1. Not finding help here? Sign up for a free 30min 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!

Schrodinger equation problem

  1. Oct 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the time-dependent Schrodinger equation for a free particle in two spatial dimensions
    1zwnbds.jpg
    Using the method of separation of variables, determine the wave function ψ(x,y,t)

    2. Relevant equations


    3. The attempt at a solution
    Not sure how to do the separation here since it is a function of 3 variables, I'm inclined to think that Ψ(x,y,t) = f(x) g(y) j(t) can be assumed, hence substitute it in the equation and separate. Is it correct to assume that?
     
  2. jcsd
  3. Oct 25, 2015 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes. See what you get.
     
  4. Oct 25, 2015 #3

    I would get this
    b7nwgl.png
    Then what? I can't separate each variable to each side, since there are 3 variables
    Correction: in the LHS the g(t) should be g(y)
     
  5. Oct 25, 2015 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    The next step is standard: divide the equation by Ψ(x, y, t) = f(x)g(y)j(t).
     
  6. Oct 25, 2015 #5
    Ok ok. Then I get this
    2mchzfb.png
    Then what do I do with it?
     
  7. Oct 25, 2015 #6

    TSny

    User Avatar
    Homework Helper
    Gold Member

  8. Oct 25, 2015 #7
    Yes, but I'm new to it. So I believe I should set the whole equation equal to a variable lambda, then the LHS would be
    24mds9f.png
    And do the same for the RHS where I would get two other separate equations, one in terms of f and one in terms of g? or would it just give:
    2jfnehg.png ?
     
    Last edited: Oct 25, 2015
  9. Oct 25, 2015 #8

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, except that you should think of ##\lambda## as a constant rather than a variable. To see this, note that your equation in post #5 must be satisfied for all allowed values of t, x, and y. So, suppose you arbitrarily pick a value for x and y. The right hand side is then some number. The left hand side is a function of time that must equal that number for any value of t. So, in fact, the left side must be independent of t. That is, the left side must be a constant which you can call ##-\lambda##.

    See if you can construct similar arguments to show that the individual terms on the right side of the equation in post #5 must each be equal to a constant (although not necessarily the same constant.)
     
  10. Oct 25, 2015 #9
    Ok, then the terms in the RHS would be e16bdw.png
    Where lambda is just an arbitrary constant
    Is the problem finished like this?
     
    Last edited: Oct 25, 2015
  11. Oct 25, 2015 #10

    TSny

    User Avatar
    Homework Helper
    Gold Member

    OK. But, you should distinguish the three constants by not using the same symbol ##-\lambda## for each one. Also, there is no need to write the constants with a negative sign. In the Wikipedia link, they wrote the constant with a negative sign for later convenience for the particular partial differential equation there. In your case, you will discover whether the constants should be positive or negative as you solve the individual equations.

    The three constants are not completely arbitrary at this point. The equation in post #5 gives you a relation that the three constants must satisfy.
     
  12. Oct 25, 2015 #11
    ħ
    Ok, then I can say
    hw0jdt.png
    And so
    33vzpso.png
    And same for the LHS of the equation in #5
    16axxdf.png
     
  13. Oct 25, 2015 #12

    TSny

    User Avatar
    Homework Helper
    Gold Member

    OK. These are standard differential equations which you can solve. How is the constant ##\lambda## related to the constants ##B## and ##C##? (Use the equation in post #5)
     
  14. Oct 25, 2015 #13
    From the equation, λ is related to B and C as
    λ = B + C
     
    Last edited: Oct 25, 2015
  15. Oct 25, 2015 #14
    Then where do I go from there?
     
  16. Oct 25, 2015 #15

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes. So, λ will be determined once the values of B and C are known.
     
  17. Oct 25, 2015 #16

    TSny

    User Avatar
    Homework Helper
    Gold Member

    You have 3 ordinary differential equations: one for j(t), one for f(x), and one for g(y). Try to solve them.
     
  18. Oct 25, 2015 #17
    ο
    I get really complicated stuff. For example for f(x), I multiply the equation by dx^2 and divide by f(x), then integrate with respect to f, and the rest of the terms are constants, the integration gives f(ln(f)-1). Am I doing something wrong here?
     
    Last edited: Oct 25, 2015
  19. Oct 25, 2015 #18

    TSny

    User Avatar
    Homework Helper
    Gold Member

    You can't multiply by dx2. You can't "break apart" the second derivative that way.

    Simplify the equation by lumping all the constants together. For example, suppose you let ##k_x## be a constant such that ##k_x^2 = \frac{2mB}{\hbar^2}##. Write the differential equation in terms of ##k_x##. You will see that you get a familiar type of ordinary differential equation.
     
  20. Oct 25, 2015 #19
    I need to take the square root to get the of the second power on df^2/dx^2 ?
     
  21. Oct 26, 2015 #20

    TSny

    User Avatar
    Homework Helper
    Gold Member

    :bugeye: No.

    Write the differential equation as ##f''(x) = -k_x^2f(x)##, where the constant ##k_x## was defined in post #18. You're looking for a function ##f(x)## such that when you take its second derivative you get back the function multiplied by a negative constant.

    Are you familiar with the form of the wavefunction for a free particle moving in one dimension? You might expect a similar form for ##f(x)##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Schrodinger equation problem
  1. Schrodinger equation? (Replies: 2)

  2. Schrodinger equation (Replies: 7)

  3. Schrodinger equation (Replies: 5)

Loading...