Uniform motion
If we suppose that the relativity theory is right, a particle at rest in one inertial system can be in uniform motion in another inertial system. In the rest frame of the particle, the probability amplitude is the same for all $x$, $y$, and $z$ but varies with $t$. The magnitude of the amplitude is the same for all $t$, but the phase depends on $t$. We can get a kind of a picture of the behavior of the amplitude if we plot lines of equal phase—say, lines of zero phase—as a function of $x$ and $t$. For a particle at rest, these equal-phase lines are parallel to the $x$-axis and are equally spaced in the $t$-coordinate, as shown by the dashed lines in Fig. 7–1. In a different frame—$x'$, $y'$, $z'$, $t'$—that is moving with respect to the particle in, say, the $x$-direction, the $x'$ and $t'$ coordinates of any particular point in space are related to $x$ and $t$ by the Lorentz transformation. This transformation can be represented graphically by drawing $x'$ and $t'$ axes, as is done in Fig. 7–1. (See Chapter 17, Vol. I, Fig. 17–2) You can see that in the $x'$-$t'$ system, points of equal phase1 have a different spacing along the $t'$-axis, so the frequency of the time variation is different. Also there is a variation of the phase with $x'$, so the probability amplitude must be a function of $x'$. Under a Lorentz transformation for the velocity $v$, say along the negative $x$-direction, the time $t$ is related to the time $t'$ by \begin{equation*} t=\frac{t'-x'v/c^2}{\sqrt{1-v^2/c^2}}, \end{equation*}  so our amplitude now varies as \begin{equation*} e^{-(i/\hbar)E_0t}= e^{-(i/\hbar)(E_0t'/\sqrt{1-v^2/c^2}-E_0vx'/c^2\sqrt{1-v^2/c^2})}. \end{equation*}  In the prime system it varies in space as well as in time. If we write the amplitude as \begin{equation*} e^{-(i/\hbar)(E_p't'-p'x')}, \end{equation*}  we see that $E_p'=E_0/\sqrt{1-v^2/c^2}$ is the energy computed classically for a particle of rest energy $E_0$ travelling at the velocity $v$, and $p'=E_p'v/c^2$ is the corresponding particle momentum. You know that $x_\mu=(ct,x,y,z)$ and $p_\mu=(E/c,p_x,p_y,p_z)$ are four-vectors, and that $p_\mu x_\mu=Et-\FLPp\cdot\FLPx$ is a scalar invariant. In the rest frame of the particle, $p_\mu x_\mu$ is just $Et$; so if we transform to another frame, $Et$ will be replaced by \begin{equation*} E't'-\FLPp'\cdot\FLPx'. \end{equation*}  Thus, the probability amplitude of a particle which has the momentum $\FLPp$ will be proportional to \begin{equation} \label{Eq:III:7:5} e^{-(i/\hbar)(E_pt-\FLPp\cdot\FLPx)}, \end{equation}  where $E_p$ is the energy of the particle whose momentum is $p$, that is, \begin{equation} \label{Eq:III:7:6} E_p=\sqrt{(pc)^2+E_0^2}, \end{equation}  where $E_0$ is, as before, the rest energy. For nonrelativistic problems, we can write \begin{equation} \label{Eq:III:7:7} E_p=M_sc^2+W_p, \end{equation}  where $W_p$ is the energy over and above the rest energy $M_sc^2$ of the parts of the atom. In general, $W_p$ would include both the kinetic energy of the atom as well as its binding or excitation energy, which we can call the “internal” energy. We would write \begin{equation} \label{Eq:III:7:8} W_p=W_{\text{int}}+\frac{p^2}{2M}, \end{equation}  and the amplitudes would be \begin{equation} \label{Eq:III:7:9} e^{-(i/\hbar)(W_pt-\FLPp\cdot\FLPx)}. \end{equation}  Because we will generally be doing nonrelativistic calculations, we will use this form for the probability amplitudes. Note that our relativistic transformation has given us the variation of the amplitude of an atom which moves in space without any additional assumptions. The wave number of the space variations is, from (7.9), \begin{equation} \label{Eq:III:7:10} k=\frac{p}{\hbar}; \end{equation}  so the wavelength is \begin{equation} \label{Eq:III:7:11} \lambda=\frac{2\pi}{k}=\frac{h}{p}. \end{equation}  This is the same wavelength we have used before for particles with the momentum $p$. This formula was first arrived at by de Broglie in just this way. For a moving particle, the frequency of the amplitude variations is still given by \begin{equation} \label{Eq:III:7:12} \hbar\omega=W_p. \end{equation}  The absolute square of (7.9) is just $1$, so for a particle in motion with a definite energy, the probability of finding it is the same everywhere and does not change with time. (It is important to notice that the amplitude is a complex wave. If we used a real sine wave, the square would vary from point to point, which would not be right.) We know, of course, that there are situations in which particles move from place to place so that the probability depends on position and changes with time. How do we describe such situations? We can do that by considering amplitudes which are a superposition of two or more amplitudes for states of definite energy. We have already discussed this situation in Chapter 48 of Vol. I—even for probability amplitudes! We found that the sum of two amplitudes with different wave numbers $k$ (that is, momenta) and frequencies $\omega$ (that is, energies) gives interference humps, or beats, so that the square of the amplitude varies with space and time. We also found that these beats move with the so-called “group velocity” given by \begin{equation*} v_g=\frac{\Delta\omega}{\Delta k}, \end{equation*}  where $\Delta k$ and $\Delta\omega$ are the differences between the wave numbers and frequencies for the two waves. For more complicated waves—made up of the sum of many amplitudes all near the same frequency—the group velocity is \begin{equation} \label{Eq:III:7:13} v_g=\ddt{\omega}{k}. \end{equation}  Taking $\omega=E_p/\hbar$ and $k=p/\hbar$, we see that \begin{equation} \label{Eq:III:7:14} v_g=\ddt{E_p}{p}. \end{equation}  Using Eq. (7.6), we have \begin{equation} \label{Eq:III:7:15} \ddt{E_p}{p}=c^2\,\frac{p}{E_p}. \end{equation}  At nonrelativistic speeds $E_p\approx Mc^2$, so \begin{equation} \label{Eq:III:7:16} \ddt{E_p}{p}=\frac{p}{M}, \end{equation}  which is just the classical velocity of the particle. Alternatively, if we use the nonrelativistic expressions Eqs. (7.7) and (7.8), we have \begin{equation} \omega=\frac{W_p}{\hbar}\quad \text{and}\quad k=\frac{p}{\hbar},\notag \end{equation}  and \begin{equation} \label{Eq:III:7:17} \ddt{\omega}{k}=\ddt{W_p}{p}=\ddt{}{p}\biggl(\frac{p^2}{2M}\biggr)= \frac{p}{M}, \end{equation}  which is again the classical velocity. Our result, then, is that if we have several amplitudes for pure energy states of nearly the same energy, their interference gives “lumps” in the probability that move through space with a velocity equal to the velocity of a classical particle of that energy. We should remark, however, that when we say we can add two amplitudes of different wave number together to get a beat-note that will correspond to a moving particle, we have introduced something new—something that we cannot deduce from the theory of relativity. We said what the amplitude did for a particle standing still and then deduced what it would do if the particle were moving. But we cannot deduce from these arguments what would happen when there are two waves moving with different speeds. If we stop one, we cannot stop the other. So we have added tacitly the extra hypothesis that not only is (7.9) a possible solution, but that there can also be solutions with all kinds of $p$’s for the same system, and that the different terms will interfere.