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Homework Help: Projecting an abstract state onto position/momentum/energy spaces

  1. Mar 21, 2007 #1
    1. The problem statement, all variables and given/known data
    Consider the quantum harmonic oscillator in the state [tex]| \psi (t) \rangle = \frac{1}{\sqrt{14}}\left( 3 | 0 \rangle \exp{\left( -\frac{1}{2}i \omega t\right)} + 2 | 1 \rangle \exp{\left( -\frac{3}{2}i \omega t\right)} + | 5 \rangle \exp{\left( -\frac{11}{2}i \omega t\right)} \right)[/tex]. What is [tex]| \psi (t) \rangle[/tex] in terms of the [tex]\psi_n (x)[/tex], [tex]\psi_n (p)[/tex] and [tex]\psi_n (E)[/tex]. Do not evaluate the specific basis vectors.


    2. Relevant equations
    [tex]\Phi (p) = \frac{1}{\sqrt{h}}\int_{-\infty}^{\infty} \psi (x) \exp{\left( \frac{-ipx}{\hbar} \right)} \; dx[/tex]


    3. The attempt at a solution
    I think expansion along the energy space is unnecessary, since the original kets are themselves eigenkets of the Hamiltonian. As for position space, is the Fourier transform [tex]\Psi(x) = \frac{1}{\sqrt{h}} \int_{\infty}^{\infty} | \psi(t) \rangle \exp{\left( \frac{i \omega x}{\hbar}\right)} \; d\omega = \langle \phi | \psi(t) \rangle[/tex], where [tex]| \phi \rangle = \exp{\left( \frac{-i \omega x}{\hbar}\right)}[/tex]?


    EDIT: Since only projection onto the position-space and momentum-space bases is necessary, would it be prudent to multiply the vector
    [tex]
    \left[ \begin{array}{c} 3\\
    2\\
    0\\
    0\\
    0\\
    1
    \end{array}\right]
    [/tex]
    by the matrix representations of the position and momentum operators respectively?
     
    Last edited: Mar 21, 2007
  2. jcsd
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