Quantized energy levels
In a later chapter we will look in detail at a solution of Schrödinger’s equation for a particular example. We would like now, however, to show you how one of the most remarkable consequence of Schrödinger’s equation comes about—namely, the surprising fact that a differential equation involving only continuous functions of continuous variables in space can give rise to quantum effects such as the discrete energy levels in an atom. The essential fact to understand is how it can be that an electron which is confined to a certain region of space by some kind of a potential “well” must necessarily have only one or another of a certain well-defined set of discrete energies.  Suppose we think of an electron in a one-dimensional situation in which its potential energy varies with $x$ in a way described by the graph in Fig. 16–3. We will assume that this potential is static—it doesn’t vary with time. As we have done so many times before, we would like to look for solutions corresponding to states of definite energy, which means, of definite frequency. Let’s try a solution of the form \begin{equation} \label{Eq:III:16:57} \psi=a(x)e^{-iEt/\hbar}. \end{equation}  If we substitute this function into the Schrödinger equation, we find that the function $a(x)$ must satisfy the following differential equation: \begin{equation} \label{Eq:III:16:58} \frac{d^2a(x)}{dx^2}=\frac{2m}{\hbar^2} [V(x)-E]a(x). \end{equation}  This equation says that at each $x$ the second derivative of $a(x)$ with respect to $x$ is proportional to $a(x)$, the coefficient of proportionality being given by the quantity $(2m/\hbar^2)(V-E)$. The second derivative of $a(x)$ is the rate of change of its slope. If the potential $V$ is greater than the energy $E$ of the particle, the rate of change of the slope of $a(x)$ will have the same sign as $a(x)$. That means that the curve of $a(x)$ will be concave away from the $x$-axis. That is, it will have, more or less, the character of the positive or negative exponential function, $e^{\pm x}$. This means that in the region to the left of $x_1$, in Fig. 16–3, where $V$ is greater than the assumed energy $E$, the function $a(x)$ would have to look like one or another of the curves shown in part (a) of Fig. 16–4. If, on the other hand, the potential function $V$ is less than the energy $E$, the second derivative of $a(x)$ with respect to $x$ has the opposite sign from $a(x)$ itself, and the curve of $a(x)$ will always be concave toward the $x$-axis like one of the pieces shown in part (b) of Fig. 16–4. The solution in such a region has, piece-by-piece, roughly the form of a sinusoidal curve. Now let’s see if we can construct graphically a solution for the function $a(x)$ which corresponds to a particle of energy $E_a$ in the potential $V$ shown in Fig. 16–3. Since we are trying to describe a situation in which a particle is bound inside the potential well, we want to look for solutions in which the wave amplitude takes on very small values when $x$ is way outside the potential well. We can easily imagine a curve like the one shown in Fig. 16–5 which tends toward zero for large negative values of $x$, and grows smoothly as it approaches $x_1$. Since $V$ is equal to $E_a$ at $x_1$, the curvature of the function becomes zero at this point. Between $x_1$ and $x_2$, the quantity $V-E_a$ is always a negative number, so the function $a(x)$ is always concave toward the axis, and the curvature is larger the larger the difference between $E_a$ and $V$. If we continue the curve into the region between $x_1$ and $x_2$, it should go more or less as shown in Fig. 16–5. Now let’s continue this curve into the region to the right of $x_2$. There it curves away from the axis and takes off toward large positive values, as drawn in Fig. 16–6. For the energy $E_a$ we have chosen, the solution for $a(x)$ gets larger and larger with increasing $x$. In fact, its curvature is also increasing (if the potential continues to stay flat). The amplitude rapidly grows to immense proportions. What does this mean? It simply means that the particle is not “bound” in the potential well. It is infinitely more likely to be found outside of the well, than inside. For the solution we have manufactured, the electron is more likely to be found at $x=+\infty$ than anywhere else. We have failed to find a solution for a bound particle. Let’s try another energy, say one a little bit higher than $E_a$—say the energy $E_b$ in Fig. 16–7. If we start with the same conditions on the left, we get the solution drawn in the lower half of Fig. 16–7. It looked at first as though it were going to be better, but it ends up just as bad as the solution for $E_a$—except that now $a(x)$ is getting more and more negative as we go toward large values of $x$. Maybe that’s the clue. Since changing the energy a little bit from $E_a$ to $E_b$ causes the curve to flip from one side of the axis to the other, perhaps there is some energy lying between $E_a$ and $E_b$ for which the curve will approach zero for large values of $x$. There is, indeed, and we have sketched how the solution might look in Fig. 16–8. You should appreciate that the solution we have drawn in the figure is a very special one. If we were to raise or lower the energy ever so slightly, the function would go over into curves like one or the other of the two broken-line curves shown in Fig. 16–8, and we would not have the proper conditions for a bound particle. We have obtained a result that if a particle is to be bound in a potential well, it can do so only if it has a very definite energy. Does that mean that there is only one energy for a particle bound in a potential well? No. Other energies are possible, but not energies too close to $E_c$. Notice that the wave function we have drawn in Fig. 16–8 crosses the axis four times in the region between $x_1$ and $x_2$. If we were to pick an energy quite a bit lower than $E_c$, we could have a solution which crosses the axis only three times, only two times, only once, or not at all. The possible solutions are sketched in Fig. 16–9. (There may also be other solutions corresponding to values of the energy higher than the ones shown.) Our conclusion is that if a particle is bound in a potential well, its energy can take on only the certain special values in a discrete energy spectrum. You see how a differential equation can describe the basic fact of quantum physics. We might remark one other thing. If the energy $E$ is above the top of the potential well, then there are no longer any discrete solutions, and any possible energy is permitted. Such solutions correspond to the scattering of free particles by a potential well. We have seen an example of such solutions when we considered the effects of impurity atoms in a crystal.