1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Charge Density in Quantum well Sub-bands.

  1. Mar 9, 2016 #1
    1. The problem statement, all variables and given/known data
    I am trying to calculate the charge density in the first subband (n=1) of the quantum well of length L as shown in the below figure.
    here 4_zpslffcxddg.jpg is the electron wave function for the first sub-band and its value (from relevant equation 1) is given as
    Ψ12(x) = 2/L*sin2(πx/L)

    from Schrodinger equation total electron charge density is given by
    n(x) = N1×Ψ12(x)
    where N1 is the electron charge density in the first subband(n=1).

    hence for charge density at x = L/2 the above equation becomes
    n(L/2) = N1×Ψ12(L/2) = N1×2/L*sin2(π/2) --result 1

    But from wave function normalization,that is the probability of finding electron between 0 and L is one(from relevant equation 2)
    Ψ12(L/2) ≈1, then the total electron charge density at x=L/2 is n(L/2) = N1×1 --result 2

    2. Relevant equations

    wave-function for the electrons in the quantum well sub bands
    wave function normalization(that is the probability of finding electron between 0 and L is one)

    3. The attempt at a solution
    When i made attempts for the solutions i got two results result 1 and result 2 as above.
    Now my question is which is the correct result from these two..?
  2. jcsd
  3. Mar 9, 2016 #2
    I think the the wave function provide you the info about the probability of finding the charge(electron) in the one dimensional well in a certain state- so the density can be calculated by multiplying the charge with modulus square of the wave function between a point x and x+dx =that should be density at x- naturally as you show the nature of wave function its modulus square will be maximum at x= L/2 but the charge will not be always found at this point - as the total probability should be unity for a normalized wave function .
    and at other points between 0 and L there exists finite probability density.
    so you can not have two answers at a point-density should / will be unique.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted