The Particle in a Box

Otherwise you saw a realization of one of the classic formal problems in quantum mechanics: the particle in a box. Imagine that a very small, light particle like an electron is trapped in the box formed by the thick black borders on the previous page. The principles of quantum mechanics tell us that the total energy of the particle in this box must change in fixed, discrete (quantum!) leaps. The RED horizontal lines drawn across the box show the energies a particle in this box can have. The height of each line above the box floor corresponds to the value of the energy: a higher line means a higher energy.

The primeval quantity in the simplest formulations of quantum mechanics is the wavefunction. This beast tells you everything you can know about the particle. In particular the square of the wavefunction tells you the probability of finding the particle as a function of position. The BLUE lines on the previous page show the wavefunctions that correspond to each allowed energy. Just so that the wavefunctions don't lie on top of one another, and to allow you to easily match a wavefunction with its corresponding energy, each wavefunction is drawn with its x-axis lying along the red line indicating its energy.

As you introduce and modify this new feature of the box the mighty Java engine underlying this deceptively simple picture will automatically and continuously recalculate and redisplay the allowed energies and corresponding wavefunctions of the system. Hopefully that will strike you as it strikes us as way cool.

It's interesting that this system can be solved exactly. The allowed energies and wavefunctions you see are all exact -- no approximations are made! What's also interesting is that you can, by manipulating the pedestal appropriately, explore many of the most basic concepts and phenomena in quantum mechanics, including degeneracy, quantum tunneling, tunneling splitting, and the ordering of wavefunctions by their "nodes". You can also explore smoothly the crossover from the simple particle in a box system you see initially to one of the most famous quantum systems of all: the double well. J. J. Sakurai has commented that four Nobel prizes -- including that for lasers -- were the result of considering the properties of various double well systems.

Experiment!

To return to the starting situation, if the applet gets confused, you can try pressing the Reload button on your browser while holding down the Shift key at the same time. If the applet does truly strange things, though, please contact the author so he can try to fix them.

What's a wavefunction?

Say what? I'm a doctor, Jim, not a quantum mechanic. . .

These waves are called "standing" waves because, well, they just stand around instead of travelling forward like waves at the beach. In other words, they have their crests and troughs always in the same places, and the areas midway between the crests and troughs (the "nodes") where the water level is undisturbed also always in the same places.

Here's the largest possible bathtub wave, which has a crest at each end:
Here's another, which has a crest at each end and in the middle:

The possible energies of standing waves in the bathtub are limited to certain, specific (quantum!) values by the requirement that each wave fit exactly into the bathtub.

Now the energies of the bathtub waves, like the wavefunctions of the particle in the box, increase with the number of wave crests that fit into the tub. If you swish the water very gently and slowly, for example, then the only wave you'll see will be the ``sloshing'' wave that has a crest exactly at each end of the tub and a node in the middle. As you increase your agitation -- whee! -- you'll see waves that have more and more crests in the middle, at least until you slop all the water out onto the floor. . .

The wavefunctions you see on the previous page are sort of like the standing waves in the bathtub. In fact, the allowed energies are calculated pretty much by figuring out which waves "fit" into the box, as noted below. The meaning of the peaks and troughs of the wavefunction is somewhat like the waves in the bath, in the sense that something is going on at the peaks and troughs but not at the nodes. In the case of wavefunctions, the square of the wavefunction at any point is equal to the probability that the particle is at that point: hence the particle is likely to be found at peaks or troughs in the wavefunction, and is never found at the nodes.

Kinetic energy

The rest of the energy of the particle is kinetic energy, which is to say the energy of its actual motion. The faster the particle bounces around the box, the larger its kinetic energy.

How is the kinetic energy reflected in the wavefunction? It turns out the kinetic energy of the particle at any point is just equal to the relative magnitude of the curvature of the wavefunction at that point. (This is true for real waves as well -- the water is moving fastest where the wave is most sharply curved, which is at the very tip of the crest.) You can see that wavefunctions at higher energies must be more sharply curved on average, because that gives a shorter distance between successive wave peaks and hence more wave peaks fitting into the box. This reflects the physical fact that to increase the energy of the particle in the box we must increase its kinetic energy, since the potential energy is fixed by where the floor of the box is. The average kinetic energy of the particle for any of the allowed total average energy is just the difference between the red line and the floor of the box.

Now the curvature of a function is just the change in slope of the function as one moves from left to right through any point. Since the slope of a function is its first derivative, the curvature must be the second derivative. [Calculus rears its ugly head! You can click here to skip the math that follows.]

That is, if we represent the graph of the wavefunction by a function (x), then the curvature of this graph is given at any point x by:

d2 ---- dx2

(x)

And the relative magnitude of the curvature (the magnitude of the curvature divided by the magnitude of the function itself) is:

[ [ [

d2 ---- dx2

(x)

] ] ]

÷

(x)

This expression gives the kinetic energy of the particle if it happens to be at the point x. To get the average kinetic energy of the particle we need to multiply by the probability that the particle actually is here. That probability is given, as we said, by the square of the wavefunction. Hence the average kinetic energy of the particle is given by multiplying the above expression by ², like so:

K(x) =

1 --- 2

(x)

d2 ---- dx2

(x)

Notice one of the new factors of went into canceling the denominator in the previous expression. We put the ½ in here just to make the units come out conveniently. You could define units of energy where this constant would not appear, if you wanted to.

