5.5: Exercises
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Exercises
Exercise 5.5.1
In Section 5.3, we derived the vector potential operator, in an infinite volume, to be
ˆA(r,t)=∫d3k∑λ√ℏ16π3ϵ0ωk(ˆakλei(k⋅r−ωkt)+h.c.)ekλ.
Since [ˆakλ,ˆa†k′λ′]=δ3(k−k′)δλλ′, the creation and annihilation operators each have units of [x3/2]. Prove that ˆA has the same units as the classical vector potential.
Exercise 5.5.2
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 5.5.3
The density of photon states at energy E is defined as
D(E)=2∫d3kδ(E−Ek),
where Ek=ℏc|k|. Note the factor of 2 accounting for the polarizations. Prove that
D(E)=8πE2ℏ3c3,
and show that D(E) has units of [E−1V−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).