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).