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Does measuring the energy of the particle cause its wave function (psi) to change if psi isn't an eigenfunction of the Hamiltonian? quite confused about this one on when the wave function changes when taking a measurement and when it doesn't. please help :)

Lalo
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Yes, the wave function is changed in this case.

Suppose the particle has a wave function which is not an eigenfunction of the Hamiltonian, i.e. it can be written as a linear combination of the energy eigenfunctions $\psi_(x)$: $$\psi(x)=\sum_i c_i \psi_i(x).$$ You measure the energy and get one of the eigenvalues $E_i$ as the result. Then the wavefunction after this measurement is the eigenfunction $\psi_i(x)$ corresponding to this eigenvalue $E_i$: $$\psi(x)=\psi_i(x).$$ This is called the wave function collapse.

Thomas Fritsch
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  • +1 - though just for clarity and because it looks like OP may be a complete beginner, may be worth mentioning that the $\psi_i(x)$ in this case are the energy eigenfunctions. In the sense that $\psi_i(x)=\langle x|\psi_i\rangle$ and $H|\psi_i\rangle=E_i|\psi_i\rangle$. – Charlie Oct 21 '23 at 17:31
  • @Charlie You are right, I clarified the answer. – Thomas Fritsch Oct 21 '23 at 17:52
  • Thank you for your answer, if the wavefunction was initially already an eigenfunction of the Hamiltonian, would measuring the energy of the particle still change the wave function in this case? – Lalo Oct 21 '23 at 17:56
  • @Lalo No, in this case $\psi(x)$ does not change. It is equal to $\psi_i(x)$ before and after the measurement. – Thomas Fritsch Oct 21 '23 at 17:58