History

• De Broglie's suggestion that "particles" have wave properties proved very fruitful. The problems remaining were to find the appropriate wave equation, and to understand what its solutions had to do with experimental observations. These were very difficult problems.

• Before they were solved, Heisenberg in 1924 proposed a different approach, called "matrix mechanics". In this theory observable quantities like position and momentum were represented by infinite-dimensional matrices which did not commute (i.e., xp was not the same as px).

• Heisenberg's theory was useful for calculating "transition probabilities" which give the relative intensities of spectral lines resulting from various transitions between atomic energy level.

• It was very difficult to use for other problems, such as determining what those energy levels should be. Even to solve the problem of the hydrogen atom was far from easy.

• In 1926 Schrödinger proposed a wave equation which provided a much simpler (and to most physicists more intuitive) approach to many problems. One conceptual difficulty arose when it became clear that the solutions of the wave equation were complex-valued functions.

• Schrödinger showed that, in terms of predictions, his "wave mechanics" gave the same answers as Heisenberg's matrix mechanics. From that time on, most physicists preferred the wave approach.

• Nevertheless, problems of interpretation remained. When Born showed that the solution of Schrodinger's wave equation could be interpreted as a non-observable object used only as an intermediary to calculate relative probabilities of the various allowed outcomes of an experiment, and when Bohr and Heisenberg solidified this interpretation (the so-called "Copenhagen interpretation") Schrödinger objected strongly. So did Einstein. To the ends of their lives those two (among a few others) argued that there must be a more direct way to describe nature.

• This is still an open question, although most physicists adopt the Copenhagen line and stop worrying about whether there is a more "intuitive" approach.

• Schrödinger's theory is non-relativistic. Attempts to make a relativistic wave mechanics led to some startling new discoveries, such as antiparticles. The current theory along these lines is called "quantum field theory".

Wave Equation for a Free Particle

• Whatever the correct wave equation might be, it should contain the relations between wave and particle properties that seem to apply for particles (or waves) moving freely through space. That is, we should have p = h/l and E = hf.

• A wave with definite wavelength and frequency should thus correspond to a particle with definite momentum and energy.

• Classical wave theories use the function describing a wave with definite wavelength and frequency, namely, a harmonic wave of the type

Here as usual k = 2p/l and w = 2pf.

• We replace the wave properties by the particle properties:

• Now for a non-relativistic free particle E = p²/2m. Note that

• It follows that

• Perhaps this is the appropriate wave equation to describe a free particle in quantum theory. But how does one introduce interactions?

The Schrödinger Equation

• In classical physics with conservative forces, the total energy consists of the kinetic energy p²/2m plus the potential energy U(x).

• Schrödinger decided to try simply adding the potential energy (times the wavefunction) to the left side of the free particle equation. This gave the result (in one dimension)

This is the Schrödinger Equation.

• Mathematically this is an interesting kind of equation:
• It is linear, so the superposition principle will apply.
• It is first order in time, so the only initial condition required to specify the solution is the value of Y at t = 0 (at all points in space). A knowledge of Y at a particular time can thus give a complete specification of the state of the system at that time.
• Because of the i on the right side, Y cannot be real (or pure imaginary). It must be a complex valued function. This means that Y cannot be directly measurable, and a problem of interpretation arises.

Stationary States

• The appearance of U in the Schrödinger equation (S-E) means that the particle is not free. Nevertheless, it may have a definite and conserved energy. According to our assumptions, this should mean a definite frequency of the wave.

• We thus expect the time dependence for such situations to be simply e^-iwt, where E and w are related in the usual way:

• We are thus led to look for solutions of the S-E having the structure

• Substituting this into the S-E gives the equation satisfied by F:

This is called the Time-Independent Schrödinger Equation (TISE).

• There are several notable aspects of this equation:
• The solutions F(x) can be taken to be real valued functions.
• In one dimension it is an ordinary linear differential equation of second order.
• Depending on what "boundary conditions" F must obey, there may or may not be any solutions for a particular value of E. This raises the possibility that only certain values of E are allowed. (This kind of problem is called an "eigenvalue" problem.)

• The states where the energy has a definite value (and hence the time dependence factors out as above) are called stationary states. We will see later what is "stationary" about them.

Interpretation of the Wavefunction

• To understand what information is contained in Y and how to extract predictions from a knowledge of Y, we start with a wave theory we understand already: electromagnetic waves.

• Solutions of the wave equation in the e-m case are the E and B fields of the wave. Since the B field is determined by the E field, we consider only the latter.

• One usually wants to know about the energy transport in the wave. This is given by the Poynting vector

• The direction of this vector is the direction of propagation of energy, and its magnitude is the intensity (power per unit area) in the wave. Since B is perpendicular to E, and the magnitudes are related by E = cB, the intensity is

• Knowing I one can predict the amount of wave energy falling on a given surface per unit time.

• But suppose we consider the radiation to be photons. Then what does I tell us? It still gives the amount of energy per unit area per unit time. But now that energy is divided among a large number of photons, each with energy hf. So I tells us how many photons will fall on unit area of the surface, per second.

• But suppose we have a very faint source, so that only a single photon is emitted per second. Then what does I tell us? It cannot tell us that this single photon will be spread all over the surface, since experimentally (the photoelectric effect) that does not happen. The photon will strike a relatively small location on the surface.

• The only reasonable interpretation is that I gives the relative probability that a photon will strike the surface here or there. That is, the value of I at a particular point on the surface is proportional to the probability of a photon landing there.

• Note that to obtain this probability distribution we take the squared amplitude of the function that obeys the wave equation.

Probability Waves

• Now we apply this lesson to the wavefunctions that satisfy the S-E. Usually we are thinking in terms of a single particle, or only a few, so the probability interpretation is more useful to us than intensities.

• We postulate that a solution of the S-E gives us the "probability amplitude" describing the particle's possible behavior.

• In particular, the probability of locating the particle at time t is given by

• We will learn how to calculate probabilities of other interesting things later.

• Of course the particle must be somewhere, so we must require the normalization condition:

This puts severe restrictions on the kinds of solutions to the S-E that can actually correspond to allowed states of the particle.

• Note that for stationary states the time dependence cancels when we take the squared magnitude:

• The probability distribution for such states does not change with time. That is why they are called "stationary".

• The interpretative postulates outlined here give the core of the Copenhagen interpretation of quantum theory.

Three Dimensions

• To extend the discussion to more than one dimension is easy. The potential energy becomes U(x,y,z) and we add partial derivatives with respect to y and z. The S-E then becomes

• There is a corresponding change in the TISE.

• The set of derivatives in () above is usually denoted by a special symbol:

• The normalization condition becomes

• Of course, coordinate systems other than cartesian can be used. For central force problems (such as the hydrogen atom) it is better to use spherical coordinates.