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Wavefunctions and expectation values

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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?

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

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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 ...

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