Heisenberg Uncertainty Principle

From the de Broglie hypothesis, we can quickly predict the outcome of many experiments with elementary particles. The first experiment we shall consider demonstrates something very unusual about elementary particles known as the Heisenberg Uncertainty Principle. This principle what we mean by ``small'' in the idea of the classical limit and is useful in giving quick estimates of quantum mechanical effects.

The basic idea comes from what we already know about the intensity of waves after passing through any opening. The notes on interference show that the intensity of waves passing through a finite slit exhibit the phenomenon of diffraction: no matter how straight the waves approach the opening, on the other side the intensity spreads out over a range of angles. From the de Broglie hypothesis, we expect the same behavior for the probability of finding particles. This has some unusual implications.

Figure 5 shows an experiment where particles of momentum $p$ are sent directly toward a slit of width $a$ and then collected on a observation screen at a large distance $R \gg a$ from the slit. From the de Broglie hypothesis we have the immediate result that particles arrive at the screen with all possible random angles $\theta$ with the same pattern we found for the interference pattern from a single slit (from LN ``Wave Phenomena II: Interference'')

$${\mathcal P}={\mathcal P}_{max} \frac{\sin^2\frac{\Delta \Phi}{2}}{\left(\frac{\Delta \Phi}{2}\right)^2},$$ (8)

where $\Delta \Phi \equiv k a \sin\theta$ with $k$ now given by the de Broglie hypothesis to be $k=p/\hbar$. Figure 6 shows the probability pattern.

Figure 5: Particle of momentum $p$ sent directly through slit of width $a$

Figure 6: Probability of observing particle on screen at deflection angle $\theta$

For this type of experiment, the classical physics prediction is that all particles should arrive on the screen at the angle $\theta=0$. Consistent with the concept of the classical limit, this is the most probable outcome. However, some particles will hit the screen at almost all angles, with the exception of just a few special points where the probability just happens to be zero. This means that every time a baseball is thrown through a door, even in vacuum so that there is nothing to disturb the path of the ball, there is some chance that it will take a sharp turn on the other side.

To see how this is not inconsistent with the observations of our daily lives, it helps to figure out what range of angles are most likely. The probabilities in Figure 5 tend to be rather small after the first minima bounding the central maximum. In fact, over 90% of the area under the curve is contained in the main peak between these first minima. Thus, with better than 90% certainty, we can say that the particle will arrive at angles in the range $\left\vert \Delta \Phi/2 \right\vert < \pi$, so that $\left\vert \Delta \Phi \right\vert < 2 \pi$. Using the definitions of $\Delta \Phi$ and $k$ above, this gives $\left\vert k a \sin\theta \right\vert = \frac{pa}{\hbar} \left\vert \sin\theta \right\vert < 2 \pi$. Thus, the reasonably expected range of angles is

$$\left\vert \theta \right\vert \approx \left\vert \sin\theta \right\vert \le \frac{2 \pi \hbar}{pa},$$ (9)

where remind ourselves of the small angle approximation. The key to Eq. (9) is that $\hbar$ is extremely small in standard units, approximately $10^{-34}$ J$\cdot$s, so that the range of likely angles thus will be very small for normal objects. For instance, if we toss ($v \approx 10$ m/s) a baseball ($m \approx 0.1$ kg) through a door ($a \approx 1$ m), we find $pa=mva=1$ kg$\cdot$m$^2$/s = 1 J$\cdot$s and thus $\vert\theta\vert < 2\pi \times 10^{-34}$ radians, an angle so small that we never notice it in everyday life. On the other hand, if the objects and distances involved are small enough so that $pa$ approaches $\hbar = 10^{-34}$ J$\cdot$s, then a wide range of angles becomes likely. This answers the question of how small is ``small enough'' so that classical physics begins to break down.

Heisenberg realized that this phenomenon of spreading out after being restricted through an aperture is very general and found a very useful way of expressing the result in terms of classical concepts about particles. The slit in Figure 5 restricts the particles to a range of values $\Delta y = a$ in the $y$-direction. The fact that the particles can then be found on the screen at angles $\theta$ means that after being restricted by the slit in the $y$-direction, the particles now pick up random (!) momenta $p_y$ in the $y$-direction of value which the geometry of Figure 5 determines to be $p_y = p \sin\theta$.⁴ The likely range in these random momenta is thus $\Delta p_y = p \vert \sin \theta \vert = \frac{2 \pi \hbar}{a}$, where we have used the likely range of angles from Eq. (9). Note that this range becomes greater and greater in inverse proportion as $a$ decreases. Heisenberg described this effect by saying that the act of constraining a particle to a region of size $\Delta y$ along the $y$-direction results in an uncertainty in its momentum $\Delta p_y$ so that $\Delta y \, \Delta p_y$ is at least as big as some constant. (He said ``at least'' as big because it is always possible to introduce additional sources of uncertainty beyond the fundamental limit.) In this case, multiplying our results, we find the constant to turn out to be $2 \pi \hbar$.

A few technical notes are in order. In defining the uncertainty $\Delta p_y$, we used the range of momenta for which we have 90% confidence. The precise definition used in advanced courses in quantum mechanics is to use the standard deviation statistic that we use in the analysis of exam scores, which gives a somewhat narrower measure for $\Delta p_y$. Also, a ``smoother'' slit where the particle has some chance of making it through the edges can also lessen the uncertainty in $\Delta p_y$. Finally, we could just as easily aligned our slit with the $x$-axis. When all of this is taken into account, the precise mathematical statement has been shown to be that we have separately for each component $x$, $y$ and $z$ that

$$\Delta x \, \Delta p_x \ge \frac{\hbar}{2}$$ (10)

$$\Delta y \, \Delta p_y \ge \frac{\hbar}{2}$$

$$\Delta z \, \Delta p_z \ge \frac{\hbar}{2},$$

but that there is no restriction when mixing directions,

$$\begin{eqnarray*} \Delta x \, \Delta p_y & \ge & 0 \\ \Delta x \, \Delta p_z & \ge & 0 \\ \Delta y \, \Delta p_x & \ge & 0 \\ & \mbox{etc.} \end{eqnarray*}$$

More typically, one never quite reaches the fundamental limit in Eq. (10) and one can make reasonably good estimates in problems by setting $\Delta x \Delta p_x, \Delta y \Delta p_y, \Delta z \Delta p_z \sim \hbar$.