Up to this point, Basic Physics has taken its examples from introductory-level classical physics. Probability has applications in classical physics, but it plays a more essential role in quantum physics. We can take examples from quantum mechanics without raising the mathematical or computational level appreciably, but for those with no previous knowledge of the subject, the brief outline given below may not be adequate.
In order to explain phenomena such as electron diffraction we have to find a way to give wave properties to a material particle. The Schrodinger equation starts with the classical relation between a particle's energy and its momentum:
And the complex exponential form of the equation of a wave:
The complex form of the equation describing a wave is used for mathematical convenience in the sections on ellipsometry, but the complex form is a necessity here. A particle wave gets its frequency from the photon's energy-frequency relation, E = hf , and its wavelength (k = 2p /l ) from the photon's momentum- wavelength relation, l = h/p. With these substitutions, and with 'h-bar' (h with a stroke through it) in place of h/(2p), the equation describing a particle wave becomes:
Schrodinger noted that for a wave of this kind, the following two equations are equivalent:
This last equation is the time-dependent Schrodinger equation. It enables the motion of a particle to be represented as the motion of a wave packet. We will deal with the time-dependent equation later in Wave Motion.
The problem of finding time-independent solutions (stationary states or standing waves) is mathematically much simpler. For a standing wave, Y(x,t) = Y(x) F(t), and Schrodinger's equation becomes:
This is the time-independent Schrodinger equation. The three-dimensional form of the equation is what is used to find the energy levels of electrons in atoms. We will work in one dimension and find the stationary states of particles in potential wells.
The function we have written Y(x) is usually written y (x), and is known as the wave function or probability amplitude. Its modulus squared is the probability density, and is the quantum mechanical analog of the classical probability density derived in the last section. The time dependence of the wave function, F(t), is a complex exponential, not a sine or cosine, so the probability density does not oscillate like a standing wave on a string. For a probability density to be physically meaningful it must be normalized, i.e. its integral with respect to x must give unity. There are only special values of E (the eigenvalues) for which normalized solutions of the Schrodinger equation exist. The problem is to find them.
Whatever one may think of the above 'plausibility argument' for the Schrodinger equation, the equation itself is not complicated. The equation says that the second derivative of Y with respect to x is equal to a function of x and Y. This is numerically no different from Newton's law for a driven oscillator where the second derivative of x with respect to t is equal to a function of t and x. This is exactly the type of equation the Feynman algorithm is designed to solve. The equation involves Planck's constant, h, but energy can be expressed in units that involve h, so h need not appear in the numerical equation. The problem, as we noted above, is to find E.
The graphical interpretation of the Schrodinger equation is not difficult either. The equation says that inside a well (where E>U) the curvature of Y(x) is toward the axis (like a sine wave), and outside the well the curvature is away from the axis (like an exponential). Usually a function that keeps curving away from the axis will end up going to infinity. In the Schrodinger equation, the curvature is proportional to the magnitude of the function, so Y(x) flattens out as it approaches the axis, and for an appropriate value of E the approach to the axis will be asymptotic.
When expressed graphically, the problem of finding an energy eigenvalue is not all that different from finding the initial velocity for a baseball to travel a specified distance before hitting the ground. The algorithm used for the baseball, the bisection algorithm, works equally well for the Schrodinger equation. Finally, to calculate probabilities we need to integrate the probability density using the trapezoid algorithm. All of the tools we need to tackle the Schrodinger equation are things we have used previously.
In order to start the Feynman algorithm with a trial value for the energy, we need to know Y and dY/dx at x=0. In order to specify initial values, we must first define the problem we are trying to solve. We will look at the examples dealt with classically in the last section: the bouncing ball, and the harmonic oscillator. The harmonic oscillator potential is symmetric, and functions with the positive and negative x-axes as asymptotes will be either symmetric or antisymmetric (have either dY/dx=0 or Y=0 at x=0). If the floor is impenetrable, Y for the bouncing ball must go to zero there. (This is required by the second derivative in the equation - in terms of Newton's law, which also has a second derivative, a bouncing ball has a discontinuity in its velocity, but not in its position).
We can put the Schrodinger equation into dimensionless form by using x/A in place of x for the harmonic oscillator, and y/H in place of y for the bouncing ball. A and H are related to the energy through kA2/2 and mgH, so the equation for the harmonic oscillator can be written:
with E measured in units of
and the equation for the bouncing ball can be written:
with E measured in units of
The program uses the symbol x in place of z, and the plot covers the region from the origin out to three times the distance to the turning point. The connection with Newton's law is emphasized in the program by using v for the first derivative and a for the second derivative. The program finds the energy level and the probability of finding the particle outside the well for the three lowest energy states of the bouncing ball and the harmonic oscillator. The trial wavefunctions are plotted until they go off scale, but they could be stopped when they begin to slope away from the axis outside the well (this is the endpoint for the trapezoid algorithm). The analytic energy values for the harmonic oscillator (1, 3, and 5 in our units) provide a check on the accuracy of the numerical solution.
The dimensionless scale we have used is convenient for finding energies, but we get a better picture of the stationary states if we plot probability density on a scale that does not superimpose the turning points. The next program makes such a plot for the first three states of the bouncing ball. Both the classical and quantum probability densities are plotted for each state. The plots are normalized, i.e. the areas under all six curves are the same. (The initial slope in the quantum calculation is set to the value that produces a normalized curve.) Note that the classical probability density begins to look like the average of the quantum probability density for higher energy states.
Our final applet deals with the double well: a square potential well with infinitely high sides and a central barrier. (The well width is 400 units, the barrier width is 17 units, and the barrier height is 200 times the ground state energy of a well with no barrier.) This system is of interest because the energies of the two lowest states differ by very little. In The Quantum Wave Equation we show that the superposition of such states leads to "beats" similar to those resulting from the superposition of two sound waves of slightly different frequency. Here we must modify the bisection algorithm to distinguish between states with slightly different energies. This is done using symmetry: the ground state goes to zero at the well boundaries, but not in between, whereas the second state crosses the axis once, and symmetry requires that this be at the centre of the well. The algorithm performs its test at the centre of the well, looking for zero slope or amplitude. The two functions it finds are closely related to the second state for a well with no barrier, and their energies are just below and just above its energy (4 expressed in units of the ground state energy with no barrier).