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?