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Basic questions about QM computations

  1. Dec 8, 2006 #1
    1. How can we calculate expectation values of an arbitrary [tex]Q[/tex], even if [tex]\psi[/tex] is not an eigenfunction of [tex]Q[/tex]?

    2. (Fourier transform related) Suppose I have piecewise wavefunction. [tex]\psi_{I}[/tex] at [tex](-\infty,-L)[/tex], [tex]\psi_{II}[/tex] at [tex](-L,+L)[/tex] and [tex]\psi_{III}[/tex] at [tex](L,+\infty)[/tex]. I can compute entire [tex]\phi(k)[/tex] by taking the Fourier transform's integral from [tex]-\infty[/tex] to [tex]+\infty[/tex]. But what if I try to calculate [tex]\phi(k)[/tex] only between [tex](-L,L)[/tex]? Is it [tex]\phi_{II}(k) = \int_{-L}^L \psi(x) e^{ikx}dx[/tex]?

    3. After I solved the time-independent SE, I get a series of solutions. I plug the time dependent part after I find [tex]E_n[/tex]s, and [tex]\Psi(x,t) = \sum c_n \psi(x)e^{-i E_n t/\hbar}[/tex], where [tex]\sum c_n = 1[/tex] But how on earth do I get [tex]c_n[/tex]. Is there a realistic example (i.e., I'm not talking about examples "let's say [tex]c_0 = 0.3[/tex] and [tex]c_2 = 0.7[/tex], calculate bla bla bla") where [tex]c_n[/tex]s are computed by us?

    Thanks in advance!
    Last edited: Dec 8, 2006
  2. jcsd
  3. Dec 9, 2006 #2


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    3. The coefficients are computed from the normalization condition for the wave-function. The typical examples are the H-atom amd the 1-dim harmonic oscillator.

    2. Yes.

    1. Expand [itex] \psi [/itex] in terms of (possibly generalized) eigenfunctions of Q.

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