How do I expand out ψ0 in this case? Watch
The eigenstates are all orthogonal so you can just take the inner product of the wavefunction with each state vector in turn to find the coefficient c_n (with the extra normalization factor tagged on). Since we're working with an infinite potential well here, the eigenstates are sine functions so this is just like a Fourier sine series if you've covered that in maths.
It's just a straight line, quantum mechanics tells us that depending on the potential there are some points on the line where you can NEVER find the particle (in this case the infinite potential walls mean the particle is confined inside a 1D "box", i.e. a small section of this infinite line)
Hmmm, this is what I initially thought as well but would we not need to take into account that the wavefunction is zero outside the potential well? Otherwise the question becomes fairly trivial just by considering the hint and comparing coefficients as you suggested (cf. finding the Fourier series of a function that is sin(x) on the interval [0,L] but zero outside this interval)
Hmmm, this is what I initially thought as well but would we not need to take into account that the wavefunction is zero outside the potential well? …
Even with my two engineering degrees I still feel like Penny to your Sheldon.
I am not sure about your question. The problem (itself) mentions that “Its normalized energy eigenstates un and eigenvalues En are…” (as far as I can recall from a decade ago memory) it already implies that the wavefunction is zero outside the potential well. You can only normalize a wavefunction based on given boundary condition(s). Again I may be wrong.
What is considered fairly trivial by experts or well-learnt students, it may not be for a beginner.