Can Planck’s constant be derived from Maxwell’s equations?

Planck’s constant cannot of course be derived from Maxwell’s equations.

However, looking at the question from a slightly different standpoint: if one decides that light is quantized, then there is a thought experiment in classical optics that motivates the form of the Planck law, i.e. that the light energy quantum has to be proportional to its frequency. Again, the value of the proportionality constant cannot be derived from Maxwell’s equations, but I think the following is interesting insofar that it is Maxwell’s equations together with special relativity that show the quantisation law has to have a certain form.

Our thought experiment is about light in a perfect optical resonator comprising two perfectly parallel mirrors with plane waves bouncing between them, as I have drawn below.

Squashing PhotonFigure 1: “Squashing” a Photon in a Perfect Resonant Cavity

We now “squash the light” by bringing the mirrors together: we accelerate the right hand one instantly to $v$ metres per second moving towards the other, which is kept still. Some time later, we stop the crush, again decelerating from $v$ metres per second to rest instantaneously.

Physical Overview

If one works through the calculation one finds, of course, that the work done pushing the mirrors shows up as energy in the cavity field. But, at the same time, the pulse bouncing in cavity keeps its original functional form – but the argument of the functional form $k \,z – \omega \, t$ gets scaled up so that the constant pulse shape is shrunken to perfectly fit into the shrinking cavity. This is Doppler blueshifting in another guise – the Fourier (the covariant wavenumber space) representation is simply being uniformly dilated and the scale factor is the same scale factor applying to the energy of the field. Alternatively, we could imagine draining energy from the light by letting the cavity expand “adiabatically” and do work against the outside force. Then, of course, we’d get Doppler redshifting; again the Doppler scale factor is the same scale factor applying to the field’s dwindling energy. This is the central point:

The Doppler shifting factor is the same as the energy scaling factor

Now, supposing we think of this field’s classical energy as arising from any number of “photons” (say $N$) all in exactly the same state at the beginning of the experiment. Presumably if we squash slowly enough so that adiabaticity holds (see the Wiki Page on the “Adiabaticity Theorem”), one might reasonably construe the field as still being in an $N$-photon number state afterwards. Whence: if we truly can assume the same number of photons, each in the same state which varies throughout the experiment, at the beginning and end, then:

Each photon’s energy must be proportional to its frequency

And it all seems to come wholly from the form of the Lorentz transformation and Maxwell’s equations.

It’s worth noting, when appealing to the Born-Fock Adiabaticity Theorem, that this result is independent of the mirror speed $v$. We can wind the mirrors together as slowly as we like, so there is at least a plausibility to this idea. Of course, there is some circular reasoning here – one has to define quantum states properly to meaningfully talk about adiabaticity and, before that, one has to assume the Planck result – or some other postulate, to build a second quantised theory to make the idea of an $N$-photon number state rigorous; even once one has done that, I must admit I can’t even see how to go about writing a second quantised description of a cavity with a moveable mirror, maybe that’s a new question. But, if one imagines going back in time to Planck’s day, one might imagine a thought experiment like this might have been taken as motivating $E = h \nu$. The idea of the electomagnetic field’s second quantisation didn’t begin to take shape until Dirac thought of it twenty-six years after Planck proposed his law in 1900. So, before Dirac’s ideas, physics had to think in terms like the above thought experiement that seem from our hindsight-enlightened viewpoints to be begging the quesiton. Maybe indeed some early twentieth century worker came up with this thought experiment.

Some Details

Here are some further details in my thought experiment. The calculations are straightforward, but messy and complicated.

Firstly we consider a one-dimensional electromagnetic wave scattering from a perfect reflector in the plane $z = 0$. To the left of the reflector, Maxwell’s equations can be fulfilled by one-dimensional plane waves with the form:

$$\begin{array}{lcl}
\mathbf{E}\left(z, t\right) &=& \left[\,f_0\left(z – c\,t\right) – f_0\left(- \left(z+ c\,t\right)\right)\,\right] \; \mathcal{U}\left(-z\right) \;\hat{\mathbf{x}}\\
\mathbf{B}\left(z, t\right)  &=& \frac{1}{c}\left[\,f_0\left(z – c\,t\right) + f_0\left(- \left(z+ c\,t\right)\right)\,\right] \; \mathcal{U}\left(-z\right) \; \hat{\mathbf{y}}\\
\mathbf{J}_s\left(0, t\right)  &=& 2 \;\sqrt{\frac{\epsilon_0}{\mu_0}} \,f_0\left( – c\,t\right) \hat{\mathbf{x}}\\
\mathbf{F}_s\left(0, t\right)  &=& 2 \,\epsilon_0 \,\left(_0f_1\left( – c\,t\right)\right)^2 \hat{\mathbf{z}}
\end{array}\quad\quad\quad\quad(1)$$

