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Basic Infinite Potential Well, Tunnelling, etc.

  1. Nov 24, 2009 #1
    I have an exam later today, and I would really apreciate it if someone could check my work or at least point out flaws in my process here. Thanks!


    1. The problem statement, all variables and given/known data
    At x=0, a proton with a kinetic energy of 10 eV is travelling in the x direction (potential energy = 0). At x=1nm, it encounters a potential barrier of height 12eV and width 0.2 nm. The potential returns to 0 at 1.2nm.

    Calculate the transmission and reflection probabilities


    2. Relevant equations

    T = [1 + [V2sinh2(kL)] / [4E (V - E)]

    R = 1 - T

    where V = 12, E = 10, L = 2*10-10,
    and k = [tex]\sqrt{}2m (V-E)[/tex]/hbar

    3. The attempt at a solution

    well its all straightforward, and I got k = 7.24 * 109,
    which means kL is 1.45.

    when I plug everything in, I get T = .12 and R = .88

    I've checked it 3 times and it comes out the same every time, but this seems like a very high transmission probability, so I dont know if i made a mistake somewhere.



    1. The problem statement, all variables and given/known data
    An electron is in a 1D potential well approximated by V = 0 for 0nm < x < 2nm and V = infinity for all other x

    what is the electron wave function for this lowest energy state?

    what is the energy of the electron in this state


    2. Relevant equations
    shroedinger wave equation was used to derive this wave function


    3. The attempt at a solution

    well I got [tex]\Psi[/tex] = Asin(kx), where kL = n[tex]\pi[/tex], so
    [tex]\Psi[/tex] = Asin(n[tex]\pi[/tex]x / L)

    when i normalize this, i get A = [tex]\sqrt{}2/L[/tex], so

    [tex]\Psi[/tex] = [tex]\sqrt{}2/L[/tex]sin(n[tex]\pi[/tex]x / L)

    so I plug in L and i end up with a numerical equation of
    [tex]\Psi[/tex] = 31623sin (5[tex]\pi[/tex] * 108 x)

    this doesnt seem right to me.

    for the energy i just plug the values into the equation:

    E = h2n2[tex]\pi[/tex]2 / 2mL2

    and i get .094 eV, which also seems wrong.
     
    Last edited: Nov 24, 2009
  2. jcsd
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