Higher-Order Wave Equations and Matter Waves |
The most common "wave equation" in one space and one time dimension is |
where V is a constant equal to the phase velocity of a monochromatic wave. This has the general solution f(x+Vt) + g(x-Vt) for arbitrary functions f and g. Among the particular solutions of this equation are the familiar monochromatic solutions of the form |
where A is an arbitrary constant. The second partial derivatives of this function with respect to x and t are -Ak^{2}cos(kx-wt) and -Aw^{2}cos(kx-wt) respectively, which implies V = w/k, showing that this particular y is strictly a function of x-Vt, i.e., it is a retarded wave. The amplitude of this function is purely real (assuming the constants and variables are real-valued), with a value that oscillates up and down corresponding to the projection onto the real axis of a point moving around in a circle in the complex plane. We could just as well include the imaginary component, since the complex function |
is also a solution of (1). In fact, this simple function is a solution of a wide range of wave equations, some of which turn out to have important physical applications. In general, for any positive integers r,s, consider the partial differential equation |
where a is an arbitrary constant. Substituting the trial solution Ae^{i(kx-wt)} into this equation, evaluating the derivatives, and dividing through by e^{i(kx-wt)}, we get |
This shows that (3) is a solution of every one of the higher-order wave equations |
The constant a in equation (4) is |
The phase velocity of the wave is still V = w/k, so if r = s we have a = (-1/V)^{r}, and hence V = -1/a^{1/r}. (Note that if m is even we can select the negative root to give a positive phase speed.) |
The energy E and momentum p (in one dimension) of a photon can be expressed in terms of the angular frequency w and wave number k respectively as |
where h is Planck's (reduced) constant. Substituting into the relativistic relation between energy, momentum, and rest mass m gives |
where c is the speed of light in vacuum. Consequently, if the phase speed of the photon equals c, meaning that V = w/k = c, the right hand side vanishes, so the rest mass m is zero. On the other hand, if the phase speed V of the wave is not equal to c, the rest mass will be non-zero. This suggests that we might regard a particle of mass m and velocity V as a "matter wave" with the frequency w and wave number k proportional to the energy and momentum respectively, in accord with the above equations for E and p. |
Returning to the generalized wave equation, we can express the constant k and w in terms of the energy and momentum of the matter wave, and write the generalized wave equation in the form |
For a photon we have E/p = c, so in order to eliminate E and p from the wave equation, i.e., to arrive at a constraint that is independent of the wave, we need r = s, in which case (6) reduces to the form |
With s = 0 this reduces to the trivial identity y = y, whereas with s = 2 it is the standard wave equation for electromagnetic waves. By construction, the simple wave function (3) satisfies this equation for every value of s, and so too (fortuitously) does the purely real wave function (2). In addition, each of these equations is consistent with all three of the conditions E = hw, p = hk, and E = pc, where c is the phase speed of the wave. |
Now, for a particle of rest mass m moving at the speed v, the momentum and energy are given by |
From the relations E = hw and p = hk this implies that the phase velocity V is |
Of course, if v = c this gives the phase velocity V = c, which applies to the case of a photon. However, for a massive particle with v less than c, this formula may seem surprising at first, because it implies that the phase velocity of a matter wave exceeds c, and in fact goes to infinity as the speed v of the particle goes to zero. This might seem to suggest that the wave velocity has no significance, and that we cannot associate the wave with the particle. Fortunately, this apparent problem disappears when we remember that the actual transmittal of energy in a wave is propagated not at the phase velocity necessarily, but at the group velocity v_{g},. (For a detailed discussion of the various velocities associated with a wave, see Phase, Group, and Signal Velocity.) The group velocity is given by |
so in fact the matter wave does propagate at the same speed as the corresponding particle. |
Eliminating v from the energy and momentum equations (8) leads immediately to the relation |
In the case of a photon we were able to go from equation (6) to equation (7), eliminating any explicit reference to the momentum and energy, and expressing the wave equation in terms of the constant c, because of the relation p/E = 1/c for massless particles. Thus if the indices r and s are equal, we can eliminate the ratio p/E. This clearly won't work for a particle with non-zero mass, but perhaps there is an analogous way of eliminating the explicit appearances of E and p in equation (6), possibly replacing them with some function of the invariant rest mass m of the particle. |
