quantum mechanics - Probability of energy from wavefunction

In general, given a wavefunction $\psi(x)\equiv\langle x\vert\psi\rangle$ for some system, how can one compute the probability that the system will be at a given energy level $E_n$? That is, how can one compute $\langle E_n\vert\psi\rangle$? I feel like this should be next to trivial, but the wavefunction is an expansion in the position basis and I would need the energy eigenbasis to perform the computation. Thus this reduces to a change of basis, but how is this done?

Note: In this system I have already solved the TISE and found the energy spectum.

Answer

The energy eigenstates can be expressed in the form of wavefunctions as well, e.g. $\psi_n(x) \equiv \langle x | E_n \rangle$. Then, you can compute the inner product of the two wavefunctions by integrating their product:

$$\langle E_n | \psi \rangle = \int_{-\infty}^\infty \langle E_n|x\rangle \langle x|\psi \rangle \, dx = \int_{-\infty}^\infty \psi_n^*(x) \, \psi(x) \, dx$$