Introduction to Quantum Mechanics · Problem 2.45

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · node_theorem

Problem

In this problem you will show that the number of nodes of the stationary states of a one-dimensional potential always increases with energy. Consider two (real, normalized) solutions (

\psi_n

and

\psi_m

) to the time-independent Schrödinger equation (for a given potential

V(x)

), with energies

E_n > E_m

. (a) Show that

\frac{d}{dx}\left(\frac{d\psi_m}{dx}\psi_n - \psi_m\frac{d\psi_n}{dx}\right) = \frac{2m}{\hbar^2}(E_n - E_m)\psi_m\psi_n.

(b) Let

x_1

and

x_2

be two adjacent nodes of the function

\psi_m(x)

. Show that

\psi_m'(x_2)\psi_n(x_2) - \psi_m'(x_1)\psi_n(x_1) = \frac{2m}{\hbar^2}(E_n - E_m)\int_{x_1}^{x_2}\psi_m\psi_n\,dx.

\psi_n(x)

has no nodes between

x_1

and

x_2

, then it must have the same sign everywhere in the interval. Show that (b) then leads to a contradiction. Therefore, between every pair of nodes of

\psi_m(x)

\psi_n(x)

must have at least one node, and in particular the number of nodes increases with energy.

Related theory

The Prime Number Theorem And Dirichlet S Theorem

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