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Hamiltonian problem concerning the simple harmonic oscillator

  1. Aug 30, 2009 #1
    1. The problem statement, all variables and given/known data

    use the hamiltonian equation H=H_x+H_y+H_z to show that wave functions of the form
    [tex]\varphi[/tex](r)=[tex]\phi[/tex]i(x)[tex]\phi[/tex]j(y)[tex]\phi[/tex]k(z)

    where the functions phi_i(x) are the energy eigenfunctions for a 1-d SHM , satisfy H*phi=E*phi , and find the followed values of E for the 3-d oscillator


    2. Relevant equations


    H=-h-bar^2/2m*del^2+1/2*K*r^2, k is a constant

    r^2=x^2+y^2+z^2

    del^2=d^2/dx^2+d^2/dy^2+d^2/dz^2
    3. The attempt at a solution

    H=d^2/dx^2(-h-bar^2/2m)+1/2*k*x^2+1. The problem statement, all variables and given/known data

    use the hamiltonian equation H=H_x+H_y+H_z to show that wave functions of the form
    [tex]\varphi[/tex](r)=[tex]\phi[/tex][SUBi[/SUB](x)[tex]\phi[/tex]j(y)[tex]\phi[/tex]k(z)

    where the functions phi_i(x) are the energy eigenfunctions for a 1-d SHM , satisfy H*phi=E*phi , and find the followed values of E for the 3-d oscillator


    2. Relevant equations


    H=-h-bar^2/2m*del^2+1/2*K*r^2, k is a constant

    r^2=x^2+y^2+z^2

    del^2=d^2/dx^2+d^2/dy^2+d^2/dz^2
    3. The attempt at a solution

    H=d^2/dx^2(-h-bar^2/2m)+1/2*k*x^2+1. The problem statement, all variables and given/known data

    use the hamiltonian equation H=H_x+H_y+H_z to show that wave functions of the form
    [tex]\varphi[/tex](r)=[tex]\phi[/tex][SUBi[/SUB](x)[tex]\phi[/tex]j(y)[tex]\phi[/tex]k(z)

    where the functions phi_i(x) are the energy eigenfunctions for a 1-d SHM , satisfy H*phi=E*phi , and find the followed values of E for the 3-d oscillator


    2. Relevant equations


    H=-h-bar^2/2m*del^2+1/2*K*r^2, k is a constant

    r^2=x^2+y^2+z^2

    del^2=d^2/dx^2+d^2/dy^2+d^2/dz^2
    3. The attempt at a solution

    H=d^2/dx^2(-h-bar^2/2m)+1/2*k*x^2+d^2/dy^2(-h-bar^2/2m)+1/2*k*y^2+d^2/dz^2*(h-bar^2/2m)+1/2*k*z^2

    not sure how to proceed with my solution but I am sure the equation i*h-bar*dphi/dt=H*ohi will play a role in helping me form my final solution
     
  2. jcsd
  3. Aug 30, 2009 #2

    kuruman

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    The problem is asking you to solve the 3-D Schrodinger equation by using the method of separation of variables. Are you familiar with this method?
     
  4. Aug 30, 2009 #3
    Somewhat, the second derivative are already explicitly given in the equation for the hamiltonian. you would not take the second derivative of phi with respect to x, y, and z and then plug them into the hamiltonian expression would you?
     
  5. Aug 30, 2009 #4

    kuruman

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    Yes I would. Then I would divide by φ(r) and see what I get.
     
  6. Aug 30, 2009 #5
    yeah, but phi is not given, only my hamiltononian equation is given and I have to show that the equation [tex]\varphi[/tex](r)=[tex]\phi[/tex]i(x)[tex]\phi[/tex]j(y)[tex]\phi[/tex]k(z)
     
  7. Aug 30, 2009 #6

    kuruman

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    φ(x,y,z) is given. You are told it is the product of harmonic oscillator solutions to the one-dimensional Schrodinger equation. Each factor in the product is a function of a single independent variable, x, y or z.
     
  8. Aug 30, 2009 #7

    gabbagabbahey

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    I don't think the problem requires him to actually solve the 3D-Schrödinger equation, only show that as long as [itex]\phi_i(x)[/itex], [itex]\phi_j(y)[/itex] and [itex]\phi_k(z)[/itex] satisfy the corresponding one-dimensional Schrödinger equations, [itex]\varphi(\textbf{r})=\phi_i(x)\phi_j(y)\phi_k(z)[/itex] will satisfy the 3D equation.

    @noblegas Just substitute [itex]\varphi(\textbf{r})[/itex] into the 3D Schrödinger equation and carry out the derivatives...what do you get?
     
  9. Aug 30, 2009 #8
    you are right.

    I got my final solution to be:

    -h-bar^2/2m*((phi_i(x)''/phi_i(x))+(phj_j(y)''/phj_j(y))+(phk_k(z)''/phk_k(z))+1/2*k*(x^2+y^2+z^2)=E
     
  10. Aug 31, 2009 #9

    kuruman

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    Correct. Now look at the piece of this equation that depends only on x. This is

    [tex]-\frac{\hbar ^{2}}{2m}\left(\frac{1}{\phi _{i}(x)}\frac{\partial ^{2} \phi_{i}(x)}{\partial x^{2}}\right )+\frac{1}{2}k x^{2}[/tex]

    Then look at the one-dimensional Schrodinger equation in x. What can the expression above be replaced with? Do the same for y and z.
     
  11. Aug 31, 2009 #10


    [tex]-\frac{\hbar ^{2}}{2m}\left(\frac{1}{\phi _{i}(x)}\frac{\partial ^{2} \phi_{i}(x)}{\partial x^{2}}\right )+\frac{1}{2}k x^{2}[/tex]=[tex]E[/tex]? Sorry about my latex
     
  12. Aug 31, 2009 #11

    kuruman

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    Which E? It can't be the same as the E in the original equation because that is the sum of three such terms. Better call it E1. Now do the same for y and z and put it together.
     
  13. Aug 31, 2009 #12
    I think it would be appropriate to call this particular expression E_x since the derivatives of phi are taking with respect to x.
     
  14. Aug 31, 2009 #13

    kuruman

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    Correct. You know from having solved the one-dimensional problem what the allowed values for Ex are. You should write them in terms of quantum number nx. Do the same for the other two directions, put it back in the 3-D equation and you should end up with an expression for the total energy E for the 3-D oscillator.
     
  15. Aug 31, 2009 #14
    Should I apply the Energy equation for the simple harmonic oscillator E=(1/2+n)*[tex]\hbar[/tex]*[tex]\varpi[/tex]
     
  16. Aug 31, 2009 #15

    kuruman

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