The Vacuum Field and Einstein's Stimulated Emission Theory

Chris Spencer
Boston University
EC 570: Lasers and Applications


Hedrick Casimir first introduced the idea of the vacuum field in 1948. Interest started when Casimir and Polder earlier in 1948 described the interaction between q perfectly conducting plate and an atom by using the Van Der Waals-London forces, then correcting it for retardation effects in the limit of large distances. Then the next issue was to show the effects between the interaction between two perfectly conducting plates using this Van Der Waals retarded interaction. It is useful to understand the properties of the paper that lead to the Casimir effect. Casimir considers a cubic cavity of a volume \(L^3\) bounded by perfectly conducting square plate with side L be parallel to the XY face and investigates the situation where plate is at a small distance from XY face and situation where the distance between is a distance \(\frac{L}{2}\) [1]. Looking at the resonant frequencies of this system, \( \frac{1}{2} \sum \hbar \omega \), the sum being over all the resonant frequencies, notice the problem that this goes to infinity and is not physical. The differences between the frequencies is well defined for the two situations. Labeling the situations of small interaction and long interaction, differences are \[\frac{1}{2}(\sum \hbar \omega )_I -\frac{1}{2}(\sum \hbar \omega)_{II}\] and the value given is the interaction between the plate and the XY face. Cavity will have dimensions \[0 \leq X \leq L, 0 \leq Y \leq L,0 \leq Z \leq a\] with the wavenumbers given by \[k_i=\frac{ \pi n_i}{L}, k_z=\frac{ \pi n_z}{a}\] where i=x,y and \(a\) is the small distance to be used between plate and XY face. The total K is given by \(K=\sqrt{k_x^2+k_y^2+k_z^2}=\sqrt{x^2+k_z^2}\).
For every \(k_x,k_y,k_z\), there is two be two standing waves unless \(n_i\) is zero, which case there is one standing wave. \(k_x,k_y\) can be regarded as continuous variables for very large L, then in polar coordinates \[\frac{1}{2} \sum \hbar \omega = \frac{ \hbar c L^2 \pi}{\pi^2 2} \sum_{(0)1}^{\infty}\int_{0}^{\infty}\sqrt{\frac{n^2\pi^2}{a^2}+x^2} x dx\] where the notation of \((0)1\) means the n=0 term is to be multiplied by \(\frac{1}{2}\). When \(a\) becomes large, the summation can be turned into an integral. Now the interaction energy between plates for the situations above is \[\frac{1}{2} \sum \hbar \omega = \frac{ \hbar c L^2 \pi}{\pi^2 2} \sum_{(0)1}^{\infty}\int_{0}^{\infty}\sqrt{\frac{n^2\pi^2}{a^2}+x^2} x dx-\int_{0}^{\infty} \int_{0}^{\infty} \sqrt{k_z^2+x^2} x dx(\frac{a dk_z}{\pi})\] For sake of brevity, the result is given for a force per cm^2 \[F=\frac{\hbar c \pi^2}{240 a^4}\] Casimir concluded that there exists a force between the two plates which is independent of material and interpreted as a zero point pressure of electromagnetic waves, this result is now known as the Casimir effect.
This result is hard to understand without a knowledge of quantum field theory or quantum electrodynamics. A brief anecdote on QFT will be given to try to understand the Casimir effect and the vacuum field. Quantum Field Theory states that all the fundamental fields are quantized at each and every point in space. A field can be seen as space being filled with interconnected vibrating balls and springs. Vibrations in a field then propagate and are governed by the appropriate wave equation for the field in question. The second quantization of quantum field theory or canonical quantization requires that each ball and spring combination be quantized. The field at each point in space is a simple harmonic oscillator and its quantization puts a quantum harmonic oscillator at each point. The vacuum we assume to be empty is not so empty after all. As it would be, the vacuum has all the properties a particle may have such as spin, energy but on average these cancel out. If these properties cancel shouldn’t the vacuum be empty? The exception is the vacuum energy. Since the vacuum field’s quantization is a quantum oscillator, we know that the ground state of a quantum harmonic oscillator is \(E_0=\frac{1}{2} \hbar \omega\). For the vacuum field, this is called the zero point energy. This is the same energy Casimir uses when calculating the force between the two plates. [2]

A visualization for the Casimir Effect between two plates