The Particle in a Box :

One of the simplest quantum mechanical problems is the so-called particle in a box. We consider a quantum mechanical particle, described by a wavefunction , in one dimension within a square well potential:

Since the potential beyond the walls of the well is infinite the particle will be confined to the box, and we only need to consider the solution of the Schrödinger equation within that space interval.

The solutions of this linear second order differential equation are:

where A and B are again integration constants defined by the boundary conditions. In our case the boundary conditions are:

The first boundary condition, eq. (4), requires that:

and therefore:

The second boundary condition, eq. (5), limits the possible parameters k because now:

Since A can not be zero (trivial because then the wavefunction wouldn't exist at all) we obtain the requirement for k:

It is customary to work with normalized wavefunctions, i.e. a wavefunction for which the integral of its absolute square over all of space is equal to 1 (which signifies that the probability to find the particle somewhere in the integration region is equal to 1). This requires that:

Evaluation of that integral yields the amplitude A as:

The solution to our problem is now complete. It consists of sin functions that depend on a free parameter n, the so-called quantum number, which is a natural number starting with 1 and running all the way up to infinity. We index our wavefunction accordingly and write it as:

Insertion of this solution in the original Schrödinger equation (eq. (2)) yields the energy eigenvalues which now also depend on the value of the quantum number n:

This shows that the quantum number determines the energy of the eigenstate the particle finds itself in. The higher the quantum number the higher the energy will be, and the higher the number of nodes (points where the wavefunction crosses through zero) will be. Find a graphic representation of the wavefunctions for n from 1 to 6 here. You will also see that the number of nodes in the wavefunction is n-1. (Don't count the zero values at the borders of the box since they are artificially pulled to zero since the potential jumps to infinity which it never does in a natural system).

If one is interested in finding out the probability density of the particle in the box, one has to plot

which is also visible in the graphical representation in both functional form and as density plots.

Important Principles:

1. The energy comes out quantized. This is a natural outcome of the boundary conditions. The spacing between neighboring energy levels varies with (2n+1) because of the step-like behavior of the potential. In cases of other functional forms of the potential the energy levels will depend differently on n.
2. The form of the wavefunction remains the same although it acquires more and more nodes as n is stepped up the ladder. This is equivalent with a shorter and shorter wavelength of the underlying wavefunction which in turn indicates increasing kinetic energy of the corresponding particle (see de Broglie's relation).
3. The highest probability to find the particle in the box is where the antinodes of the sin function appear. It is zero at those locations where the nodes appear. Now one may ask how the particle gets back and forth in the box if we have nodes in its path where it is not allowed to exist (not even for a fraction of time). However, we cannot describe the particle anymore in classical terms where it would be expected to zig-zag back and forth. Therefore the question is not meaningful anymore. The particle, if we were to measure its space coordinates over and over again would just be more likely to be found around the antinodes of the wavefunction.

4. Zero-point energy: The energy the system would approach if it were cooled down to T=0K is:

   

   This means that the particle is still able to move with a finite linear momentum and finite kinetic energy through the box which will never go away to zero! This is contrary to classical theory that predicts a total standstill of motion at T=0K.

5. The uncertainty principle is closely connected to the zero point energy. If we look at the momentum for the state n=1 we find two possibilities, i.e. a positive or a negative p, indicating the particle to move in +x or in -x direction:

   

   If we perform a large number of momentum measurements on the particle we will find it to move in +x direction half of the time with a momentum of h/2L, and in -x direction half of the time with a momentum of -h/2L. We cannot predict the outcome of just one measurement because there are both possibilities for the result. We can only specify the momentum measurement with an uncertainty of

   

   At the same time if we attempt to measure the location of the particle we may in principle find it anywhere inside the box. So, the uncertainty of the position measurement can be estimated as

   

   The product of the uncertainties of both measurements is then:

   

   which is consistent with Heisenberg's uncertainty principle.

   If the kinetic energy of the particle really could vanish to zero then

   

   and the uncertainty in that measurement would also go away:

   

   This would violate Heisenberg's uncertainty principle which is based on the duality of waves and particles. We thus acknowledge that there must be perpetual motion in a quantum system even at zero Kelvin. However, it is also obvious that one cannot extract energy from the quantum system by bringing the zero-point motion to a halt since this would violate the basic particle-wave duality of nature.

6. The correspondence principle: When we look at the probability function of the particle in the lowest quantum state (n=1) we find a squared sin-function. This is in contrast to classical expectation in which the particle would have an even probability to be found anywhere in the box. As we increase the quantum number the quantum-mechanical probability function acquires more and more nodes. At some point the nodes are so numerous, and found so close to each other that it will be impossible to distinguish them from each other experimentally. In that case a position measurement will average over the probability function within an interval dx determined by experimental resolution. If we keep dx constant and probe the probability of the particle in the box we will find an even distribution throughout the box which corresponds to the classical picture.

   The correspondence principle is important in the sense that it provides the link between the quantum-mechanical picture and the classical picture. It states that when the value of the quantum number gets very high the system behaves classically.

Properties of the Wavefunction:

1. Normality of the wavefunction: The normalization condition (eq. (10)) assures that the wavefunction is normalized.

2. Orthogonality of the wavefunction: All eigenfunctions are orthogonal to each other, i.e. fulfill the following requirement:

   

   Both properties (normality and orthogonality) combined are called orthonormal. The orthonormality condition for the wavefunction can be written with the Kronecker delta:

   

   where

   

3. Symmetry of the wavefunction: Inspection of our wavefunctions proves that they are either symmetric or antisymmetric against reflection through a mirror plane put at the center of the box. The reason for this is that the Hamilton operator is invariant under that symmetry operation. In general, the wavefunctions have to be symmetric or antisymmetric under any transformation that leaves the Hamilton operator unchanged.

The particle in a box in higher dimensions:

The particle in a box problem can readily be extended from 1 dimension into higher spatial dimensions (i.e. 2D and 3D). Since the potential inside the box is always zero the Schrödinger equation

is easily separated in two or three separate equations dependent on the spatial coordinates x, y, and z that have the exact same form as eq. (2). The solutions are the same for each individual dimension, and the total wavefunction is then a product of the individual wavefunctions:

while the energy is a sum of the individual energies:

For a threedimensional representation of the wavefunctions and probability functions of the two-dimensional particle in a box follow the links. The wavefunctions of the threedimensional particle in a box is more difficult to visualize since we lack the forth dimension to plot its amplitude. The plot just shows the surface of the wavefunction at some arbitrarily chosen value. These surface areas may be viewed as to contain the space in which the amplitude of the wavefunction exceeds a predetermined value.