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.
c) Is this expectation value of energy an energy eigenvalues of this system? Why or why not?© BrainMass Inc. brainmass.com June 3, 2020, 9:15 pm ad1c9bdddf
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 ...
The expert examines wave functions and expectation values. Energy measurements are determined.