Grading the Electromagnetic Field

And a brief application of graded vector spaces to Statistical Mechanics.

We study the Quantum Mechanics of the simple harmonic oscillator not because it is a realistic description of phenomena, but rather as a gateway to understanding such systems.

Where can we apply the physics of the simple harmonic oscillator?

We could attempt to model a molecular bond as a harmonic oscillator - and this does indeed work - although for single bonds we’d also need to worry about rotational modes¹.

Nevertheless, the main point is that the energy eigenstates of a physical system bound to a finite region of space² by a conservative force form a discrete spectrum. When the energy of that system is unbounded above, the spectrum is infinite. When the energy of that system is fixed - say when considering the rotations of a three-dimensional harmonic oscillator at fixed amplitude³ - that spectrum is finite.

There is one application of the quantum harmonic oscillator that represents a physical system rather well, as a model of electromagnetic waves.

From Classical to Quantum Optics

High school students are often taught geometric optics, an assortment of rules that describe how light waves move through material. Reflection and refraction are devised by empirical rules and find notable applications in the study and history of Astronomy.

With Maxwell’s insight that light amounts to waves in the electromagnetic field⁴, these empirical rules can be derived from the laws of Classical Electrodynamics.

The Quantum Mechanics of the electromagnetic field in the absence of sources involves the study of electromagnetic waves. As with ordinary matter, if electromagnetic waves were confined to a compact region of spacetime - like a cube of length L - its frequencies would be confined to a discrete spectrum, given by the energy eigenvalues

\(E = \hbar\omega = \frac{\hbar}{2\pi\lambda}.\)

Here the wavelength λ can be written as half integral⁵ multiples of L,

\(\omega = \frac{1}{2\pi nL}.\)

As we take the size of that box to fill all of three-dimensional space, the allowed frequency range - and energy the spectrum of energy eigenvalues - becomes effectively continuous⁶.

It’s worth pointing out here that Max Planck used such a model - with one oscillator per values of n - to derive his famous spectrum of thermal radiation, which helped kick start the study of Quantum Mechanics.

The Electromagnetic Field Operator

The classical solutions to the simple harmonic oscillator and the wave equation are essentially the same. Hence, the algebra of operators is also the same. For example, the classical, source-free electric field may be represented by plane waves, given a vector-valued function⁷

\(\vec{E}_{n}(x,y,z,t) = \vec{u}_{n}\exp\left(-i\vec{k}\cdot \vec{x} + i\omega_{k} t\right) + \vec{u}_{n}^{\star}\exp\left(i\vec{k}\cdot \vec{x} - i\omega_{k} t\right).\)

Here the wave equation⁸ relates the magnitude of the wave vector k the angular frequency

\(c|\vec{k}| = \omega_{k},\)

Where c is the speed of light. Multiply both sides by ℏ and we have the relationship between the energy and momentum ℏk.

It is worth distinguishing this from the standard (nonrelativistic), massive free particle we studied last time⁹,

\(E_{k} = \frac{(\hbar |k|)^{2}}{2m}.\)

so we might say that a slow moving, massive particle has a quadratic dispersion relationship between wave energy and wave momentum, whereas a relativistic particle - like a photon - has a linear one.

Now, the electromagnetic wave vector itself has a magnitude inversely proportional to the wavelength, and the direction points in the direction of the wave¹⁰.

By linearity of the electromagnetic field, an electromagnetic wave can be represented by an appropriate linear combination of such functions. These is a essentially Fourier Series expansion of the electric field.

In the Quantum Mechanics of light, the electric field is promoted to an operator, and these Fourier modes come equipped with ladder operators.

\(\vec{E}_{n}(x,y,z,t) \rightsquigarrow a_{\vec{k}} \exp\left(-i\vec{k}\cdot \vec{x} + i\omega_{k} t\right) + a^{\dagger}_{\vec{k}}\exp\left(i\vec{k}\cdot \vec{x} - i\omega_{k} t\right).\)

The algebra of operators is exactly the same as you’d expect,

\(\left[\,a_{\vec{k}} \,,\, a^{\dagger}_{\vec{k}}\,\right] = 1.\)

The difference here is that each mode is indexed by a three-dimensional wave vector k, suggesting that we have - for each energy level ℏω - three independent oscillators to consider.

More succinctly, we can use the three-dimensional Dirac delta function¹¹ to write the commutator for the full algebra of operators for all such Electromangetic waves.

\(\left[\,a_{\vec{k}} \,,\, a^{\dagger}_{\vec{p}}\,\right] = \delta^{(3)}(\vec{k} - \vec{p}).\)

The Representation of the Electromagnetic Field

As we saw with the case of N harmonic oscillators, operators with different values of the vector k commute with one another. Hence the full Hilbert space of quantum states separates.

