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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.

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

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I agree with the comment from ZeroTheHero: The standard text book examples will help you. Yet, it is also reasonable to discuss the following (rather simple but useful) computation scheme:

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: $$ \sum\limits_n H_{mn}\, c_n = E \,c_m \quad . $$

All in all, for the given case, we just have to choose a basis from which we build our Hamiltonian matrix and then have to diagonalize this matrix (numerically) in order to find the eigenvalues and eigenvectors. The elements of the Hamiltonian of course depend on the problem you are studying.

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In any practical case, you first need to choose some kind basis. Then the Schrödinger-equation (together with the boundary conditions) will define a dynamical problem, which will have a solution if you give the initial conditions.

Obviously there are mathematical nontrivialities. Using either the spatial coordinate eigenfunction-; or momentum basis is not quite trivial from a mathematical point of view, and there are initial conditions which are outside the range of the Hamiltonian.

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