# Projecting an abstract state onto position/momentum/energy spaces

1. Mar 21, 2007

### v0id

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

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

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 $$\Psi(x) = \frac{1}{\sqrt{h}} \int_{\infty}^{\infty} | \psi(t) \rangle \exp{\left( \frac{i \omega x}{\hbar}\right)} \; d\omega = \langle \phi | \psi(t) \rangle$$, where $$| \phi \rangle = \exp{\left( \frac{-i \omega x}{\hbar}\right)}$$?

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

Last edited: Mar 21, 2007