© Matt Strassler [August 28, 2012]
This is article 2 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
In the previous article (Ball on a Spring [Classical]), which you should read first, the key results were that oscillatory motion of a ball on a spring, in the pre-quantum physics of Newton and his friends, takes the form
- z(t) = z0 + A cos [ 2 π ν t ]
- z is the position of the ball as a function of the time t
- z0 is the equilibrium position of the ball (i.e. where it would sit if it were not oscillating)
- A is the amplitude of oscillation (which we are free to choose to be as large or small as we want)
- ν is the frequency of oscillation (which depends only on the strength of the spring K and the mass of the ball M, and does not depend on A)
Also, the total energy stored in the oscillation is
- E = 2 π2 ν2 A2 M
By adjusting A as we wish, we can store any amount of energy in the oscillation.
In quantum mechanics, things change. At first glance (and that’s the only glance we need, really, but I’ll say a bit more in a moment) there’s really only one thing that changes, and that is the statement that “we are free to choose [the amplitude] to be as large or small as we want.” It turns out this isn’t true. And correspondingly, the energy stored in the oscillation cannot be chosen arbitrarily.
Quantization of the Amplitude of Oscillation
Max Planck, the famous turn-of-the-20th-century physicist, was the one who discovered that there was something quantum about the universe, and he introduced a new constant of nature, called Planck’s constant, h. Every time you see something in quantum mechanics, you will see h appear. (Quantitatively, h = 6.626068 × 10-34 m2 kg / s — tiny in the units of ordinary human life.) And here we go:
The quantum ball on a spring can only oscillate with amplitude
- A = (1/2 π) √ 2 n h / ν M
where n is an integer, i.e. 0, or 1, or 2, or 1798, or 2,348,979, etc. Oscillation is not arbitrary, it is quantized: and we may call n the number of quanta of oscillation. (Singular: quantum; plural: quanta.) And a definition: we will say that a ball oscillating with n quanta is in the nth excited state; if it has zero quanta we say it is in its ground state.
To give you a sense for what this means, the first five excited states, and the ground state, are (very naively — don’t take the picture very seriously!) illustrated in Figure 1. Note the smallest possible oscillation is the n=1 state. That’s one quantum of oscillation; you can’t have a fraction of a quantum. The ball cannot oscillate less than this, except to be in its non-oscillating ground state, n=0.
Otherwise things are (at first glance) pretty much the same. To be sure, the story of quantum mechanics is quite a bit murkier than just this! But we can get the physics basically right while skirting the murkiness of quantum mechanics for now.
Now why can’t we easily tell that oscillation is quantized? Because in systems of daily life, the quantization is far too small. For a realistic ball and spring — let’s say the ball has a mass of 50 grams (about 1/9 of a pound) and the frequency of the oscillator is once per second. Then an oscillation with just one quantum (n=1) corresponds to the amplitude
- A = (1/2 π) √ 2 h / ν M = 1.8 x 10-16 m
That’s a couple of ten thousandths of a millionth of a millionth of a meter, or about 10 times smaller than a proton! A single quantum of oscillation wouldn’t even make the ball move by a distance of an atomic nucleus! No wonder we can’t observe this quantization!! If the ball moves an amount that we can see, it has an enormous number of quanta of oscillation — and for such large values of n, we can make A be anything we want, as far as we can tell; see Figure 2. We can’t measure A nearly well enough to notice such fine restrictions on its precise value.
Comment: Notice this is partly due to the huge mass of the ball. If the ball were made from 100 iron atoms, and about a thousandth of a millionth of a meter in radius, its minimum amplitude of oscillation would be about a millionth of a meter, i.e. a thousand times larger than the ball’s radius. That’s big enough that you could observe it with a microscope. But such a small ball, subject to typical atomic-scale forces, would also typically oscillate much more often than once a second, and higher frequency would mean lower amplitude for a single quantum of oscillation; so even with a smaller ball, it’s still not so easy to detect the quantization of nature.
Quantization of the Energy of Oscillation
Now let’s take the quantization of the amplitude, and put it into the formula for the energy of oscillation that we had at the very start of this article, E = 2 π2 ν2 A2 M. Plugging in the allowed values for A from our red formula above, we find an amazing result
- E = n h ν (naive!)
a remarkably simple answer! The energy stored in an quantum ball-on-a-spring is (naively) proportional to n, the number of quanta of oscillation, times Planck’s famous constant h, times the frequency ν of the ball-on-the-spring. Even more amazing, this simple formula is almost correct, too! What’s correct about it?
- the energy required to increase the number of quanta in the oscillator by one (n → n+1) is h ν.
- in any oscillator we’d encounter in ordinary life, a single quantum of energy is so small that we’d never know energy was quantized.
[Let’s check the last statement. For a ball and spring that oscillates once per second, a single quantum of energy is 6.6 × 10-34 Joules, i.e. 0.000,000,000,000,000,000,000,000,000,000,000,66 Joules. And a Joule is about the energy you would exert in raising an apple from the ground up to waist height… not that much to start with! So this is an astonishingly tiny amount of energy. Only in small molecules, and even smaller systems, can the frequency of oscillation be so large as to allow the quantization of energy to be detectable.]
It turns out that our energy formula isn’t entirely right. With a real quantum mechanics calculation, one finds that the correct formula for the energy is
- E = (n + 1/2) h ν (correct!)
We won’t often need to pay attention to that little shift of n by 1/2. However, this shift is very, very interesting — it’s where all the murkiness of quantum mechanics starts. Isn’t it curious? Even when there are no quanta of oscillation in the oscillator at all — when n = 0 — there’s still some amount of energy in the oscillator. This is called zero-point energy, and it is due to a basic jitter, a basic unpredictability, that is at the heart of quantum mechanics. (See Figure 3 for a [inevitably schematic and inaccurate] picture that attempts to illustrate how the jitter is responsible for the zero-point energy; note the ball moves around randomly, even in the ground state.) And we’ll get back to the zero-point energy at some point later… because it will lead us to some of the deepest problems in all of physics.
Quantum Mechanics: Into the Murk [under construction, and not our first priority]