Classical waves

In quantum mechanics, everything is described as a wave. In the classical description of the world, we often think in terms of “particles” — little billiard balls. This concept has no fundamental place in quantum mechanics, though particle-like behavior can emerge from the underlying waves.

To illustrate the difference between quantum and classical mechanics, it is useful then to compare classical waves to quantum waves.

Waves on strings

As an example of a classical wave, consider a string instrument, like a violin or a guitar. The strings are of equal length, and fixed at both ends. When a string is plucked or bowed, it begins to vibrate as a wave. At a moment in time, the wave can be described as a displacement away from the resting position at each point along the string. As time progresses, the displacements evolve in a particular way as given mathematically by a “wave equation.” Waves, or portions of the wave, can often be characterized by an amplitude — the maximum extent of displacement — and a phase — the particular location of the peaks and troughs of the wave. The phase is most clearly defined in comparison to another wave. For example, swapping the peaks and troughs of a wave (i.e. flipping it over) results in a wave with opposite phase.

Classical waves have a number of properties familiar from our experience with things like string instruments, jump ropes, slinkies, water waves, and the like:

• Waves propagate.

  Imagine two people, Alice and Bob, stretching a jump rope between them. If Alice jerks her end up and down, a pulse-like wave will travel from one end to the other. A pulse-like wave is sometimes referred to as a “wave packet.” A wave packet could be a single up-and-down, or a pulse containing some number of wiggles.

• Waves reflect.

  If we continue to imagine that pulse, or wave packet, traveling down the jump rope, what happens when it gets to the other end? If Bob holds his end still, the wave will reflect off that fixed end and start to travel back towards Alice. Partial reflection is also possible. Imagine a very light rope tied to a very heavy rope and stretched between Alice and Bob. A pulse started at the light end, when reaching the junction between the ropes, will only launch a small wave in the heavy part, and mostly reflect the pulse back down the light part. This scenario is shown in Fig. 1.1. Alternatively, a pulse started at the heavy end will mostly propagate onto the light end but also partially reflect.

• Waves can be described as superpositions.

  If we take a snapshot of a wave at a moment in time, we can imagine the particular displacements not as a single wave, but as a sum of two (or more) distinct waves. As a trivial example, the two component waves could each be half of the total displacement at each point. A less trivial example could be a jump rope where Alice wiggles her end, then a short time later wiggles again (before the first wave has reached Bob.) Then there will be two distinct pulses traveling towards Bob. If we like, we can consider the two pulses as separate waves that we can add together. The example described above (and shown in Fig. 1.1) where a pulse partially reflects and partially transmits at a junction between light and heavy ropes can be viewed as one pulse splitting into a superposition of two pulses.

  The notion of envisioning a wave as the sum of two component waves is just a simple mathematical operation. In some cases, however, this division can be particularly useful. Specifically, this separation into component waves is useful if the behavior of the wave is linear, meaning that two waves differing only by a constant factor in their amplitudes evolve in time in the same way. Linearity is often a good approximation when the amplitude of the wave is sufficiently small. In this case, the evolution of the two component waves can be considered separately, and added back together at any time. So in the case of the two pulses traveling down a jump rope, each pulse can be considered separately as if they are independent entities. Each pulse propagates and reflects as it would alone, and then at any moment in time the two components can be added back together to find what the total wave looks like.

  There are unlimited ways that a wave can be split up into a superposition of different components. A particular choice of constituent waves is called a basis. For example, a rope with two pulses traveling down it could be written as a sum of two individual pulses \(d(x,t) = d_{a}(x,t) + d_{b}(x,t)\). Here the basis functions are \(d_{a}\) and \(d_b\). But we could define two new basis functions \(d_+ = (d_a + d_b)/2\) and \(d_{-} = (d_a - d_b)/2\). The former has two positive pulses, and the latter has one positive and one negative pulse. In the basis of \(d_+\) and \(d_-\), the total wave is just \(d(x,t) = 2 d_+(x,t)\). On the other hand, if we have a wave with just a single pulse \(d(x,t) = d_a(x,t)\), we could write the single pulse as the superposition of \(d(x,t) = d_+(x,t) + d_-(x,t)\). In this case, the positive and negative \(d_b\) components cancel out, leaving only \(d_a\). While this choice of basis may seem to be over-complicating things, we will see in the quantum case, that it is sometimes more convenient to work in one basis or another.

• Waves interfere.

  If we split a wave into a superposition of two parts that do not overlap, as in the two-pulse jump rope example, then it is straightforward to see how they add together. But now imagine what happens after the first pulse reflects off Bob’s end, and starts traveling back toward Alice, and towards the second pulse. At some point, the two pulses will cross each other. When this happens, the two pulses will either add together or subtract. If the two pulses have an identical shape, and have the same sign of displacement when they meet, then the two components interfere constructively, and the amplitude is increased. On the other hand, if the two components have opposite sign, then there will be a moment in time with zero amplitude, and the waves are said to interfere destructively. If the two pulses do not have the same shape, then interference can yield a more complicated wave when the two are added together, with the possibility of constructive interference in some places, and destructive interference in others.