The Schrödinger Equation

V(x) ²(x)

Putting the kinetic and potential energy terms together we have the complete average energy at any point x:

(x)

[ [ [

1 --- 2

d2 ---- dx2

(x) + V(x) (x)

] ] ]

Notice we've factored out a factor of from each term. If we sum (i.e. integrate) this expression over all positions x we will have the total average energy of the particle:

E

=

dx

(x)

[ [ [

1 --- 2

d2 ---- dx2

(x) + V(x) (x)

] ] ]

Now, we have a great principal of mechanics which tells us what the wavefunction will be if we have an expression for the energy. The fancy name for this principle is the variational principle, which can also be considered as no more than the second law of thermodynamics in disguise. A simpler name might be the ``Principle of Laziness.'' The idea is simply that the wavefunction adjusts itself until the particle has the least possible total average energy.

It can be shown that the right-hand side of the expression for the energy above will be a minimum if and only if the expression in brackets is given like so:

1 --- 2

d2 ---- dx2

(x) + V(x) (x) = E (x)

For every x. This is the celebrated Schrödinger equation. It basically gives a prescription for the relationship between the curvature of the wavefunction at any point and the potential energy at that point.

Boundary conditions and quantization

If we use Schrödinger's equation we can generate a wavefunction for any energy whatsoever. Where the quantization of the energy comes in, that is, the forbidding of all but certain values of the energy, is that we have to make sure that the wavefunction reaches a certain value at the boundary of the problem. This is a "boundary condition", and it is to these that the wavefunction owes its quantum nature, just as the walls of the bathtub enforce the quantization of waves there, as indeed the open sea would not.

In the case of the square well, our boundary condition is that the wavefunction be zero as soon as it reaches the right-hand wall. If we require this, it turns out we can only choose certain specific values for the initial slope and curvature in slope of the wavefunction at the left-hand wall. This in turn means we can only use wavefunctions corresponding to certain energies. Quantization of the energy!

Classically forbidden zones

But not quantum mechanically. Quantum mechanics only requires that the total average energy summed up over all possible positions of the particle be equal to the total average energy. It's OK to have a negative kinetic energy at some point as long as it's balanced by a positive contribution somewhere else. Thus the quantum particle is allowed to visit "classically forbidden" regions of the box, i.e. those inside the pedestal when the top of the pedestal exceeds the total energy. You will note, however, that the wavefunction is greatly damped in these regions, and more so as the top of the pedestal is raised further, indicating that the probability of the particle being in a classically forbidden region is greatly reduced.

This gives rise to the famous phenomena of quantum tunneling. Suppose a particle is trapped in a well, as in, say, one of the two wells you create by raising the pedestal high. Quantum mechanics says the particle can get into the pedestal even if that region is classically forbidden, as you can see by doing the experiment on the previous page and looking at the wavefunction.

Now if the classically forbidden region is of a finite width, and there is a classically allowed region on the other side (as there is in this system, for example), then a particle trapped in the first allowed region can be found after a while in the other side, having apparently traversed a region of space in which it was "not allowed". This is "tunneling", so named because one imagines the particle tunnneling through the barrier offered by the pedestal, say, instead of going over the top, which it can't do because it doesn't have enough energy.

Radioactivity

But interestingly the force binding protons and neutrons together falls off very rapidly with distance (unlike gravity, say, or a magnetic field). Hence there is a certain point, not far from a nucleus, where protons and neutrons are once again classically allowed. Thus a tunneling situation exists, with the barrier between the inside and the outside of the nucleus being the attraction of any given proton or neutron for all the rest inside the nucleus.

And indeed, on regular occasions groups of protons and neutrons are observed to tunnel out of the nucleus -- that is, the nucleus undergoes radioactive decay -- at temperatures here on Earth which are far too low to give protons and neutrons enough energy to go "over the wall". (In case you are wondering, the reason the same thing does not happen with the electrons in an atom is that the force binding the electrons to the nucleus falls off too slowly with distance, hence there is no region outside the atom which is classically allowed -- there is no place for the electron to tunnel to! However the electron can tunnel in to the nucleus, and this phenomenon, called K-shell capture, is a mode of radioactive decay for certain atoms.)

Tunneling is thought to be important to the function of certain enzymes in the body, and, according to Hawking, may hold the key to how the universe itself came into being (the idea seems to be that the state of no universe is classically allowed, as is the state of an expanding universe of a finite size, but there is no "classical" way to get between the two situations. The solution to the problem of Where Did the Universe Come From might be simply that it tunneled out of nonexistence.)

Tunneling splitting

Eventually, as the pedestal becomes high (and wide) enough, the even levels merge with (become degenerate with) the next higher odd levels. Since any linear combination of degenerate solutions to the Schrödinger equation is itself a solution, we can redescribe the wavefunctions at this point as independent solutions to the two separate square well problems.

If we now lower or narrow the pedestal from this point, tunneling begins to couple the two problems. Performing degenerate perturbation theory we would predict a slight tunneling splitting between the otherwise degenerate energy levels in each well. This you see, developing first of course for the higher energy levels. As you lower or narrow the pedestal, the splitting becomes more pronounced (and degenerate perturbation theory no longer works). Eventually the spacing between adjacent pairs of levels becomes nearly equal, and equal to that of the larger pure square well problem. At this point, we re-enter the regime of validity of perturbation theory, only now we must perturb away from the problem of the single large well, instead of from two narrow wells.

Gauge transformation of the 1st kind

This verifies the physical intuition that changing the zero of energy (e.g. by building a "false floor" across the bottom of the square well) does not change the physics of the situation at all.

For example, a basic picture of the atom or molecule is that (negatively charged) electrons are trapped in the neighborhood of one or more (positively charged) atomic nuclei because of the attraction the electrons feel for the nuclei. Just like the particle in the box on the previous page, these electrons can have only certain energies, which correspond to the 1s, 2p etc. orbitals of freshman chemistry.