where $\mathbf{E}$ and $\mathbf{B}$ are respectively the electric field and magnetic induction, $f$ any arbitrary pulse shape, $c$ the freespace lightspeed, $\mathbf{J}_s$ surface current (in amp\`eres per metre) in the perfect reflector, $\mathbf{F}_s$ force per unit area on the conductor and $\mathcal{U}$ the Heaviside step function. The force is most straightforwardly calculated by the method of virtual work; to understand the calculation from the Lorentz force formula, one must calculate the scattering from a metal with finite conductivity $\sigma$ as in Method 3 of my answer here, integrate the body force density $\mathbf{J} \wedge \mathbf{B}$ and then take the limit as $\sigma \rightarrow \infty$, the skin depth $\delta \rightarrow 0$ and the body current density thus becomes a surface current. This result differs by a factor of two from the “blithe” result  $\mathbf{J}_s \wedge \mathbf{B}$ gotten by applying the Lorentz force formula without heed to the limiting process that defines a perfect conduction and current sheet. Tacitly, an assumption has been made that the plane’s conductivity $\sigma$ fulfills $\sigma \gg \omega_{max} \epsilon$  where $\omega_{max}$ is the highest frequency of a “significant” Fourier component of $f_0()$.

Now we want to know what happens when the perfect reflector is shifted leftwards so that its velocity is $-v \, \hat{\mathbf{z}}$. The outcome can of course be found by calculating the fields seen by an observer moving uniformly at velocity $v\,\hat{\mathbf{z}}$. Upon making the relavent Lorentz transformation on Eq.(1), one finds:

$$\begin{array}{lcl}
\mathbf{E}\left(z, t\right) &=& \left[\sqrt{\frac{c-v}{c+v}}\,f_0\left(\sqrt{\frac{c-v}{c+v}}\left(z – c\,t\right)\right) – \sqrt{\frac{c+v}{c-v}}\,f_0\left(- \sqrt{\frac{c+v}{c-v}} \left(z+ c\,t\right)\right)\,\right] \; \mathcal{U}\left(-\left(z+v\,t\right)\right) \; \hat{\mathbf{x}}\\
\mathbf{B}\left(z, t\right)  &=& \frac{1}{c} \left[\sqrt{\frac{c-v}{c+v}}\,f_0\left(\sqrt{\frac{c-v}{c+v}}\left(z – c\,t\right)\right) + \sqrt{\frac{c+v}{c-v}}\,f_0\left(-  \sqrt{\frac{c+v}{c-v}} \left(z+ c\,t\right)\right)\,\right] \; \mathcal{U}\left(-\left(z+v\,t\right)\right) \; \hat{\mathbf{y}}\\
\end{array}
\quad\quad\quad\quad(2)$$

These equations are more meaningful if we rewrite them so that $f_1\left(u\right) = \sqrt{\frac{c-v}{c + v}} \; f_0\left(\sqrt{\frac{c-v}{c + v}} \; u\right)$, i.e. we rescale amplitudes and arguments so that:

$$\begin{array}{lcl}
\mathbf{E}\left(z, t\right) &=& \left[\,f_1\left(z – c\,t\right) – \frac{c+v}{c-v}\,f_1\left(-\frac{c+v}{c-v} \left(z+ c\,t\right)\right)\,\right] ] \; \mathcal{U}\left(-\left(z+v\,t\right)\right) \; \hat{\mathbf{x}}\\
\mathbf{B}\left(z, t\right)  &=& \frac{1}{c} \left[\,f_1\left(z – c\,t\right) + \frac{c+v}{c-v}\,f_1\left(- \frac{c+v}{c-v} \left(z+ c\,t\right)\right)\,\right] ] \; \mathcal{U}\left(-\left(z+v\,t\right)\right) \; \hat{\mathbf{y}}\\
\end{array}
\quad\quad\quad\quad(3)$$

and the reflected waves $\frac{c+v}{c-v}\,f_1\left(-\frac{c+v}{c-v} \left(z+ c\,t\right)\right)$ are given in terms of the incident waves $f_1\left(z- c\,t\right)$. This form of the equations underlies the wonted causal relationships in such a system: the rightwards running wave $f_1\left(z- c\,t\right)$ at any point in the region $z < 0$ will meet the reflector in the future, so that this wave must be uninfluenced by the reflector until that time of meeting. Its shape and scaling must therefore simply be a delayed version of what left its source somewhere far out in the region $z < 0$. The scattered wave $\frac{c+v}{c-v}\,f_1\left(-\frac{c+v}{c-v} \left(z+ c\,t\right)\right)$ has already met the reflector and has been Doppler shifted by it (witness that the argument has been multiplied by the squared Doppler factor $\frac{c+v}{c-v}$, so that wavelengths are shrunken by the factor $\frac{c-v}{c+v}$) and its intensity boosted by the factor $\left(\frac{c+v}{c-v}\right)^2$. Positive work must be done on the reflector to push it leftwards at constant speed against the photonic pressure.

