© Matt Strassler [September 1, 2012]
This is article 5 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
Reminder: The Quantum Ball on a Spring
Back in the first article in this series we studied the ball (of mass M) on a spring (of strength K), and recalled that its oscillations have
- a motion formula: z(t) = z0 + A cos [ 2 π ν t ]
- energy: E = 2 π2 ν2 A2 M
- an equation of motion: d2z/dt2 = – K/M (z – z0)
where the equation of motion forces ν = √ K/M / 2π, but allows the amplitude A to be of any positive size. Then in the second article, we saw that what quantum mechanics does to the oscillations (among many more subtle things) is effectively restrict the amplitude — it can’t be just anything. Instead the amplitude is quantized; it has to take one of an infinite number of discrete values.
- A = (1/2 π) √ 2 n h / ν M
where n=0, or 1, or 2, or 3, or 44, or any integer greater than or equal to zero. In particular, A can be as small as (1/2 π) √ 2 h / ν M , but it can’t be anything smaller, except zero. We say that n is the number of quanta of oscillation in the ball’s motion. The energy of the ball is also now quantized:
- E = (n+1/2) h ν
The most important fact, for us, is that the energy required to add one quantum of oscillation to the ball’s motion is h ν; we may say that each quantum carries energy hν.
The Quantum Wave
For waves, it’s basically the same; we have, for a wave of frequency ν and wavelength λ oscillating with amplitude A around an equilibrium position Z0,
- a motion formula: Z(x,t) = Z0 + A cos (2π [ν t – x/λ])
- the energy per wavelength: 2 π2 ν2 A2 Jλ
(where Jλ is a constant that depends only on, say, the rope, if these are waves on a rope), and several possible equations of motion, of which we chose two to study
- Class 0: d2Z/dt2 – cw2 d2Z/dx2 = 0
- Class 1: d2Z/dt2 – cw2 d2Z/dx2 = – (2 π μ)2 (Z-Z0)
Again quantum mechanics restricts the amplitude A, which we might have thought could be of any size we liked, to discrete values. Just as for an oscillation of a spring,
- a (simple) wave of a given frequency and wavelength is made from n quanta
- the allowed values of the amplitude A are proportional to √ n ;
- the allowed values of the energy E are proportional to (n+1/2)
More precisely, just as for a ball on a spring
- the allowed values of the energy are E = (n+1/2) h ν
- each quantum of a wave carries energy h ν
The formula for A is a bit more complicated, because here we have to know how long the wave is, and an exact formula would be messy, so let me just write one that gets the right idea. We got most of our formulas studying waves that are infinite, but any real wave in nature has a finite length. If the wave is roughly of length L, and therefore has L/λ crests, then the amplitude is approximately
- A = (1/2 π) √ 2 n h λ / ν L Jλ
which is proportional to √ n h / ν just as for the spring, but depends on L; a longer wave has a smaller amplitude, arranged just such that each quantum of the wave always has energy h ν.
And that’s it, as illustrated in the figure below (which you may wish to compare with Figures 1 and 3 in the article on the Quantum Ball on a Spring.)
What does this mean for our Class 0 and Class 1 waves?
Since waves that satisfy an equation of Class 0 can have any frequency, they can correspondingly have any energy. Even with a tiny amount of energy ε, you can always make a single quantum of a Class 0 wave with frequency ν = ε/h. For such small energy, that quantum wave will have very low frequency and very long wavelength, but it can exist.
Waves that satisfy an equation of Class 1 are different. Since there is a minimum frequency νmin = μ that such waves can have, there is a quantum of lowest energy
- Emin = h νmin = h μ
If your tiny amount of energy ε is less than this, you cannot make a quantum of this sort of wave. Quanta of Class 1 waves with finite wavelength and larger frequency all have E ≥ h μ .
To Sum Up
Before we account for quantum mechanics, the amplitude of a wave, just like the amplitude of a ball on a spring, can vary continuously; you can make it as large or as small as you want. But quantum mechanics implies there’s a smallest possible non-zero amplitude for a wave, just as for an oscillation of a ball on a spring; and in general only discrete values of the amplitude are allowed. The allowed amplitudes are such that for both an oscillating ball on a spring and a wave of any class with a definite frequency ν
- to add a single quantum of oscillation requires energy h ν
- with n quanta of oscillation, the energy of oscillation is (n+1/2) h ν
Now it’s time to apply this knowledge to fields, and observe when and how the quanta of the waves in these fields could be interpreted as what we call the “particles” of nature.