\mathbf{k}\lambda} \, e^{i(\mathbf{k}\cdot\mathbf{r} - \omega_{\mathbf{k}} t)} + \mathrm{h.c.}\Big)\, \mathbf{e}_{\mathbf{k}\lambda}.$$

Since $[\hat{a}_{\mathbf{k}\lambda}, \hat{a}^\dagger_{\mathbf{k}'\lambda'}] = \delta^3(\mathbf{k}-\mathbf{k}') \delta_{\lambda\lambda'}$, the creation and annihilation operators each have units of $[x^{3/2}]$. Prove that $\hat{\mathbf{A}}$ has the same units as the classical vector potential.

Exercise $\PageIndex{2}$

Repeat the spontaneous decay rate calculation from Section 5.4 using the finite-volume versions of the creation/annihilation operators and the vector potential operator (5.4.3). Show that it yields the same result (5.4.16).

Exercise $\PageIndex{3}$

The density of photon states at energy $E$ is defined as

$$\mathcal{D}(E) = 2\int d^3k\; \delta(E-E_{\mathbf{k}}),$$

where $E_{\mathbf{k}} = \hbar c |\mathbf{k}|$. Note the factor of 2 accounting for the polarizations. Prove that

$$\mathcal{D}(E) = \frac{8\pi E^2}{\hbar^3c^3},$$

and show that $\mathcal{D}(E)$ has units of $[E^{-1}V^{-1}]$.

Further Reading

[1] F. J. Dyson, 1951 Lectures on Advanced Quantum Mechanics Second Edition, arxiv:quant-ph/0608140.

[2] A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, 2010). [cite:zee]

[3] L. L. Foldy and S. A. Wouthuysen, On the Dirac Theory of Spin $1/2$ Particles and Its Non-Relativistic Limit, Physical Review 78, 29 (1950).