Electromagnetic Radiation in a Cubic Box · USAPhO 2011 #A4
Physics · quantum-mechanics
Problem
In this problem we consider a simplified model of the electromagnetic radiation inside a cubical box of side length L. In this model, the electric field has spatial dependence
E(x,y,z)=E0sin(kxx)sin(kyy)sin(kzz)
where one corner of the box lies at the origin and the box is aligned with the x, y, and z axes. Let h be Planck's constant, kB be Boltzmann's constant, and c be the speed of light.
a. The electric field must be zero everywhere at the sides of the box. What condition does this impose on kx, ky, and kz? (Assume that any of these may be negative, and include cases where one or more of the ki is zero, even though this causes E to be zero.)
b. In the model, each permitted value of the triple (kx,ky,kz) corresponds to a quantum state. These states can be visualized in a state space, which is a notional three-dimensional space with axes corresponding to kx, ky, and kz. How many states occupy a volume s of state space, if s is large enough that the discreteness of the states can be ignored?
c. Each quantum state, in turn, may be occupied by photons with frequency ω=f/2π=c∣k∣, where
∣k∣=kx2+ky2+kz2
In the model, if the temperature inside the box is T, no photon may have energy greater than kBT. What is the shape of the region in state space corresponding to occupied states?
d. As a final approximation, assume that each occupied state contains exactly one photon. What is the total energy of the photons in the box, in terms of h, kB, c, T, and the volume of the box V? Again, assume that the temperature is high enough that there are a very large number of occupied states. (Hint: divide state space into thin regions corresponding to photons of the same energy.)
Note that while many details of this model are extremely inaccurate, the final result is correct except for a numerical factor.