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QM: Splitting up the Hamiltonian

  1. Feb 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    I have a Hamiltonian given by:

    [tex]
    H = H_x + H_y = -\frac{\hbar^2}{2m}(d^2/dx^2 + d^2/dy^2).
    [/tex]

    Now I have a stationary state on the form [itex]\psi(x,y)=f(x)g(y)[/itex]. According to my teacher, then the Hamiltonian can be split up, i.e. we have the two equations:

    [tex]
    H_x f(x) = E_xf(x) \qquad \text{and}\qquad H_y g(y)=E_yg(y).
    [/tex]

    I can't see why this must be true. Inserting in the time-independent Schrödinger-equation doesn't give me these expressions. What am I missing here?

    Thanks in advance.

    Best regards,
    Niles
     
  2. jcsd
  3. Feb 12, 2009 #2
    If you put it in the time-indep Schrodinger equation, and then if you divide both sides by f(x)g(x), and then if you take one of the two terms on the left hand side to the right hand side, you get : a LHS that is a function of x only, and RHS that's function of y only. This can only be true if both sides are equal to some constant. Try to take it from there.
     
  4. Feb 12, 2009 #3
    Ahh, yes. I see.

    Thanks for that.
     
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