Free Electron Motion

According to Equation (1129), the relativistic Hamiltonian of a free electron takes the form

$$H = c\, \mbox{\boldmath$\alpha$} \cdot{\bf p} + \beta\,m_e\,c^2.$$ (1176)

Let us use the Heisenberg picture to investigate the motion of such an electron. For the sake of brevity, we shall omit the suffix $t$ that should be appended to dynamical variables that vary in time, according to the formalism of Section 3.2.

The above Hamiltonian is independent of ${\bf x}$ . Hence, momentum ${\bf p}$ commutes with the Hamiltonian, and is therefore a constant of the motion. The $x$ component of the velocity is

$$\dot{x} = \frac{[x,H]}{{\rm i}\,\hbar} = c\,\alpha_1,$$ (1177)

where use has been made of the standard commutation relations between position and momentum operators. This result is rather surprising, since it implies a relationship between velocity and momentum that is quite different from that in classical mechanics. This relationship, however, is clearly connected to the expression $j_x = \psi^\dag c\,\alpha_1\,\psi$ for the $x$ component of the probability current. The operator $\dot{x}$ , specified in the above equation, has the eigenvalues $\pm c$ , corresponding to the eigenvalues $\pm 1$ of $\alpha_1$ . Since $\dot{y}$ and $\dot{z}$ are similar to $\dot{x}$ , we conclude that a measurement of a velocity component of a free electron is certain to yield the result $\pm c$ . As is easily demonstrated, this conclusion also holds in the presence of an electromagnetic field.

Of course, electrons are often observed to have velocities considerably less than that of light. Hence, the previous conclusion seems to be in conflict with experimental observations. The conflict is not real, however, because the theoretical velocity discussed above is the velocity at one instance in time, whereas observed velocities are always averages over a finite time interval. We shall find, on further examination of the equations of motion, that the velocity of a free electron is not constant, but oscillates rapidly about a mean value that agrees with the experimentally observed value.

In order to understand why a measurement of a velocity component must lead to the result $\pm c$ in a relativistic theory, consider the following argument. To measure the velocity we must measure the position at two slightly different times, and then divide the change in position by the time interval. (We cannot just measure the momentum and then apply a formula, because the ordinary connection between velocity and momentum is no longer valid.) In order that our measured velocity may approximate to the instantaneous velocity, the time interval between the two measurements of position must be very short, and the measurements themselves very accurate. However, the great accuracy with which the position of the electron is known during the time interval leads to an almost complete indeterminacy in its momentum, according to the Heisenberg uncertainty principle. This means that almost all values of the momentum are equally likely, so that the momentum is almost certain to be infinite. But, an infinite value of a momentum component corresponds to the values $\pm c$ for the corresponding velocity component.

Let us now examine how the election velocity varies in time. We have

$${\rm i}\,\hbar\,\dot{\alpha}_1 =\alpha_1\,H-H\,\alpha_1.$$ (1178)

Now $\alpha_1$ anti-commutes with all terms in $H$ except $c\,\alpha_1\,p^1$ , so

$$\alpha_1\,H+H\,\alpha_1 = \alpha_1\,c\,\alpha_1\,p^1 + c\,\alpha_1\,p^1\,\alpha_1 = 2\,c\,p^1.$$ (1179)

Here, use has been made of the fact that $\alpha_1$ commutes with $p^1$ , and also that $\alpha_1^{\,2}=1$ . Hence, we get

$${\rm i}\,\hbar\,\dot{\alpha}_1 = 2\,\alpha_1\,H-2\,c\,p^1.$$ (1180)

Since $H$ and $p^1$ are constants of the motion, this equation yields

$${\rm i}\,\hbar\,\ddot{\alpha}_1 = 2\,\dot{\alpha}_1\,H,$$ (1181)

which can be integrated to give

$$\dot{\alpha}_1(t) = \dot{\alpha}_1(0)\,\exp\left(\frac{-2\,{\rm i}\,H\,t}{\hbar}\right).$$ (1182)

It follows from Equation (1177) and (1180) that

$$\dot{x}(t) = c\,\alpha_1(t) = \frac{{\rm i}\,\hbar\,c}{2}\,\dot{\... ...,\exp\left(\frac{-2\,{\rm i}\,H\,t}{\hbar}\right)H^{\,-1} + c^2\,p_x\,H^{\,-1},$$ (1183)

and

$$x(t) = x(0)- \frac{\hbar^2\,c}{4}\,\dot{\alpha}_1(0)\,\exp\left(\frac{-2\,{\rm i}\,H\,t}{\hbar}\right)H^{\,-2} + c^2\,p_x\,H^{\,-1}\,t.$$ (1184)

We can see that the $x$ -component of velocity consists of two parts, a constant part, $c^2\,p_x\,H^{\,-1}$ , connected with the momentum according to the classical relativistic formula, and an oscillatory part whose frequency, $2\,H/h$ , is high (being at least $2\,m_e\,c^2/h$ ). Only the constant part would be observed in a practical measurement of velocity (i.e., an average over a short time interval that is still much longer than $h/2\,m_e\,c^2$ ). The oscillatory part ensures that the instantaneous value of $\dot{x}$ has the eigenvalues $\pm c$ . Note, finally, that the oscillatory part of $x$ is small, being of order $\hbar/m_e\,c$ .