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Position wave function of energy eigenstates in 1D box

  • #1

Homework Statement


Consider a particle which is confined in a one-dimensional box of size L, so that the position space wave function ψ(x) has to vanish at x = 0 and x = L. The energy operator is H = p2/2m + V (x), where the potential is V (x) = 0 for 0 < x < L, and V (x) = ∞ otherwise.

Find the position-space wave functions ψn(x) ≡ <x|ψn> of the energy eigenstates |ψn>. Make sure that your wave functions are normalized, so that <ψnn> = 1

Homework Equations



maybe

<x'|x|ψ> = x'<x'|ψ>
1 =

The Attempt at a Solution



I'm not 100% sure what the question is asking for but I'm guessing it's asking me to find the RHS of

ψn(x) ≡ <x|ψn>

I can write the RHS as a sum since energy eigenstates are discrete

<x|ψn> = <x|1|ψn> = ∑ψni2

I'm not sure if this is the right step but it's all I could come up with.
 

Answers and Replies

  • #2
Chandra Prayaga
Science Advisor
649
148
Are you familiar with the Schrodinger equation for such a system?
 
  • #3
Are you familiar with the Schrodinger equation for such a system?
Yes, for a mass moving in 1D, the Schrodinger equation gives

n(x) + Vψn(x) = Eψn(x)

H = Hamiltonian

So if I solve the equation for a 1D box I would get something like

ψn(x) = Asin(kx)+Bsin(kx)

Do I then solve for k,A, and B?
 
  • #4
I am unsure of what it means by position wave functions of energy eigenstates. Does it mean Solve for ψn(x) for the definite levels of energy levels?
 
  • #5
BvU
Science Advisor
Homework Helper
2019 Award
13,009
3,005
Yes. You have the solution, now apply the boundary conditions and normalization. Only specific ##k## do the trick and they can be numbered from n=0 to infinity.

(I assumed you wrote A sin .. + B cos .. ?)
 

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