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Homework Help: Basic QM - Trying to catch up in a class

  1. Aug 25, 2007 #1
    I haven't taken one of the recommended courses for a class I am in, so I'm playing a little catch-up here.

    Q: The wave function of an electron in a one-dimensional infinite square well of width a,x -> (0,a) at time t=0 is given by:

    [tex] \psi (x,0) \sqrt{\frac{2}{7}}\psi_1(x) + \sqrt{\frac{5}{7}}\psi_2(x) [/tex]

    where [tex] \psi_1(x) [/tex] and [tex] \psi_2(x) [/tex] are the ground and first excited stationary state of the system respectively, [tex] \psi_n(x)=\sqrt{\frac{2}{a}}\sin (n\pix/a), \,\,\, E_n = n^2\pi^2 \bar h^2/(2ma^2), \,], n=1,2,... [/tex]

    a) Write down the wavefunction at time t in terms of [tex] \psi_1(x) [/tex] and [tex] \psi_2(x) [/tex].

    b) You measure the energy of an electron at time t=0. Write down possible values of the energy and the probability of measuring each.

    c) Calculate the expectation value of hte enrgy in the state [tex] \psi(x,t) [/tex] above.

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    I am lost here. I am stuck on (a) right now, and really do not know where to go. To me it looks like [tex] \psi_n(x) [/tex] is the wavefunction already solved from the schrodinger equation (with the boundary conditions 0 and a). So I need to express the total wavefunction in terms of two states psi_1, and psi_2. This to me seems like another condition that would come from a differential equation something with a time dependence. I am lost...

    Would someone nudge me in a proper direction?

    thanks,
     
  2. jcsd
  3. Aug 25, 2007 #2

    dextercioby

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    Science Advisor
    Homework Helper

    Hint for 1. [tex]\psi (t)=U(t)\psi=e^{\frac{1}{i\hbar}tH} \psi [/tex]

    Daniel.
     
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