Take heed that the wonted electromagnetic field boundary conditions do not hold for moving boundaries. The discontinuity in the tangential electric field components can be understood as follows: as the reflector and its surface current advances leftwards, it is quelling the field in its wake altogether. Thus, if we imagine a thin loop  whose plane is normal to both the reflector and the magnetic induction and with width $\Delta z$ in the $z$ direction and length $\ell$ along the direction of the magnetic field, the magnetic flux through this loop goes from $\left|\mathbf{B}\right| \ell \Delta z$ in time $\Delta z / v$ as the reflector passes by the loop, hence there must be a difference $\left|\Delta \mathbf{E}\right|$ between the electric fields along the loop’s long sides, i.e. $\left|\Delta \mathbf{E}\right| \ell =  \left|\mathbf{B}\right| \ell v$ as $\Delta z \rightarrow 0$, hence the discontinuity $2 \, v f(0) / (c-v)$ in the electric field. Again, the electrodynamics of this discontinuity are better understood by doing the calculations at a finite conductivity (thus removing the discontinuity) and passing to the infinite conductivity limit.

Now we shift the reflector to an arbitrary $z$-position $a$:

$$\begin{array}{lcl}
\mathbf{E}\left(z, t\right) &=& \left[\,f_1\left(z – c\,t – a\right) – \frac{c+v}{c-v}\,f_1\left(-\frac{c+v}{c-v} \left(z+ c\,t – a\right)\right)\,\right] \; \mathcal{U}\left(a-z-v\,t\right) \; \hat{\mathbf{x}}\\
\mathbf{B}\left(z, t\right)  &=& \frac{1}{c} \left[\,f_1\left(z – c\,t – a\right) + \frac{c+v}{c-v}\,f_1\left(- \frac{c+v}{c-v} \left(z+ c\,t – a\right)\right)\,\right] \; \mathcal{U}\left(a-z-v\,t\right) \; \hat{\mathbf{y}}\\
\end{array}
\quad\quad\quad\quad(4)$$

then transform the functional notation so that $f\left(t – \frac{z}{c}\right) = f_1\left(z – c\,t – a\right)$:

$$\begin{array}{lcl}
\mathbf{E}\left(z, t\right) &=& \left[\,f\left(t – \frac{z}{c}\right) – \frac{c+v}{c-v}\,f\left(\frac{c+v}{c-v} \left(t + \frac{z}{c}\right) – \frac{2 \,a}{c – v}\right)\,\right] \; \mathcal{U}\left(a-z-v\,t\right) \; \hat{\mathbf{x}}\\
\mathbf{B}\left(z, t\right)  &=& \frac{1}{c} \left[\,f\left(t – \frac{z}{c}\right) + \frac{c+v}{c-v}\,f\left(\frac{c+v}{c-v} \left(t+ \frac{z}{c}\right) -\frac{2 \,a}{c – v}\right)\,\right] \; \mathcal{U}\left(a-z-v\,t\right) \; \hat{\mathbf{y}}\\
\end{array}
\quad\quad\quad\quad(5)$$

and imagine a second, still reflector at $z = 0$ so as to consider a one-dimensional cavity resonator as shown in the drawing. The cavity resonator is “shrinking” and the light within it is being “squashed”. Boundary conditions very like those in Eq.(1) hold, thus implying the “loop condition”:

$$f\left(\frac{c-v}{c+v}\, u + \frac{2}{c+v}\, a\right) = \frac{c+v}{c-v}\,f\left(u\right)\quad\quad\quad\quad(6)$$

and the field’s intensity and frequency both grow exponentially together i.e. vary like $\left(\frac{c+v}{c-v}\right)^n$with the cavity circulation number $n$.