First, we should note that when we carry over the equation E = hw from photons to massive particles there is an ambiguity of interpretation of the quantity of energy represented by "E". Since a photon has no rest mass, its total energy and kinetic energy (i.e., total minus rest mass energy) are the same, so this is unambiguously the quantity represented by E. However, for a massive particle the total and kinetic energies are different, so we need to decide which of these quantities is applicable for a matter wave. In other words, we must decide if we are interested in the wave that corresponds to the total energy of the particle, or in the wave that corresponds to just the kinetic energy associated with the motion of the particle. Since these two energies differ only by a constant (the rest mass of the particle), they propagate with the same group velocities, but they have very different phase velocities. We saw in equation (9) that the phase velocity V of the total energy of a particle moving at the speed v is c^{2}/v, but the phase velocity of the particle's kinetic energy is given by |
assuming the momentum is taken to be the same in both cases. Now, it might be argued that we are mixing incompatible quantities here, because the wave number k is still taken to be p/h, meaning that it is proportional to the momentum of the total mass, not just of the kinetic mass. We might have attempted to interpret p in the relation p = hk as just the "kinetic momentum", i.e., the velocity times the difference between the total relativistic mass and the rest mass, and then we would (again) have the phase velocity V = c^{2}/v. However, if we did this, the group velocity would no longer equal v, because the difference between the total momentum and the "kinetic momentum" is a function of v. Moreover, by analogy with the energy, we note that each particle can be associated with a certain "rest momentum" just as it has a certain rest mass-energy, but the rest momentum of a particle is identically zero, so there is no distinction between its total momentum and its kinetic momentum. |
Consequently, we conclude that it's reasonable to treat the wave of a particle of rest mass m and speed v as having the wave number k = p/h and the frequency w = E_{kinetic}/h . Hence the equation (6) governing this wave (free of external constraints) is expected to be of the form |
If v is small compared with c, then the momentum is essentially p » mv and the kinetic energy is E_{kinetic} » (1/2)mv^{2}. Eliminating v from these two equations, we have the relation |
Therefore, in order to create a general constraint out of (10) by eliminating the ratio p^{r}/E_{kinetic}^{s}, we must have r = 2s, in which case the wave equation becomes |
Again, with s = 0 this reduces to the trivial identity y = y, whereas with s = 1 it gives the Schrodinger equation for the complex wave function of a free particle (i.e., with no potential field). By its construction, this wave equation for any non-negative integer s is satisfied by the complex-valued wave function (3). However, it is satisfied by the real-valued wave function (2) only for even values of s. |
Equation (12) is based on the non-relativistic equation (11). If instead we use the exact relativistic expressions for momentum and kinetic energy, equation (11) would become |
which confirms that this ratio is essentially equal to 2m for non-relativistic velocities. In general it equals the rest mass plus the relativistic mass of the particle. |
Focusing on the traditional non-relativistic Schrodinger equation (12), it's interesting to consider the consequences of setting the index s to a value other than 1. The complex-valued wave function (3) is a solution of every one of these equations, but the equations are nevertheless distinct, as illustrated by the fact that although every solution with s = 1 is also a solution with s = 0, the converse is not true. Also, with s = 2 the equation is satisfied by the real-valued wave function (2), even though this is not a solution with s = 1. Of course, the choice of s = 1 makes the left hand side equal to the Laplacian, which we expect on physical grounds, but it would be interesting to know what (if any) phenomena could be modeled by the higher-order wave equations, both in the case of massless (r = s) and massive (r = 2s) particles. At the very least, the higher-order wave equation (10) is satisfied by any function (or linear combination of functions) of the form |
where q is a constant phase offset and f is a polynomial of degree r-1 in x and degree s-1 in t. |
Needless to say, the above does not constitute an a priori derivation of the Schrodinger equation of a free particle. A more natural and well-motivated approach can be given in terms of the Hamiltonian operator whose eignevalues are the possible energy levels of the particle. Nevertheless, it's interesting to see how the general outline of Schrodinger's equation is essentially fixed by the relation between the energy and momentum of a particle as represented by a wave with the same group velocity as the particle. This approach also clarifies that a matter wave actually has two distinct phase velocities, one (c^{2}/v) corresponding to the total energy of the particle, and the other (~v/2) corresponding to just the kinetic energy of the particle. |