Schrödinger's Cormorant

6.5 The Rules of Boxing

The Particle in a 1-dimensional box: a test system for the Schrödinger equation.

What Nefertiti had forgotten, of course, is our proposal from way back in Chapter 1, that it is not the waviness of a particle per se that leads to quantised energy levels but the boundaries applied to that wave. We suggested that – as with any type of wave – a stable stationary wave can only be created if it meets specific conditions: this is what limits the possible waves and consequently, the possible energies. To investigate this, Erwin suggests that we build a box to keep our particle in. A sketch of our construction, defined in terms of potential energy, is in figure VI-ii:

Along the bottom of the box the potential is zero – it’s just like free space there – so we should expect the wavefunction to resemble the one we just found for a free particle. However when you get to the perpendicular walls, they are infinitely high. That means that getting into the region outside the “allowed” region in the middle would require the particle to have infinite potential energy which is clearly impossible. So the message is clear: the particle is trapped. In general, additional constraints on the wavefunction like these infinite walls are called boundary conditions. Let’s see how these ones affects our wavefunction.

In order to do this, we need to introduce an additional principle of quantum mechanics that Erwin proposed. “Smooth curves” muses Erwin. Nefertiti's patience has now worn wafer-thin but Erwin hastily clarifies his meaning: “A wavefunction cannot have discontinuities; it is not physically reasonable to suppose a step change in probability density at some specific point in space.” Let’s explore the implications of this for our free particle wavefunction, ψ(x) = a sin(kx) + b cos(kx), now we've introduced boundary conditions.

Question 6-5

The result is not just one, but a series of possible sinusoidal wavefunctions, each of which has an integral number of half-wavelengths between the limiting walls:

Visually, we can see that how we have come around to something very close to the standing waves that Nefertiti set up long ago, on her guitar string:

Remember that these are just the spatial parts of the overall wavefunction, ψ(x). The full wavefunction, Ψ(x,t) is that of a standing wave: the wave does not translate but oscillates in a time-dependent fashion. You can see that for the n=1 and 2 wavefunctions below:

n=1

n=2

The physical constraints imposed by anchoring the string at its ends are, in effect, boundary conditions that have the same effect as Erwin’s continuity requirement has on limiting the possible particle wavefunctions. We can say that the particle can exist in different states (which we could, if we were so inclined, call eigenstates of the Hamiltonian), each defined by a different wavefunction (an eigenfunction of the Hamiltonian) and with a defined energy (an eigenvalue ….).

Our next job is to find out how the energy of a particle depends on which eigenstate it is in.

Question 6-6

This question shows you that each allowed wavefunction specifically defines the energy of a particle in a box:

This exercise has led us to not just one, but several key features of this quantum mechanical system:

Now we have put in boundary conditions (in this case, our walls), quantisation has appeared. Only a very specific set of wavefunctions are both solutions to the Schrodinger equation and consistent with these boundary conditions. Hence there are defined states in which the particle can exist.
Each of these wavefunctions is defined by an integer (which we have called n in this instance). This is called a quantum number.
As the quantum number increases, so does the energy of a particle with the associated wavefunction. Thus there are quantised energy levels for a particle, which is what we started out - way back in Chapter 1 - looking to explain.
These energies are kinetic energies (since the potential energy is zero within the box). As the value of the quantum number n increases, the kinetic energy increases. This reflects an increase in the curvature of the wavefunction, which in turn correlates with an increase in the number of nodes.
The lowest energy state (n=1) does not have zero energy. The minimum energy that a particle must have is called its zero point energy. We already came across this idea in Chapter 5, as a consequence of the uncertainty principle.

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