Explore BrainMass

Explore BrainMass

    Wavefunctions and expectation values

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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?

    © BrainMass Inc. brainmass.com June 3, 2020, 9:15 pm ad1c9bdddf
    https://brainmass.com/physics/energy/wavefunctions-expectation-values-173258

    Attachments

    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.

    $2.19

    ADVERTISEMENT