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Particle moving in potential

  1. Sep 26, 2013 #1
    1. The problem statement, all variables and given/known data
    The wave function for a particle of mass m moving in the potential
    V =
    { ∞ for x=0
    { 0 for 0 < x ≤ a
    { V0 for x ≥ a

    is
    ψ(x)
    { Asin(kx) for 0 < x ≤ a
    { Ce-Kx for x ≥ a

    with

    k = √[(2mE/h2)]

    K =√[((2m(V0-E)/h2)]

    where h is h-bar in both equations, and remains so throughout this thread.

    (a) Apply the boundary conditions at x = a and obtain the transcendental equation which determines the bound state energies E.
    (b) If
    √[(2mV0a2)/h2] = 3pi

    determine the allowed bound state energies. Express your answers in the form of a numerical factor multiplying the dimensional factor (h2/(2ma2)).

    (c) Given that the normalization factor for the wave function corresponding to the nth energy En is

    An = √[2/a]*√[(Kna)/(1+(Kn*a))]

    normalize the wave function for the lowest energy state.
    (d) What is the probability that a measurement of the position of a particle in the ground state will give a result ≥ a?
    2. Relevant equations



    3. The attempt at a solution
    I have completed parts (a) and (b) and I believe they are correct, here is a brief outline of what I did.
    At x = a: ψ(a)

    Asin(ka) = Ce-Ka (1)

    dψ/dx

    kAcos(kx) = -KCe-Ka (2)

    Dividing (2) by (1):

    kcot(ka) = -K
    For the transcendental: z = ka, z0 = a√[(2mV0)/h2]

    ∴ -cot(z) = √[(z0/z)2-1]

    Now setting z0 = 3pi (part b) and graphing -cot(z) and √[(z0/z)2-1] on the same plot, I find that intersections occur just below zn = npi

    z = ka = a√[(2mE/h2)] ≈ npi

    ∴ Solving for En = (n2pi2h2)/(2ma2)

    This is where I hit a road block. I believe the lowest energy state is E1 because it is just below pi (or 1pi), but I'm not sure what I'm supposed to do with this. I want to plug this into Kn, but that creates quite the mess. As for the second portion of ψ(x), I can just normalize that and solve for C using Kn, and I think I have a handle on part (d) as well. Any hints on part (c) or just checking my work so far would be much appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 26, 2013 #2

    Simon Bridge

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    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You mean it is the first non-zero energy eigenstate? ##\small E_0=0## and all.

    You have the equations for the wavefunctions, you have decided that n=1 is the lowest one, so the lowest energy wavefunction must be ##\psi_1## - which you found an expression for in part a and b.

    Part c asks for the normalized wavefunction - which requires the normalization factor A1: which you have an equation for.

    What's the problem?

    You could always check by normalizing the hard way.
     
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