Suppose at $t = 0$, the rightwards running cavity wave’s functional form is $g_+(z), \;0\leq z \leq a$ and that there is no leftwards running wave. The wave’s lagging (leftmost) edge meets the right reflector (i.e. that which was at position $z = a$ at time $t = 0$) at time $t = a / (c + v)$. Likewise, the wave’s leading edge is boosted in amplitude by a factor $(c+v)/(c-v)$ and meets the left reflector (at $z = 0$) slightly later at time $t = a / c$. So, at this time, the wave is now wholly backwards (leftwards) running, its whole length still fits into the shortened cavity and it still has the same functional form, but with a “squashed” $z$-dependence; its functional form is now $\frac{c+v}{c-v} g_+\left(a – \frac{c+v}{c-v}\,z\right)$ for $0 \leq z \leq \frac{c-v}{c+v} a$, whilst the cavity’s length is now $\frac{c-v}{c} a$, i.e. longer than the wave’s extent. Now we repeat the reasoning for the wave scattering from the left reflector. This time there is no Doppler shift or amplitude boost, and the time taken for the wave’s leading edge to run from the left to the right reflector is $\frac{c-v}{c+v} \frac{a}{c}$, i.e. exactly the wave’s temporal duration and this duration in turn is exactly the time taken for the wave’s lagging edge to reach $z = 0$. Thus, after a total time $t = 2\frac{a}{c + v}$ the wave has returned to its original shape, albeit that its amplitude has been boosted by a factor $\frac{c+v}{c-v}$, its functional form is now $g_+(\frac{c+v}{c-v}\,z),\; 0\leq z \leq \frac{c-v}{c+v}\, a$, the wave’s length $\frac{c-v}{c+v}\, a$ so that it fits exactly into its new cavity length $a^\prime = \frac{c-v}{c+v}\, a$. We can repeat the analysis for a backwards running wave $g_-(z), \;0\leq z \leq a$ and assume that there is no forwards running wave. The result is naturally the same: after one circulation time $t = 2\frac{a}{c + v}$, the wave has returned to being a wholly backwards running wave, its amplitude has been boosted by the factor $\frac{c-v}{c+v}$ and its argument has been shrunken (blueshifted) so that it fits exactly into the shrunken cavity, which now has a length  $a^\prime = \frac{c-v}{c+v}\, a$. Thus, if the cavity begins with forward and backwards running variations $g_+(z),\; g_-(z)$ respectively for $0\leq z \leq a$, the following parameters define $n^{th}$ cavity round trip:

$$\begin{array}{llcl}
n^{th}\, \mathrm{Round\,Trip\,Time}:& t_n & = & 2 \frac{a}{c + v} \left(\frac{c-v}{c+v}\right)^{n – 1}\\
\mathrm{Time\,Till\,Completion}:& T_n & = & \sum\limits_{j = 1}^n t_n = \frac{a}{v} \left(1 – \left(\frac{c-v}{c+v}\right)^n\right)\\
\mathrm{Blueshift\,(Frequency\,Scale)}:& \nu_n & = & \left(\frac{c+v}{c-v}\right)^n\\
\mathrm{Cavity\,Length}:& L_n & = & \left(\frac{c-v}{c+v}\right)^n = \nu_n^{-1}\\
\mathrm{Amplitude\,Scale}:& a_n & = & \left(\frac{c+v}{c-v}\right)^n = \nu_n\\
\mathrm{Intensity\,Scale}:& i_n & = & \left(\frac{c-v}{c+v}\right)^{2\,n} =  \nu_n^2\\
\mathrm{Total\,Cavity\,Energy}:& E_n & = & \frac{\epsilon_0}{2}\,\int_0^{\frac{a}{L_n}} \nu_n^2 \left(g_+\left(\nu_n\,z\right)^2+g_-\left(\nu_n\,z\right)^2\right) \mathrm{d}z\\
& &= &\nu_n \frac{\epsilon_0}{2}\,\int_0^a \left(g_+\left(z\right)^2+g_-\left(z\right)^2\right) \mathrm{d}z = \nu_n\,E_0\\
\mathrm{Total\,Cavity\,Energy\,Scale}:& e_n & = &  \left(\frac{c+v}{c-v}\right)^n = \nu_n\\
\mathrm{Photonic\,Pressure\,Scale}:& p_n & = &  \left(\frac{c-v}{c+v}\right)^{2\,n} =  \nu_n^2\\
\end{array}
\quad\quad\quad\quad(7)$$

thus the light within the cavity is infinitely blueshifted and power and pressure needs of this process increase without bound as the cavity approaches zero length. Note that analogous results can be gotten for a reflector speed $v\left(t\right)$ that varies with time. In this case, the functional forms $g_+(z)$ and $g_-(z)$ are in general nonuniformly stretched and shrunken to account for the variation of speed within each circulation period. The results in Eq.(7) are replaced by effective average definitions, but the fundamental results that the total cavity energy and mean blueshift are both inversely proportional to the cavity length are the same and independent of the detailed time variation. So, no matter how one gets there, the cavity energy and mean blueshift depend only on the current cavity length.