The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [47]
Specifically for the square box, therefore, the electron waves are precisely the same shape as the waves on a guitar string; they are sine waves with a very particular set of allowed wavelengths. Now we can go ahead and invoke the de Broglie equation from the last chapter to relate the wavelength of these sine waves to the momentum of the electron via p = h/λ. In which case, the standing waves describe an electron that is only allowed to have certain momenta, given by the formula p = nh/(2L), where all we did here was to insert the allowed wavelengths into the de Broglie equation.
And so it is that we have demonstrated that the momentum of our electron is quantized in a square well. This is a big deal. However, we do need to take care. The potential in Figure 6.3 is a special case, and for other potentials the standing waves are not generally sine waves. Figure 6.5 shows a photograph of the standing waves on a drum. The drum skin is sprinkled with sand, which collects at the nodes of the standing wave. Because the boundary enclosing the vibrating drum skin is circular, rather than square, the standing waves are no longer sine waves.2 This means that, as soon as we move to the more realistic case of an electron trapped by a proton, its standing waves will likewise not be sine waves. In turn this means that the link between wavelength and momentum is lost. How, then, are we to interpret these standing waves? What is it that is generally quantized for trapped particles, if it isn’t their momentum?
We can get the answer by noticing that in the square well potential, if the electron’s momentum is quantized, then so too is its energy. That is a simple observation and appears to contain no important new information, since energy and momentum are simply related to each other. Specifically, the energy E = p2/2m, where p is the momentum of the trapped electron and m is its mass.3 This is not such a pointless observation as it might appear, because, for potentials that are not as simple as the square well, each standing wave always corresponds to a particle of definite energy.
Figure 6.5. A vibrating drum covered in sand. The sand collects at the nodes of the standing waves.
The important difference between energy and momentum emerges because E = p2/2m is only true when the potential is flat in the region where the particle can exist, allowing the particle to move freely, like a marble on a table top or, more to the point, an electron in a square well. More generally, the particle’s energy will not be equal to E = p2/2m; rather it will be the sum of the energy due to its motion and its potential energy. This breaks the simple link between the particle’s energy and its momentum.
We can illustrate this point by thinking again about the ball in a valley, shown in Figure 6.4. If we start with the ball resting happily on the valley floor, then nothing happens.4 To make it roll up the side of the valley, we’d have to give it a kick, which is equivalent to saying that we need to add some energy to it. The instant after we kick the ball, all of its energy will be in the form of kinetic energy. As it climbs the side of the valley, the ball will slow down until, at some height above the valley floor, it will come to a halt before rolling back down again and up the other side. At the moment it stops, high up the valley side, it has no kinetic energy, but the energy hasn’t just magically vanished. Instead, all of the kinetic energy has been changed into potential energy, equal to mgh, where g is the acceleration due to gravity at the Earth’s surface and h is the height of the ball above the valley floor. As the ball starts to roll back down into the valley, this stored potential energy is gradually converted back into kinetic energy as the ball speeds up again. So as the ball rolls