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!

Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the ODE

  1. Nov 24, 2012 #1

    fluidistic

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    I must get the first eigenvalues of the time independent Schrödinger's equation for a particle of mass m inside a cylinder of height h and radius a where ##h \sim a##.
    The boundary conditions are that psi is worth 0 everywhere on the surface of the cylinder.

    2. Relevant equations
    ##-\frac{\hbar ^2}{2m} \triangle \psi =E \psi##.
    Laplacian in cylindrical coordinates.

    3. The attempt at a solution
    I've used separation of variables on the PDE, seeking for the solutions of the form ##\psi (\rho, \theta , z)=R(\rho) \Theta (\theta ) Z(z)##.
    I reached that ##\frac{Z''}{Z}=\text{constant}=-\lambda ^2##. Assuming that the Z function is periodic and worth 0 at the top and bottom of the cylinder, I reached that ##Z(z)=B \sin \left ( \frac{n\pi n}{h} \right )## where n=1,2, 3, etc.
    Then I reached that ##\frac{\Theta''}{\Theta} = -m^2## where m=0, 1, 2, etc (because it must be periodic with period 2 pi). So that ##\Theta (\theta )=C \cos (m \theta ) +D \sin (m \theta)##.
    Then the last ODE remaining to solve is ##\rho ^2 R''+\rho R'+R \{ \rho ^2 \left [ \frac{2mE}{\hbar ^2} - \left ( \frac{n\pi}{h} \right )^2 \right ] -m^2 \}=0##. This is where I'm stuck.
    It's very similar to a Bessel equation and Cauchy-Euler equation but I don't think it is either. So I don't really know how to tackle that ODE. Any idea? Wolfram alpha does not seem to solve it either: http://www.wolframalpha.com/input/?i=x^2y%27%27%2Bxy%27%2By%28x^2*k-n^2%29%3D0.
     
  2. jcsd
  3. Nov 24, 2012 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Look's like Bessel's equation. See http://www.efunda.com/math/bessel/bessel.cfm

    Of course, you'll need to rescale ##\rho## to simplify the expression inside your { }.
     
  4. Nov 24, 2012 #3

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Hmm ok.
    Hmm what do you mean exactly? I have an equation of the form ##\rho ^2 R''+\rho R' +R(\rho ^2 p^2 -m^2)## where p is a constant for a given n.
    Rescaling rho means to get ##p^2=1##?
     
  5. Nov 24, 2012 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Yes. Define a new independent variable (##x##, say) in terms of ##\rho## such that you get the standard form of Bessel's equation.
     
  6. Nov 25, 2012 #5

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    I try ##x=\rho p## so ##\rho =x/p## but then the ODE changes to ##\frac{x^2R''}{p^2}+\frac{xR'}{p}+R(x^2-m^2)=0##. I could multiply by ##p^2## but I would not get the standard form of the Bessel equation. I don't see how I could rescale the factor in front of rho ^2 without rescaling the coefficients in front of R'' and R.
     
  7. Nov 25, 2012 #6

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    You need to take care of the rescaling in the derivatives, too. For example, ##dR/d\rho = \left(dR/dx\right)\left(dx/d\rho\right)##
     
  8. Nov 25, 2012 #7

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Oh right, I totally missed this!
    So indeed now I recognize a Bessel equation!!!
    Therefore I get that the solutions of the form ##\psi =R ( \rho ) \Theta (\theta ) Z(z)=B_n \sin \left ( \frac{n\pi z}{h} \right ) [C_m \cos (m\theta ) + D_m \sin (m \theta )]J_m \left ( \rho \sqrt {\frac{2mE}{\hbar ^2} - \frac{n^2 \pi ^2}{h^2}} \right )##.
    So the solution that satisfies the boundary condition is a linear combination of those.
    I'm not 100% sure about what they mean by "eigenvalues". Eigenfrequencies? Lowest energies possible? (They only want the first 3 eigenvalues).
    I'm pretty sure this will concern the cases (1) n=1 and m=0 and m=1. (2) n=2, m=0. But I'm not sure what they are asking me.
     
  9. Nov 25, 2012 #8

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    I think they want the three lowest energies. They are called eigenvalues because they are eigenvalues of the time independent Schrodinger equation ##H|\psi> = E|\psi>##
     
  10. Nov 25, 2012 #9

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    You don't seem to have used the boundary condition at ρ=a.
     
  11. Nov 25, 2012 #10

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Ok thanks!
    Hmm I don't know how to get that information.
    I have a feeling I should add a subscript "n" under "E" in ##\psi =R ( \rho ) \Theta (\theta ) Z(z)=B_n \sin \left ( \frac{n\pi z}{h} \right ) [C_m \cos (m\theta ) + D_m \sin (m \theta )]J_m \left ( \rho \sqrt {\frac{2mE}{\hbar ^2} - \frac{n^2 \pi ^2}{h^2}} \right )## and then isolate ##E_n## but not sure to what I should equate the equation.
     
  12. Nov 25, 2012 #11

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Follow haruspex's lead.
     
  13. Nov 25, 2012 #12

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Oh right guys sorry. And thanks for helping. I did not see haruspex's post.
    So if ##x_p## is the p'th zero of the Bessel function then ##E_n= \left ( \frac{\hbar ^2}{2m} \right ) \left [ \left ( \frac{x_p}{a} \right ) ^2 +\left ( \frac{n^2 \pi ^2}{h^2} \right ) \right ]##. I guess I'll have to check if I can replace "p" by "n". It's not obvious to me at a first glance.
     
  14. Nov 25, 2012 #13

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Ok I've thought a bit on this. The first 3 lowest energy values are when ##x_p=x_0##. So ##E_1= \frac{\hbar ^2}{2m} \left [ \left ( \frac{x_0}{a} \right ) ^2 + \frac{\pi ^2}{h^2} \right ]##, ##E_2= \frac{\hbar ^2}{2m} \left [ \left ( \frac{x_0}{a} \right ) ^2 + \frac{4\pi ^2}{h^2} \right ]## and ##E_3= \frac{\hbar ^2}{2m} \left [ \left ( \frac{x_0}{a} \right ) ^2 + \frac{9\pi ^2}{h^2} \right ]##.
    I'm not very confident because I don't know if ##\frac{\hbar ^2 }{2m} \left [ \left ( \frac{x_1}{a} \right ) ^2 + \frac{\pi ^2}{h^2} \right ] <\frac{\hbar ^2}{2m} \left [ \left ( \frac{x_0}{a} \right ) ^2 + \frac{9\pi ^2}{h^2} \right ]## for example.
     
  15. Nov 25, 2012 #14

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    You'll need to consult a Table of Roots

    The problem states that the height of the cylinder is approx. equal to the radius: [itex]h\approx a[/itex], which should help figure out the lowest three energies.
     
    Last edited: Nov 25, 2012
  16. Nov 25, 2012 #15

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    Great and thank you once more.
    I get from lower to upper: ##E_{1,0}##, ##E_{1,1}## and ##E_{2,0}## where the subscript are ##E_{n,p}##.
     
  17. Nov 25, 2012 #16

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    I think that might be correct.
     
  18. Nov 25, 2012 #17

    fluidistic

    User Avatar
    Gold Member

    Re: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the

    :approve:
    Thanks for all.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Getting the eigenvalues of the Schrödinger's equation in a cylinder. Stuck on the ODE
  1. Schrödinger equation (Replies: 2)

  2. Schrödinger equation (Replies: 3)

Loading...