# Further Interpretation of the Wave Function

• K. T. Hecht
Part of the Graduate Texts in Contemporary Physics book series (GTCP)

## Abstract

Consider a quantum-mechanical system with a Hamiltonian that has a discrete spectrum only, with allowed energies, E n , and eigenfunctions, ψ n . In general, the state of this quantum system can be specified by a wave function
$$\Psi (\vec r,t) = \sum\limits_n {{C_n}} {\psi _n}(\vec r),{e^{ - \frac{1}{\hbar }{E_n}t}}$$
(1)
describing a system for which the energy is not uniquely specified. If it is a single particle,
$$< \Psi ,\Psi > = 1 = \sum\limits_n {c_n^ * {c_n} < {\psi _n},{\psi _n} > + } \sum\limits_{n \ne m} {c_n^ * {c_m} < {\psi _n},{\psi _m} > {e^{\frac{1}{\hbar }({E_n} - {E_m})t}}.}$$
(2)
Because $$<{\psi_n},{\psi_n}>=0$$for $$n \ne m,$$
$$<\Psi,\Psi>=\sum\limits_n{|{C_n}}{|^2},$$
(3)
. Similarly,
$$<E>=<\Psi,H\Psi>=\sum\limits_n{|{C_n}}{|^2}{E_n}$$
(4)
$$<{E^k}>=<\Psi,{H^k}\Psi>=\sum\limits_n{|{C_n}}{|^2}E_n^k$$
(5)
. It is natural to interpret $$|{c_n}{|^2}$$ as P(E n ), the probability the particle be found in the state with energy E n . Without a coupling of our system to an outside field, that is, without an outside perturbation, these P(E n ) are independent of the time.