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I was given a wave equation.
I know that probability amplitude is the eigenvalue of an observable operating in a state.
$$H| \psi\rangle = h| \psi\rangle$$
where $h$ is the probability amplitude of $H$ on state $|\psi\rangle$.
The only thing I know about expectation value is it should be written as
$$\langle H \rangle,$$
but as far as I can understand, and correct me if I'm wrong, aren't they the same thing?
There are different eigenvalues $h_n$ with different probabilities (lets call them $\phi_n$).
Your expectation value is then $\sum_n h_n\cdot \phi_n$
If you have only one eigenvalue it's obviously identical to the expectation value
In what you wrote above $h$ is not the probability amplitude and for most states the equation you wrote: $$ H|\psi\rangle=h|\psi\rangle $$ It is only true if $|\psi\rangle$ happens to be an eigenstate of the Hamiltonian.
$H$ can be written in the form $$ H = \sum_j h_j|j\rangle\langle j| $$ where the $j$s are just numbers that label each possible outcome of a measurement, they could be $1,2,3,4\dots$ or $\dots -1,-2,-3,0,1,2,3\dots$ or soemthing else depending on how it is helpful to label the outcomes. The $h_j$ are the set of possible values of $H$ and they are not limited to be between 0 and 1, unlike probabilities.
The probability amplitude of the $j$th outcome is $\phi_j = \langle j|\psi\rangle$ and has a square magnitude between 0 and 1. By contrast $$ H|\psi\rangle=\sum_j h_j \phi_j |j\rangle\langle j| $$ and $h_j \phi_j$ need not have an amplitude between 0 and 1. $h_j$ could be 1,000,000 and $\phi_j = i/2$ in which case their product doesn't have a square amplitude between 0 and 1.
The expectation value is $$ \langle H\rangle=\langle\psi|H|\psi\rangle=\sum_j h_j |\phi_j|^2. $$ Each term in that sum is the product of the probability for that outcome and the value of that outcome.
You might want to read a book about quantum mechanics like Christopher Isham's "Lectures on Quantum Theory: Mathematical and Structural Foundations".
$\psi\psi^* $ is the probability amplitude according to the Copenhagen interpretation, which is the most commonly accepted interpretation of the wave function.
H is the Hamiltonian.
h is usually the symbol for Planck's constant and would be the eigenvalue in the equation of the question.
