Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

5.5: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercises

Exercise 5.5.1

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

ˆA(r,t)=d3kλ16π3ϵ0ωk(ˆakλei(krωkt)+h.c.)ekλ.

Since [ˆakλ,ˆakλ]=δ3(kk)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)=2d3kδ(EEk),

where Ek=c|k|. Note the factor of 2 accounting for the polarizations. Prove that

D(E)=8πE23c3,

and show that D(E) has units of [E1V1].

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).


This page titled 5.5: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?