I started reading, today, Chap 13: Propagation in a Crystal Lattice of Feynman's Lectures III.

But, I couldn't understand some of his writings as:

If you have a harmonic oscillator which is coupled to another harmonic oscillator, and that one to another, and so on …, and if you start an irregularity in one place, the irregularity will propagate as a wave along the line. ___The same situation exists if you place an electron at one atom of a long chain of atoms.___$^1$

Usually, the simplest way of analyzing the mechanical problem is not to think in terms of what happens if a pulse is started at a definite place, but rather in terms of steady-wave solutions. There exist certain patterns of displacements which propagate through the crystal as a wave of a single, fixed frequency. Now the same thing happens with the electron—and for the same reason, because it’s described in quantum mechanics by similar equations.

You must appreciate one thing, however; the amplitude for the electron to be at a place is an amplitude, not a probability. If the electron were simply leaking from one place to another, like water going through a hole, the behavior would be completely different. For example, if we had two tanks of water connected by a tube to permit some leakage from one to the other, then the levels would approach each other exponentially. But for the electron, what happens is amplitude leakage and not just a plain probability leakage. And it’s a characteristic of the imaginary term—the $i$ in the differential equations of quantum mechanics—which changes the exponential solution to an oscillatory solution. What happens then is quite different from the leakage between interconnected tanks.$^2$

$\bullet ^1$ What does Feynman mean by saying the same situation in case of electrons? Meant to say, why did he compare the leakage of electrons with infinitely coupled oscillator? What is the relationship between these two cases?

$\bullet ^2$ What would happen if the amplitude is not an amplitude, but a probability? Why would the situations be different then? What's the difference between amplitude leakage & probability leakage??

[...] any state $|ϕ⟩$ of the electron in our one-dimensional crystal can be described by giving all the amplitudes $⟨n|ϕ⟩$ that the state $|ϕ⟩$ is in one of the base states—which means the amplitude that it is located at one particular atom. Then we can write the state $|ϕ⟩$ as a superposition of the base states $$|ϕ⟩=\sum_n|n⟩⟨n|ϕ⟩.\tag{13.1}$$ Next, we are going to suppose that when the electron is at one atom, there is a certain amplitude that it will leak to the atom on either side. And we’ll take the simplest case for which it can only leak to the nearest neighbors—to get to the next-nearest neighbor, it has to go in two steps. We’ll take that the amplitudes for the electron jump from one atom to the next is $iA/ℏ$ (___per unit time___$^3$). For the moment we would like to write the amplitude $⟨n|ϕ⟩$ to be on the $n$th atom as $C_n.$ Then Eq. $(13.1)$ will be written $$|ϕ⟩=\sum_n|n⟩C_n.\tag{13.2}$$ If we knew each of the amplitudes $C_n$ at a given moment, we could take their absolute squares and get the probability that you would find the electron if you looked at atom $n$ at that time. What will the situation be at some later time? By analogy with the two-state systems we have studied, we would propose that the Hamiltonian equations for this system should be made up of equations like this: $$iℏ\frac{dC_n(t)}{dt}=E_0C_n(t)−AC_{n+1}(t)−AC_{n−1}(t).\tag{13.3}$$ The first coefficient on the right, $E_0$, is, physically, the energy the electron would have if it couldn't leak away from one of the atoms. (It doesn't matter what we call $E_0$; as we have seen many times, it represents really nothing but our choice of the zero of energy.) The next term represents the ___amplitude per unit time that the electron is leaking into the $n$th pit from the $(n+1)$st pit___$^3$; and the last term is the amplitude for leakage from the $(n−1)$st pit. As usual, we’ll assume that $A$ is a constant (independent of $t$).

$\bullet ^3$ Now what is amplitude per unit time? Change in amplitude per unit time could make sense. But amplitude per unit time seems to make no sense to me. Can anyone please explain me what Feynman meant by amplitude per unit time?


1 Answer 1

  1. He is saying that for both the coupled harmonic oscillators and the electron in the chain of atoms:

if you start an irregularity in one place, the irregularity will propagate as a wave along the line

That's basically just what it means to be coupled. If one starts doing something weird its neighbor will be affected so it will start doing something weird.

  1. I will go into this with more detail by giving an in depth discussion of his two tank example.

  2. It's basically a change in amplitude per unit time as you said. You are supposed to think of negative change in amplitude in the $n$th pit and a positive change in amplitude in the $(n+1)$th pit. Thus it is as if amplitude is being transferred from one pit to the other, so you can talk about the amplitude per unit time transferred.

Two tank example

Suppose we have two tanks with a small tube connecting them. What will happen? Feynman was saying the classical and quantum versions look different. Let's look at the clasiscal tank system first

Classical tank system

The state of the classical system can be characterized by a single parameter $\Delta$ giving the difference in the amount of water in the two tanks. We can express the rate of change of the system with the time derivative of $\Delta$, $\dot{\Delta}$. There will be some equation which gives the rate of change of the system in terms of its current state. This equation will have the form $\dot{\Delta} = f(\Delta)$. Thus we can see we have a one dimensional dynamical system. Now one dimensional dynamical systems are boring, and one big way they are boring is that they can't have oscillitory behavoir. Thus, as feynman said, all you will get is just the water levels equalizing.

Quantum tank system

Now imagine a quantum tank system. The state of either tank can be represented by a complex number, so it takes four real numbers to represent the system. Now we know the square magnitudes of these complex numbers must add to one, and we know that multiplying both numbers by the same phase factor has no physical significance, but there are still two real numbers which specify the phase of the system. One of them is the difference in square amplitude. This is analogous to the $\Delta$ of the classical system. The new degree of freedom is the difference in phase between the two amplitudes. This makes it a two dimensional dynamical system and allows for more complicated behavior.

Suppose for example the Hamiltonian was $\begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$. Then a system which at $t_0=0$ has the first tank amplitude $1$ and the second tank amplitude $0$, will, at some later time $t$, have a first tank amplitude of $\cos(t)$ and a second tank amplitude of $-\sin(t)$. Thus amplitude can be transferred between the two tanks in an oscillatory manner, unlike the case with the classical tanks.

  • $\begingroup$ +1; some queries though: one dimensional dynamical systems are boring, and one big way they are boring is that they can't have oscillitory behavoir- I'm new to dynamical system; I couldn't conceive what it is though I checked Wikipedia but can you please tell me why they can't have an oscillatory behaviour? $\endgroup$
    – user36790
    Oct 2, 2015 at 11:27
  • $\begingroup$ ` Now we know the square magnitudes of these complex numbers must add to one`: why should they add to one? $\endgroup$
    – user36790
    Oct 2, 2015 at 11:29
  • $\begingroup$ I will answer your second question first. The square magnitudes of the complex numbers are supposed to add to one in my example because I was thinking about it like wave function, where the square amplitude represents probability, and probability adds to one. If it were really just supposed to be a water tank, then I guess you could say the sum of the square amplitudes gives you the total amount of water, but the total amount of water must be fixed, so it is still not a dynamical degree of freedom. $\endgroup$ Oct 2, 2015 at 12:35
  • $\begingroup$ A dynamical system is just a deterministic system having the property that where you will be depends only on where you are now. To have oscillations in 1D, like a mass on a spring, you would need to sometimes go left and sometimes go right when you are in the middle. But as I said, in a dynamical system you will determinstically go left every time or right every time, so you can't have oscillations in 1d. However in higher dimensions you can go in a circle. $\endgroup$ Oct 2, 2015 at 12:38

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