• Waves can interact with other waves.

  Consider a string instrument, like a guitar. Each string can support a different wave. But the waves on each string are not independent of each other – they can interact. For example, the lowest and highest strings on a guitar are both tuned to the note ‘E’ separated by two octaves. If one E string is plucked, the other E will also start to vibrate. This occurs because the sound waves produced by one string in the air or in the body of the instrument are transmitted to the other string, causing it to vibrate. Therefore, to understand the behavior of multiple interacting waves, we must take into account not only the wave equations for the individual waves, but also include the effects of interactions that couple different waves together.

Figure 1.1

Video example of a wave on a light rope (left) tied to a heavy rope (right). A pulse launched from the light end is only able to transmit a small pulse into the heavy side, and is mostly reflected.

Classical waves on a string, oscillating in only one direction, are straightforward to visualize. At a moment in time, a simple plot of displacement vs. position serves to visualize the wave. To fully describe the wave, we would also need to know the velocity of the string at each position. This could be shown as a second curve on the plot. Alternatively, the velocity becomes apparent if we have a movie of the displacement curve over time. For multiple strings, we can easily make multiple plots of displacement (and velocity) vs. position for each string. These could be plotted on the same set of axes in different colors, or on vertically stacked axes for easy comparison. For example, the displacement of two different strings can be represented by \(d_1(x,t)\) and \(d_2(x,t)\), and both could be plotted on the same axes at some time \(t_0\). We could then generate frames for a movie by varying the time and replotting at some time step. This plotting method is shown for waves on two strings at the bottom of Fig. 1.2.

An alternative way to visualize two classical waves is to multiply the two waves with separate x-axes: \(D(x_1,x_2,t) = d_1(x_1,t)\cdot d_2(x_2,t)\). The new function \(D\) could now be plotted as a square 2D plot, where the product of displacements at each \((x_1,x_2)\) point is represented by color. Such a plot can be understood as showing the first wave along the horizontal axis, and the second wave along the vertical axis. Points along the diagonal where \(x_1=x_2\) represents the same \(x\) position for the two waves. The same two waves are also shown using this plotting method at the top of Fig. 1.2. This way of plotting offers little to no benefit over simply plotting multiple 1D plots. However, we will see that the 2D plot is essential for visualizing two quantum waves. This difference between classical and quantum waves is perhaps the single most important difference between the quantum and the classical, and the key to understanding many surprising features of quantum mechanics.

Figure 1.2

Video examples of two 1D classical waves evolving in time, illustrating wave packets, propagation, reflection, superposition, and interference. The red wave moves slowly to the right. The blue wave can be viewed as a superposition of two pulses, both initially traveling to the right. The first pulse reflects at the right side, and is inverted. The two pulses interfere destructively, move past each other, and finally the second pulse reflects.

Klein-Gordon waves

Before turning to the topic of quantum waves, let’s take a look at one, slightly more complicated, type of classical wave. These waves occur on a stretched string, as above, but now also with some additional springy force pushing or pulling the string back towards its resting position. Imagine a tennis net hanging from a cable stretched between two posts. If the net is tautly attached along its length to the ground, then when a wave travels along the top cable the net will be stretched and pull the cable back down. (To make a better analogy, the net should also push back up when compressed, unlike a real tennis net made of strings. Possibly a sheet of rubber instead of a net would behave like that.) A wave on the cable in this scenario is called a Klein-Gordon wave (and the wave equation that describes it is called the Klein-Gordon equation.)

Because of the “net” always pulling on the cable, there is a minimum frequency for a Klein-Gordon wave. Even if there is no wave at all and the whole straight cable is just pulled upwards and released, the force supplied by the net will cause it to oscillate up and down at some frequency set by the stretchiness of the net. If some waves are present in the cable then the combined wave on the cable and the stretching of the net leads to a higher frequency of oscillation.

Another interesting property of Klein-Gordon waves is that waves with different wavelengths propagate at different speeds. This does not occur for an ideal wave on a string (like the jump rope examples), but is not uncommon for other types of waves, such as water waves. A consequence of this (for reasons not explained here) is that when a pulse of waves is launched, the individual peaks and troughs within the pulse travel at a different speed than the pulse itself. This can be seen in the case of water waves: if a pebble is dropped into a pond, the waves spread out in a circle, but the peaks and troughs of the wave appear to move faster than the spread of the disturbance’s edge. The peaks and troughs disappear when they reach the edge of the disturbance, which only spreads out more slowly. For Klein-Gordon waves, the effect is the other way — the peaks and troughs travel slower than the pulse. You can see this in the example videos below by keeping track of one of the peaks in the wave. Does it stay in the same place relative to the pulse, or does it move through the pulse over time?

Let’s look at an extreme example where the minimum frequency caused by the net is much higher than any additional frequency caused by the cable itself. This would happen when the net is very stiff, and/or the cable is loose, and/or the waves on the cable are very long.