For each wave vector k, the relevant algebra of operators is spanned by

\(a_{\vec{k}},\; a^{\dagger}_{\vec{k}},\; 1,\)

and its Hilbert subspace of physical states H is spanned by the set of polynomials in the creation operator,

\(H[\vec{k}] \cong \mathbb{C}[a^{\dagger}_{\vec{k}}].\)

As usual, this Hilbert subspace is graded by the nonnegative integers and can be decomposed into one-dimensional subspaces,

\(H[\vec{k}] = \bigoplus_{n\in\mathbb{N}} \mathbb{C}a^{\dagger n}_{\vec{k}},\quad \mathbb{C}a^{\dagger n}_{\vec{k}} = \mathsf{span}_{\mathbb{C}}\left\{ a_{\vec{k}^{\dagger n}}\right\}.\)

Therefore the full Hilbert space is the product of all these individual subspaces,

\(H = \bigotimes_{\vec{k} } H[\vec{k}].\)

Compared to what we’ve encountered so far, the full H is a pretty big space. Its nature also depends on whether the wave vectors k are discrete or continuous.

To avoid complications, it can be useful to simply assume it is discrete, and that we are working within a three-dimensional, fully reflective¹² cube of very large length L. However, for practical computations, it is much easier to work with a continuous representation, as we now demonstrate.

Wave Packets of Single Photons

H is a vector space, so physical states can be consistently put into linear combination. The physics of photons is considerably different than the physics of particles precisely because they have no mass. We have already seen that they enjoy a different dispersion relations.

Following the logic of last time, let us assemble a Gaussian wave packet of single photon states. For simplicity let us restrict to the one-dimensional case¹³.

\(\Psi(x,t) = N\int_{\mathbb{R}} dk\; \exp\left(ikx - i kc t\right) \exp\left(-\frac{(k-k_{0})^{2}}{2\sigma^{2}}\right).\)

What’s different about this construction is the time-dependence. In the case of the free particle, time entered with the term quadratic in k. Now it enters in the linear term, as

\(c|\vec{k}| = \omega_{k},\)

Putting these together we can evaluate the integral to find

\(\Psi(x,t) = \frac{1}{\zeta \sqrt{2\pi}}\exp\left(-\frac{1}{2\zeta^{2}}(x-ct)^{2}\right),\)

where the Gaussian parameter ζ here depends on both σ and k_0.

In this form, we can immediately read off that the full wave packet moves together at the speed of light, c, without spreading.

One way to understand the spreading of the massive, free particle wave packet is that particles of different energies move at different velocities. Hence the incoherence of the physically realistic wave packet was inevitable. For photons - waves in the electromagnetic field - Special Relativity tells us that light only moves at the speed of light. Energy merely changes the frequency. So wave packets of photons are coherent, at least in a vacuum.¹⁴

The Planck Distribution

The wave packet description we just discussed involved linear combination of states with a single excitation of the electromagnetic field. Hence a physically reasonable notion of a particle of light - a photon - involved a slight spread over the wavelengths of light.

But why restrict to single photon states?

Electromagnetism affords standing waves inside a metal cavity. Indeed, this is precisely what our model of a large cube of length L represents!

You might ask, what does a generic linear combination of states inside such a box look like? In this case, we can combine not only single mode excitations, but all possible excitations.

Of course, a generic state is just a generic vector in H. A physically meaningful, generic linear combination of states requires some extra data.

Let us use a one-dimensional example of our cubical construction and suppose that we have a finite amount of energy to distribute amongst all the modes in the electromagnetic field. That is, we want to distribute some total energy E amongst all possible states.

This is a well-understood problem Statistical Mechanics.

Now, from a Representation Theory point of view, recall that H is doubly graded. First each subspace of fixed k is graded by the natural numbers N, corresponding to one-dimensional subspaces of monomial powers of the creation operator for that fixed k.

Second, H is graded by k itself, which in the one-dimensional case is also graded by N, these are the harmonics of the size of the one-dimensional box.

We represent these by the positive integers n and m, respectively.

\(n\hbar\omega_{k} = \frac{2\pi n\hbar}{\lambda_{k}}= \frac{4nm\pi}{L}, \quad n,m \in \mathbb{N}, \; m> 0.\)

Our aim is to find the homogenous subspace of H with degree M, where

\(E = \frac{4\pi}{L} M = \sum_{m=1}^{M}\sum_{n=0}^{M} \kappa_{mn}\frac{4nm\pi}{L},\quad \kappa_{mn} = 0 \,\mathrm{or}\, 1.\)

Boltzmann taught us how to do this in a more physically realistic way¹⁵. Namely, we can sum over all the states, but assign to them a factor¹⁶,

\(\exp(-m\hbar\omega_{k} / T ).\)

Now T is a parameter for our system - just like k_0 in wave packet discussion above. It is akin to the average energy. Some folks would call it the temperature.

Hence we take as our linear combination the state,

\(|\Psi(T)\rangle = \int dk\; \sum_{n=0}^{\infty}\;e^{-\hbar\omega_{k}/T} a^{\dagger n}_{k}|\varnothing\rangle.\)

I’ll spare the gory details, but this end result of this logic is that the energy is distributed by frequency as

\(\epsilon(\omega) = \frac{8\pi \hbar\omega \left(\omega L/(\pi c)\right)^{3} }{\exp(\hbar\omega/T) - 1},\quad E =\int d\omega \; \epsilon(\omega).\)

It is sometimes more approach to use the normalized probability distribution for finding an excited mode of frequency ω,

\(\rho(\omega) = \frac{1}{\exp(\hbar\omega/T) - 1}.\)

This is the so-called Planck Distribution for thermal radiation at temperature T.

