Quantum Mechanics: Concepts and Applications · Problem 3.15

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · energy-eigenfunction-expansion-time-evolution

Problem

The complete set expansion of an initial wave function

\psi(x, 0)

of a system in terms of orthonormal energy eigenfunctions

\phi_n(x)

of the system has three terms,

n = 1, 2, 3

. The measurement of energy on the system represented by

\psi(x, 0)

gives three values,

E_1

and

E_2

with probability

1/4

and

E_3

with probability

1/2

. (a) Write down

\psi(x, 0)

in terms of

\phi_1(x)

\phi_2(x)

, and

\phi_3(x)

. (b) Find

\psi(x, 0)

at any later time

t

, i.e., find

\psi(x, t)