# Amplitude of an electromagnetic wave containing a single photon

Given a light pulse in vacuum containing a single photon with an energy $E=h\nu$, what is the peak value of the electric / magnetic field?

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– Ben Crowell Jun 3 '13 at 15:41

The electric and magnetic fields of a single photon in a box are in fact very important and interesting. If you fix the size of the box, then yes, you can define the peak magnetic or electric field value. It's a concept that comes up in cavity QED, and was important to Serge Haroche's Nobel Prize this year (along with a number of other researchers). In that experiment, his group measured the electric field of single and a few photons trapped in a cavity. It's a very popular field right now.

However, to have a well defined energy, you need to specify a volume. In a laser, you find an electric field for a flux of photons (n photons per unit time), but if you confine the photon to a box you get an electric field per photon. I'll show you the second calculations because it's more interesting.

Put a single photon in a box of volume $V$. The energy of the photon is $\hbar \omega$ (or $\frac{3}{2} \hbar \omega$, if you count the zero-point energy, but for this rough calculation let's ignore that). Now, equate that to the classical energy of a magnetic and electric field in a box of volume $V$:

$$\hbar \omega = \frac{\epsilon_0}{2} |\vec E|^2 V + \frac{1}{2\mu_0} |\vec B|^2 V = \frac{1}{2} \epsilon_0 E_\textrm{peak}^2 V$$

There is an extra factor of $1/2$ because, typically, we're considering a standing wave. Also, I've set the magnetic and electric contributions to be equal, as should be true for light in vacuum. An interesting and related problem is the effect of a single photon on a single atom contained in the box, where the energy of the atom is $U = -\vec d \cdot \vec E$. If this sounds interesting, look up strong coupling regime, vacuum Rabi splitting, or cavity quantum electrodynamics. Incidentally, the electric field fluctuations of photons (or lack thereof!) in vacuum are responsible for the Lamb shift, a small but measureable shift in energies of the hydrogen atom.

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This is a reasonable question to ask, but the answer is probably not what you're expecting: the electric and magnetic fields don't have well-defined values in a state with a fixed number of photons. The electric and magnetic field operators do not commute with the number operator which counts photons. (They can't, because they are components of the exterior derivative of the field potential operator, which creates/annihilates photons.) The lack of commutativity implies via Heisenberg's uncertainty principle that the field might have arbitrarily large values.

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Classically, any wavepacket won't have a single frequency. You need an infinitely long wave to get a single frequency. @user1504 is this related to your answer? – John Rennie Dec 18 '12 at 7:14
Anna V: the quantum electromagnetic field has operators analogous to the harmonic oscillator. Instead of the Hamiltonian $H = p^2 + x^2$ (I'm dropping the units here) and energy $(\frac{1}{2} + n) \hbar \omega$, we write a Hamiltonian for each frequency $\omega$: $H = \frac{\epsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2$, where $E$ and $B$ are treated as conjugate quantum operators (just like $x$ and $p$). We find creation and annihilation operators such that $E = a^\dagger + a$ and $B = a^\dagger - a$, so the energy must be $(\frac{1}{2} + n) \hbar \omega$. $\langle E \rangle$ is well-defined. – emarti Dec 18 '12 at 8:34
The peak value depends on the wave packet size. Assuming a very large size in vacuum, you get a very low amplitude, but such a photon is absorbed anyway with a resonator. It takes just more time to pump a resonator (an atom, for example). – Vladimir Kalitvianski Dec 18 '12 at 14:19
@user1504 This is not a homework question, I thought about it myself. Unfortunately I have only a little background in quantum mechanics so most of the technical detail of the answers is unclear to me at this moment. I hope later on my studies I'll catch up. Meanwhile, I think I've understood the general idea. When the photon is localized in a box we can calculate the peak values of the electromagnetic field from the box's volume and the photon's energy. When it's not localized, given only the energy, we can't do so. It can take arbitrarily large values, with some probability. Am I correct? – Andrey B Dec 18 '12 at 21:26
@andrey.baj: I'm glad to hear that my guess was wrong. And yes, you're basically correct. If you have a photon which is roughly localized in both position and momentum space, you can estimate the largest expected field values. But you'll get into trouble if you assume the photon's energy is perfectly known. – user1504 Dec 18 '12 at 21:55

If an atom emits energy hf, it emits also an angular momentum (spin). That combination is called "photon" or "wave packet". Linking the appropriate formulas from QM and E&M waves, you get the diameter of the wave packet (about λ/2) but not the length. The radius and the direction of propagation do not change as long as the wave packet is not disturbed. It is not locked in a box but propagates in vacuum.

If the coherence length L is accepted as the length of the cylindrical wave packet, you can calculate the energy density u~f³/L and the electric field strength E~sqrt(f³/L), which is constant inside the cylinder.

I got the following results: a) The Hydrogen line at 1420 MHz has FWHM≈5 kHz, L≈60,000 m, E≈1e-8 V/m

b) The Sodium D-Line has FWHM≈10 MHz, L≈6 m, E≈220 V/m

c) X-ray, λ≈1e-12 m, L≈1000λ, E≈1e16 V/m

If you choose a different shape, perhaps like a cigar, those values differ

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Where have you read that Energy+Angular momentum=Photon? – jinawee Aug 24 '14 at 11:00
users.df.uba.ar/schmiegelow/materias/FT2_2010_1C/extra/… Beth called it light. Some people insist that light consists of photons. Some say, light = electromagnetic waves. – 9herbert9 Aug 24 '14 at 14:45
You may be making a mistake relying on a 1936, experimental paper as your source for the semantics of photons. Firstly the paper predates QED and therefore misses out on a lot of important work on how a photon should be understood and secondly experimental papers tend to give only as much theory as needed for that measurement and only in the interpretation in which the measurement is clearest (I speak as an experimenter). All the qualified theorists I've talked to about this subject suggest caution in trying to impose a simple wave-packet interpretation on a photon. – dmckee Aug 26 '14 at 18:34

In a box of defined, thus finite, volume an infinitely long wave is by definition impossible. Positing an infinitely long wave would also deny the physical reality of the photon having a wavelength, as wavelength is never infinite; measured wavelengths, of visible light for instance, are extremely short, not infinite.

By defining the volume of the box, i.e. by setting a volume arbitrarily, one is in effect setting an arbitrary upper limit on wavelength. But a single photon cannot yield a value for wavelength, since there is no possibility of measuring a peak-to-peak distance between adjacent peaks in the waveform, when there is no second peak to measure to.

Energy is a derivative of amplitude, but only in a statistical sense, as an average of many photons per second, since the uncertainty principle makes measuring a single photon problematic. Its electric and magnetic field values are only a statistical average; individual photons may deviate widely from that average. Equations derived from these group averages are likewise valid only for the group, not for individual photons.

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An infinitely long wave can have a perfectly well-defined wavelength in quantum mechanics (and in fact, only infinitely long waves have well-defined wavelength, both classically and quantum mechanically). – Peter Shor Jan 27 '13 at 15:46