Energy eigenstates of a 1-dimensional particle are given by solutions to differential equations of the form $$ \left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x) = E\psi(x) $$ where $V$ is the potential, $\psi$ is the energy eigenstate, and $E$ is the energy eigenvalue.

In introductory (and even fairly advanced) texts on quantum mechanics, the following facts are asserted without proof:

  1. The set $\{E \in \mathbb{R} : E \text{ is an energy eigenvalue and } E < \sup_{x \in \mathbb{R}} V(x) \}$ is a discrete subset of the reals bounded below by $\inf_{x \in \mathbb{R}} V(x)$.
  2. If the discrete energy eigenvalues above are listed in ascending order as $E_0, \ldots, E_n$ and have corresponding energy eigenstates $\psi_0, \ldots, \psi_n$, then $\psi_n$ will have $n + 1$ local maxima.

The above results are often stated with less precision, and sometimes even conflict with each other.

What statements like 1 and 2 above are actually true, and how does one go about proving them?

  • $\begingroup$ See Courant & Hilbert volumes. $\endgroup$ – Felix Marin May 18 at 5:05
  • $\begingroup$ These hold for potentials that drop off rapidly at $\infty$, but in general there can be eigenfunctions with an infinite number of maxima. $\endgroup$ – Keith McClary May 18 at 15:39
  • $\begingroup$ You have been wasting time on bad texts. Go to Messiah, Quantum Mechanics, v I , Ch III, sec 8. $\endgroup$ – Cosmas Zachos May 18 at 20:09
  • $\begingroup$ This reference explains the discrete energies. Does it explain the number of oscillations somewhere else? $\endgroup$ – Charles Hudgins May 19 at 19:47
  • $\begingroup$ In part, but I also want to know why the eigenfunctions in the discrete spectrum oscillate a number of times that corresponds to the energy level. For example, the bound states $\psi_n$ of the quantum harmonic oscillator have $n + 1$ critical points. I was told this is a general phenomenon in my quantum mechanics course, but I've never seen a proof of it. Of course it's entirely possible it's just something special about the quantum harmonic oscillator, and I was misled. $\endgroup$ – Charles Hudgins May 21 at 15:33

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