# Particle-in-a-box eigenstates

Problem: The particle in a 1D box [0, a]

Eqs.: The general solution of the time-independent Schrödinger eq. may be written as ψ(x) = Acos(kx) + Bsin(kx), E = ħ2k2/2m. Imposing the boundary conditions ψ(0) = ψ(a) = 0 , we get immediately A = 0, ka = nπ (for any positive integer n). Using x' = x - a/2 , plus the boundary condition and elementary trigonometry, we also get ψ(x) = Bsin(kx) = Bcos(kx') , same eigenvalue. Everything is nice and dandy up to now, but, if I start from the box [-a/2, a/2] and impose boundary conditions ψ(-a/2) = ψ(a/2) = 0 , I seem to be getting B = 0, ka = nπ (but this time for odd n only)

What's the reason for this inconsistency? Either in the first case we should take only odd n also, and nobody noticed it in QM textbooks (unlikely), or I'm making some silly mistake, but I can't point my finger where; I've checked the sol. to the boundary condition system many times, it seems fine to me.

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Orodruin
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What's the reason for this inconsistency?
You are not doing the maths correctly.

You are not doing the maths correctly.
I figured as much; though

ψ(-a/2) = Acos(-ka/2) + Bsin(-ka/2) = Acos(ka/2) - Bsin(ka/2) = 0
ψ(a/2) = Acos(ka/2) + Bsin(ka/2) = 0

=> B = 0 (because symmetry, Fourier series, however you prefer) , cos(ka/2) = 0 <=> ka/2 = (2m+1)π/2

Like I said, it's some silly mistake; but whatever it is, it's in the 3 lines above.

vela
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I figured as much; though
I'm not sure what more of a response you could have gotten considering all you told us was that you got the wrong answer.

ψ(-a/2) = Acos(-ka/2) + Bsin(-ka/2) = Acos(ka/2) - Bsin(ka/2) = 0
ψ(a/2) = Acos(ka/2) + Bsin(ka/2) = 0

=> B = 0 (because symmetry, Fourier series, however you prefer) , cos(ka/2) = 0 <=> ka/2 = (2m+1)π/2

Like I said, it's some silly mistake; but whatever it is, it's in the 3 lines above.
Your mistake is in concluding B=0. If you recast the two equations in matrix notation, you have
$$\begin{bmatrix} \cos \frac{ka}{2} & -\sin \frac{ka}{2} \\ \cos \frac{ka}{2} & \sin \frac{ka}{2} \end{bmatrix} \begin{bmatrix} A \\ B \end{bmatrix} = 0.$$ To get a non-trivial solution, the determinant of the matrix has to vanish. What does that give you?

itssilva
Orodruin
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because symmetry
This is wrong. If the potential is symmetric, eigenstates are symmetric or anti-symmetric.

Your mistake is in concluding B=0. If you recast the two equations in matrix notation, you have
$$\begin{bmatrix} \cos \frac{ka}{2} & -\sin \frac{ka}{2} \\ \cos \frac{ka}{2} & \sin \frac{ka}{2} \end{bmatrix} \begin{bmatrix} A \\ B \end{bmatrix} = 0.$$ To get a non-trivial solution, the determinant of the matrix has to vanish. What does that give you?
Oh, I see; indeed, that was naive of me. Sorry, guys; I'm being way over my head for stuff that's supposed to be simple.

vela
Staff Emeritus