How do I expand out ψ0 in this case? Watch

username970964
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This is what I've done so far.
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Eimmanuel
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(Original post by Airess3)
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This is what I've done so far.
What have you really done other than posting the question?
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silentshadows
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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.
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Eimmanuel
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(Original post by silentshadows)
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.
Taking the inner product of the wavefunction may be a “more rigorous way” of finding cn, (IMO) we can just compare the coefficients.
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RickoNctd
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Is that A level physics ?
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Eimmanuel
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(Original post by RickoNctd)
Is that A level physics ?
No, it is undergraduate Quantum Mechanics question/problem.
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silentshadows
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(Original post by Eimmanuel)
Taking the inner product of the wavefunction may be a “more rigorous way” of finding cn, (IMO) we can just compare the coefficients.
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)
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RickoNctd
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(Original post by Eimmanuel)
No, it is undergraduate Quantum Mechanics question/problem.
First year ?
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silentshadows
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(Original post by RickoNctd)
First year ?
It depends where you're studying but personally we only covered this topic in second year (although a lot of the maths for Fourier series was done in first year)
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Duncan2012
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I'm still trying to work out what a 1-dimensional box looks like
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silentshadows
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(Original post by Duncan2012)
I'm still trying to work out what a 1-dimensional box looks like
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)
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Duncan2012
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(Original post by silentshadows)
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)
Even with my two engineering degrees I still feel like Penny to your Sheldon.
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Eimmanuel
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(Original post by silentshadows)
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)

(Original post by silentshadows)
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? …
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.

(Original post by silentshadows)
… Otherwise the question becomes fairly trivial just by considering the hint and comparing coefficients as you suggested …
What is considered fairly trivial by experts or well-learnt students, it may not be for a beginner.
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silentshadows
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(Original post by Duncan2012)
Even with my two engineering degrees I still feel like Penny to your Sheldon.
There's still lots of topics I find difficult, you just happened to catch me at a good time because we've just gone over this topic in class lol

(Original post by Eimmanuel)
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.
That makes sense, looking back at my notes we start with the Schrodinger equation and solve to obtain [tex] \psi (x) = A \sin (kx) + B \cos (kx) [\tex] where the constants and k are determined from the fact that the wave function must be continuous at the edges, so comparing coefficients should work fine as you initially suggested. Apologies if that last part came across as dismissive or rude, this is just the first part so I'm guessing there's more to it
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username970964
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(Original post by Eimmanuel)
What have you really done other than posting the question?
I've set out En and un but for some reason I couldn't attach the image of my working onto this website.
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