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Wavefunctions and expectation values

For a particle in a 1-dimensional box confined between 0<x<a , the initial state of a particle is given by

phi = phi_1 + 3phi_2 + 2phi_3 (all phi's are functions of x).

a) Normalize this wave function.

b) If no energy measurements are made what is the expectation value of energy of this state at a later time t?

c) Is this expectation value of energy an energy eigenvalues of this system? Why or why not?


Solution Preview

I'll assume that the psi's are the normalized eigenfunctions:

psi_n(x) = sqrt(2/a) sin(n pi x/a)

The energy eigenvalue of psi_n is:

E_n = pi^2 hbar^2 n^2/(2 m a^2) (1)

We want to normalize the wavefunction

psi = psi_1 + 3 psi_2 + 2 psi_3

This can be done without much computation using linear algebra techniques as follows. You consider the wavefunction to be a vector in a vector space (that vector space is called a Hilbert space). One defines a complex inner product on this vector space as follows:

<psi|phi> = Integral of ...

Solution Summary

The expert examines wave functions and expectation values. Energy measurements are determined.