Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linearity of Schrodinger Equation

  1. Oct 20, 2006 #1
    Could someone please address this question?
    How do you algebraically demonstrate the superposition principle revealed by the Schrodinger equation (ie. If Psi1(x,t) and Psi2(x,t) are both solutions then Psi(x,t)= Psi1(x,t)+Psi2(x,t) is also a solution.)?
     
  2. jcsd
  3. Oct 20, 2006 #2

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What do you get when you substitute Psi(x,t) into the left side of the Schrodinger equation?
     
  4. Oct 20, 2006 #3
    I am not quite sure what you are asking me to do...Psi(x,t) is already in there?
     
  5. Oct 20, 2006 #4
    No, [tex]\psi(x,t)[/tex] is not in there. However, you do know that [tex]\psi_1(x,t)[/tex] is a solution -

    [tex]-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi_1(x,t)+V(x)\psi_1(x,t)=i \hbar \frac{d}{dt}\psi_1(x,t)[/tex]


    And [tex]\psi_2(x,t)[/tex] is also in there:

    [tex]-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi_2(x,t)+V(x)\psi_2(x,t)=i \hbar \frac{d}{dt}\psi_2(x,t)[/tex]


    However, you have to show that [tex]\psi=\psi_1+\psi_2[/tex] also satisfies this dynamic equation.

    And then you can extend your result to [tex]a\psi_1 + b\psi_2[/tex] for any coefficents a,b.
     
    Last edited: Oct 20, 2006
  6. Oct 20, 2006 #5

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You have to show that

    [tex]
    -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \Psi(x,t) + V(x) \Psi(x,t) = i \hbar \frac{\partial}{\partial t} \Psi (x,t).
    [/tex]

    In the left, substitute [itex]\Psi = \Psi_1 + \Psi_2[/itex], and, using that [itex]\Psi_1[/itex] and [itex]\Psi_2[/itex] both statisfy Schrodinger's equation, work your way to the right side of the equation you must show true.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Linearity of Schrodinger Equation
  1. Schrodinger equation? (Replies: 8)

  2. Schrodinger equation (Replies: 1)

  3. Schrodinger equation (Replies: 4)

Loading...