# How do I find the wave function in a separable Hilbert Space?

I am confused as to how I would go about finding the wave function in the Hilbert Space. As I understand, a wavefunction in the Hilbert space can be represented as

$$|\Psi\rangle = \sum_{n} c_n|\psi_n\rangle$$

and the wave function can be calculated with $$i\frac{\partial |\psi\rangle}{\partial t} = H|\psi\rangle$$

I think I should use the second equation to try and find the wave function, then find a representation for it in terms of the orthonormal basis of the Hilbert Space. But I'm not exactly sure what that process would look like.

Also, I'm new to quantum mechanics and this site. If there's anything I need to do to improve my question or other resources I should look at, I'll gladly take any advice.

• it may be useful to look at elementary examples, like the particle-in-a-box or the harmonic oscillator. Jan 17, 2021 at 16:26

Consider a finite dimensional Hilbert space with a given orthonormal basis $$\{|n\rangle\}$$. We further restrict the discussion to time-independent systems, i.e. we aim to solve the time-independent Schrödinger equation: $$H \,|\Psi\rangle = E \, |\Psi\rangle \quad .$$
To do so, we can make use of the completeness relation $$\mathbb{1} = \sum\limits_n |n\rangle\langle n|$$ and multiply the equation from the left with $$\langle m|$$, which yields $$\sum\limits_n \langle m|H|n\rangle \, \langle n |\Psi\rangle = E\, \langle m|\Psi\rangle \quad .$$ If we expand the wave function in the $$n$$-basis, i.e. if we write $$|\Psi\rangle = \sum\limits_n c_n \, |n\rangle$$, then we find $$c_m = \langle m|\Psi\rangle$$. Moreover, we can interpret $$H_{mn} \equiv \langle m|H|n\rangle$$ as elements of a matrix. The above equation is nothing but the time-independent Schrödinger equation in matrix form: $$\sum\limits_n H_{mn}\, c_n = E \,c_m \quad .$$