Oscillations of an Energy Eigenstate

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?

• See Courant & Hilbert volumes. – Felix Marin May 18 at 5:05
• These hold for potentials that drop off rapidly at $\infty$, but in general there can be eigenfunctions with an infinite number of maxima. – Keith McClary May 18 at 15:39
• You have been wasting time on bad texts. Go to Messiah, Quantum Mechanics, v I , Ch III, sec 8. – Cosmas Zachos May 18 at 20:09
• This reference explains the discrete energies. Does it explain the number of oscillations somewhere else? – Charles Hudgins May 19 at 19:47
• 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. – Charles Hudgins May 21 at 15:33