[CHAT/QM-13002] LET's TALK --- NATURE OF ENERGY SPECTRUM

Nature of energy spectrum

In this talk we present a set of thumb rules which can be used to decide the nature of energy spectrum of a quantum particle in one dimension. It turns out that for this purpose it is sufficient to look at the absolute minimum of the potential and its limits as \(x\to \pm \infty\). Other details, though important, are not relevant in the present context for text book level problems. When we say nature of energy spectrum, we seek answers to following questions. "Are the energies discrete or continuous? or some levels are discrete and some others have continuous energies?" \\ "How many linearly independent solutions are there for a particular level? In other words, what is the degeneracy of a particular level?" The Schr\"{o}dinger equation being a second order differential equation and has two linearly independent solutions. A general solution for a given energy will be a linear combination and of the two solutions and has form \begin{equation} \psi(x,E) = \alpha \psi_1(x,E) + \beta \psi_2(x,E) \end{equation} where we have explicitly indicated the the energy dependence of solutions. It must also be remembered that changing the over all normalization does not lead to a new physical solution, therefore, a general solution has  only one, undetermined arbitrary constant, the ratio of \(\alpha\) and \(\beta\). In this talk we present a set of thumb rules which can be used to decide the nature of energy spectrum of a quantum particle in one dimension. It turns out that for this purpose it is sufficient to look at the absolute minimum of the potential and its limits as \(x\to \pm \infty\). Other details, though important, are not relevant for text book level problems. The energy eigenvalues are always greater than, or equal to, the absolute minimum, say \(V_\text{min}\) of the potential. We begin with the conditions imposed by physical requirements on the behaviour of the solution at infinity.

Bound states

For bound state solutions, the probability density, $|\psi(x)|^2$ must approach zero as $|x|\to \infty$. This puts two conditions on the energy eigenfunction with only one unknown constant of integration being available. So in general a solution will exist only for some energy values. Hence the energy is quantized. In addition the continuity equation can be used to prove that the bound state eigenfunctions are non degenerate. A necessary condition is that the energy should be less than the value of the potential both at \(\infty\) and \(-\infty\). Thus for \[ E < V_-\text{and} E < V_+\] we expect energy eigenvalues to be discrete and non degenerate.

Scattering solution

Suppose we wish to describe scattering of a beam particle from a potential. In this case we have an incoming beam from \(x=-\infty\) and a transmitted beam going to \(x=+\infty\). This situation requires that the solution behave like free particle at large distance. This happens when limiting values \(V_\pm\) of the potential are (finite) constants and \(E > V_\pm\). No other restriction is placed on the solution and energy values are doubly degenerate and continuous.

An intermediate case Suppose we have a case intermediate between the two cases taken up in the above. At large distances, the solution of the eigenvalue problem goes to zero on one side and behaves like plane waves on the other side at large distances. Then there is only one requirement, the solution going to zero on one side, to be imposed and the undetermined constant of integration gets fixed. This happens when \(V_+\ne V_-\) and for energies lying between \(V_+\) and \(V_-\). This results in no constraints on energy. Thus the energy is continuous and non degenerate.

Summary

Postponing a detailed discussion, we summarise a few important thumb rules on the nature of energy spectrum. In addition to the restrictions given below, the rule \(E>_\text{min}\) needs to be imposed strictly in each of the following three cases. 

1. If \(E\) is less than both $V_\pm$, in addition to being greater than $V_\text{min}$, the eigenvalue is discrete and non degenerate. This situation corresponds to bound states.
2. For energies in between \(V_+\) and \(V_-\), the energy is continuous and solutions are non degenerate. These solutions describe scattering process.
3. For \(V_+\ne V_-\) and for energies between the two values, the eigenvalues are continuous and non degenerate.