This distribution doesn’t exactly look like Gaussian, but it’s easy enough to verify that ρ(ω) satisfies the axioms of the central limit theorem. Hence, for fixed temperature T, there is a well-defined mean frequency ω.

As with the Gaussian wave packet, this linear combination of vectors in H doesn’t spread out as time evolves. Although it does spread out as the temperature changes.

Coherent versus Thermal

For the massive free particle, we saw that the typical, physically reasonable quantum state - the Gaussian wave packet - spread out in time. This is consistent with the fact that the information about said particle is being lost as time evolves.

For an individual photon, we saw that this was not the case. A wave packet of single particle excitations of the electromagnetic field retains its shape and uncertainty in space.

There are other physically reasonable states of the electromagnetic field besides the photon. Inside a radiation cavity - like a cubical box of length L - almost any state would be physically reasonable, a priori. Indeed, a physical light wave of fixed frequency should merely amount to a very large value of N, where

\(a^{\dagger N}_{\vec{k}} |\varnothing\rangle.\)

In this model anyway, there is no effective sense in which a classical electromagnetic wave and quantum electromagnetic wave qualitatively different.

But a large excitation of a very specific mode of the electromagnetic field - while more reasonable than a single plane wave model of a free, massive particle - isn’t typically the most probable physical state.

Like the former, it is sensible to draw a probability distribution around the wavenumber of that particular mode. Doing this consistently and practically involves error around the energy, which leads us to a thermal distribution of photons in a box.

In the study of Quantum Mechanics, this distinction is common. We have thermal modes and coherent modes. The former have their own probabilities error associated to the Boltzmann factor. The latter are precisely specified.

To relate these two kinds of physical models, it sometimes pays to ask the question, what happens when the temperature goes to zero? We’ll explore that next time.

1. As we’ve seen with the three-dimensional case, these too are quantized. For molecules with a single or σ-bond, these atoms can rotate collectively and independently, resulting a complex assortment of internal energy states. For a double or triple bond - including what chemists refer to as π-bonds - those independent rotations are restricted. In all cases, these energy levels are ascribed to thermal energy, and can be excited by exposing the molecules to infrared light. Conversely, transitions between vibrational and rotational molecule energy levels can be observed by looking for infrared light. As with the specific frequency of the harmonic oscillator, these are discrete energy levels that are unique to the individual molecules involved. The full spectrum of infrared light emitted by specific molecules can therefore be used to identity them.

2. For the one-dimensional case this is clear. In higher dimensions we would say something like a state restricted to a bounded subset of R^3.

3. We saw this most recently in our discussion of the functional representation of the three-dimensional harmonic oscillator:

From One Oscillator to Many

4. Of course, before Maxwell we thought of the electric and magnetic fields as separate albeit related phenomena. More precisely, Maxwell included a term that would induce an effective current from the time-rate-of-change of the electric field. This was akin to Faraday’s discovery that an effective electromotive force could be generated by the time-rate-of-change of the magnetic field. Putting these two facts together, mathematically, resulting in the wave equation, unifying our understanding of the two phenomena together with that of light.

5. Half integral because a full period of oscillation passes through the origin twice.

6. Topologically speaking, of course, this is ridiculous. It is a Physicist’s style of playing fast and loose with the mathematics. From the Physcist’s perspective, the large L limit means we can’t observationally distinguish between different, discrete energy levels. Hence, the models are indistinguishable, especially at scales much smaller than L.

7. We’re using a complex representation of the electromagnetic field here, but you could flip to sines and cosines with real vector coefficients if you prefer.

8. Here the wave equation relates the magnitude of the wave vector k the angular frequency

9. See the details here:

Uncertainty in Quantum States

10. This is true in an isotropic, linear media. In some complicated materials, the wave vector can point in a different direction, although it still serves as a vector parameter for the possible ways light can propagate.

11. If we have restricted ourselves to a compact region of space - like a box - where the values of k are restricted to a discrete spectrum, we can instead write this as the product of three Kronecker deltas instead.

12. By reflective here we mean that the value of the electromagnetic field should vanish on the surface of the box, effectively imposing Dirichlet boundary conditions on the wave equation.

13. We are cheating a little here, defining the normalization factor N to absorb the details of k_0.

14. This is in vacuum. In a dispersive media, such as glass, the index of refraction - and hence speed of light in the medium - varies with frequency. This has a natural interpretation of the photons picking up an effective mass. Hence wave packets will spread in such a medium.

15. Strictly speaking, the double partition approach is called by physicists the Microcanonical ensemble. Boltzmann’s approach here is often called the Canonical Ensemble. The latter is more approachable physically because it allows us to have some statistical error in what we mean by the total energy of the system.

16. We have absorbed the Boltzmann constant kB into the definition of the temperature T.