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I have my quantum mechanics final creeping up on me and I just have a question about something that doesn't appear to be covered in the text.

Let's say you have a wave function of the following form for a linear harmonic oscillator:

[itex] \Psi = c_1 | E_1 \rangle + c_2 | E_2 \rangle [/itex]

The basis is just the first two excited energy states. My question is how the Hamiltonian matrix is represented in this case. Is it

[itex] H = \hbar \omega \left( \begin{matrix} 0 & 0 & 0 \\ 0 & \frac{3}{2} & 0 \\ 0 & 0 & \frac{5}{2} \end{matrix} \right) [/itex]

Or do you just use the non-zero elements:

[itex] H = \hbar \omega \left( \begin{matrix} \frac{3}{2} & 0 \\ 0 & \frac{5}{2} \end{matrix} \right) [/itex]

Any help would be greatly appreciated. Thanks!

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# Matrix Representation of Operators in a Finite Basis

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