Figure 1.3

Video example of a propagating Klein-Gordon wave packet. The top panel shows the actual wave displacement. The middle panel decomposes the top wave into the two components u and v. The bottom panel plots the combined amplitude squared \(u^2+v^2\).

Here, the peaks and troughs of the displacement \(d\), shown in the top panel of Fig. 1.3, propagate much faster than the speed of the overall pulse. If we are mainly interested in the slow pulse itself, then the very fast oscillations caused by the net are rather distracting to look at, and make the equations more complicated. It would be nice to ignore them. We can accomplish just that with a little math. If the net alone causes oscillation at the minimum frequency \(m\), then we can describe the displacement \(d\) of the wave using two other effective waves \(u\) and \(v\) as

\[d(x,t) = u(x,t) \cos{(mt)} + v(x,t) \sin{(mt)}.\]

The two new waves \(u\) and \(v\), shown in the middle panel of Fig. 1.3, will describe the relatively slow motion of the wave in the cable and the propagation of the pulse, and we can ignore the fast oscillation at frequency \(m\). Just looking at the propagation of the waves \(u\) and \(v\), the situation is now much easier to look at, without the very fast oscillations present. It somewhat strange, though, that now we have two waves instead of one. But don’t worry: the same information is present as in the original wave. The extra wave is needed to retain the information about the phase of the fast component that we are now ignoring. But if we don’t care about the phase, then it turns out that a natural and useful way to combine them together is to add their squares: \(u^2+v^2\). When we do that, the waves making up the pulse disappear, and we only see the pulse traveling along. This happens because, in this example, wherever \(u\) is zero, \(v\) is at a peak or trough, and vice versa. The pulse viewed as \(u^2+v^2\) is visually similar to the jump rope pulses, but don’t forget that under the hood, oscillations are occurring.

In removing the fast oscillations at frequency \(m\), we have taken one wave \(d\) (the displacement of the wave, described by real numbers), and replaced it with two waves \(u\) and \(v\) (also described by real numbers) that encode the amplitudes of the two distinct phases of original wave. It turns out that we have a mathematical framework to simplify the task of keeping track of amplitude and phase – complex numbers. We can represent both waves \(u\) and \(v\) as a single complex wave \(\Psi\), whose real part is \(u\) and imaginary part is \(v\).

Now we can see how Klein-Gordon waves manifest the properties of waves mentioned previously. The example just shown (Fig. 1.3) has illustrated how Klein-Gordon waves propagate. We can see how a Klein-Gordon wave reflects by making one half of the net stiffer than the other half. The stiffer part acts as a “barrier” to the propagation of the wave. This is similar to the case of a light rope tied to a heavy rope, but with one important difference. Because there is a minimum frequency for the Klein-Gordon wave to propagate, if the barrier is large enough then it is possible that the wave will not be able to continue past the barrier at all. Figure 1.4 shows this situation. The wave is completely reflected. Note that though the wave cannot propagate beyond the barrier, it does extend a little ways into the barrier while it is in the process of reflecting.

Figure 1.4

Video example of a reflecting Klein-Gordon wave packet. The step plotted in gray indicates the region where the minimum frequency is increased by a stiffening of the net. The top panel shows the actual wave displacement. The middle panel decomposes the top wave into the two components u and v. The bottom panel plots the combined amplitude squared \(u^2+v^2\).

The fact that the wave extends a bit into a region forbidden to a propagating wave gives the wave the ability to “tunnel” through a sufficiently narrow barrier. We will keep the barrier the same height as in the previous example, but make it narrow so that only a small region of the net is stiffer. Now we see in Fig. 1.5 that when the wave extends a bit into the barrier, it can come out the other side. In this case, the wave is partially transmitted and partially reflected, yielding a nice example of the wave property of superposition.

Figure 1.5

Video example of a tunneling Klein-Gordon wave packet. The step up and down plotted in gray indicates the region where the minimum frequency is increased by a stiffening of the net. The top panel shows the actual wave displacement. The middle panel decomposes the top wave into the two components u and v. The bottom panel plots the combined amplitude squared \(u^2+v^2\).

As for interference, we have already seen one example of interference, when the Klein-Gordon wave reflects off of the barrier in Fig. 1.4 and Fig. 1.5. While the reflection is taking place, the wave oscillations become visible in the bottom plot of \(u^2+v^2\). These occur as the leading edge of the pulse that has already reflected interferes with the trailing edge of the pulse still heading for the barrier. A clearer view of interference, however, can be obtained if we start with two pulses heading towards each other (Fig. 1.6). When they meet, we can see that the waves \(u\) and \(v\) interfere in a way such that the peaks and troughs of both occur in the same place, resulting in visible oscillations in the combined \(u^2 + v^2\).

Figure 1.6

Video example of an interfering Klein-Gordon wave. The wave is initially in a superposition of two packets heading towards each other. The top panel shows the actual wave displacement. The middle panel decomposes the top wave into the two components u and v. The bottom panel plots the combined amplitude squared \(u^2+v^2\).