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

A property of solutions

  1. Jun 23, 2014 #1

    ShayanJ

    User Avatar
    Gold Member

    The Schrodinger equation is of the form [itex] \frac{d^2 \psi}{dx^2}+[\varepsilon-v(x)] \psi=0[/itex].
    In a lecture, the lecturer said that if we have in a point [itex] x_0 [/itex] , [itex] \psi(x_0)=\psi'(x_0)=0 [/itex], then [itex] \psi(x)=0 [/itex].(for a smooth [itex]v(x)[/itex]!)
    Can anyone give a proof of this?
    Is it only for a equation of the form given above?
    Thanks
     
  2. jcsd
  3. Jun 24, 2014 #2
    I'm new to this but I'm assuming you're referring to the time-independent non-relativistic equation. Let [itex]\psi(x_0)[/itex] represent a potential value where kinetic is 0. Let [itex]\psi'(x_0)[/itex] represent kinetic value where potential is 0. Iff [itex]\psi(x_0)[/itex] and [itex]\psi'(x_0)[/itex] are two separable wave functions that subscribe to all applicable axioms of [itex]\psi(x)[/itex], [itex]\psi[/itex] will include all potential and kinetic components. A rough analogy could be drawn from binary code. Anyway I hope this helped steer you in the right direction if it did anything at all. Everything I wrote came from a purely logical standpoint.
     
  4. Jun 24, 2014 #3

    ShayanJ

    User Avatar
    Gold Member

    I don't understand what you mean by "potential value" and "kinetic value"!
    And...[itex] \psi(x_0) [/itex] and [itex] \psi'(x_0) [/itex] are two complex numbers. Not wave functions!!!
    What do you mean by binary code?
     
  5. Jun 24, 2014 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    This can be proved by an application of the basic "existence and uniqueness" theorem for differential equations: "If [itex]\phi(x,t)[/itex] is continuous in both x and t and differentiable with respect to x in some neighborhood of [itex](t_0, x_0)[/itex] then there exist a unique function x(t) satisfying [itex]dx/dt= \phi(x, t)[/itex] and [itex]x(t_0)= x_0[/itex]."

    That can be extended to second (and higher) order equation by writing [itex]d^2\phi/dt^2= F(t,\phi, \phi')[/itex] as [itex]dx/dt= F(x, t)[/itex] by taking x to be the vector function [itex]x= (x_1, x_2)= (\phi, d\phi/dt)[/itex]. Then [itex]d^2\phi/dt^2= dx_2/dt[/itex] and of course, [itex]dx_1/dt= d\phi/dt= x_2[/itex] so that the differential equation becomes the vector equation [itex](dx_1/dt, dx_2/dt)= (x_2, F(t, x_1, x_2))[/itex].

    The "initial value conditions", [itex]\phi(x_0)= 0[/itex] and [itex]\phi'(x_0)= 0[/itex] becomes the single vector condition [itex]x(0)= (x_1(0), x_2(0))= 0[/itex].

    Obviously [itex]\phi(t)= 0[/itex] is a solution to the given differential equation and obviously [itex]\phi(0)= \phi'(0)= 0[/itex]. The "existence and uniqueness theorem" tells us that this is the only solution.

    (This theorem does NOT apply to "boundary value conditions" since we cannot create the single vector condition. The differential equation [itex]d^2\phi/dt^2= -\phi[/itex] has NO solution that satisfies [itex]\phi(0)= 0[/itex], [itex]\phi(\pi/2)= 0[/itex] but has an infinite number of solutions that satisfy [itex]\phi(0)= 0[/itex], [itex]\phi(\pi)= 0[/itex].)
     
  6. Jun 25, 2014 #5

    ShayanJ

    User Avatar
    Gold Member

    I don't understand.
    The vector ODE you mentioned can also be written as [itex] \frac{d^2 x_1}{dt^2}=F(t,x_1,\frac{dx_1}{dt}) [/itex]. This is again a second order ODE and instead of helping us, just gets us back to the main question. I don't see a proof!!!
    Another problem is, the function [itex] y=x^3 [/itex] is the answer to the 2nd order ODE [itex] \frac{d^2 y}{dx^2}-6 \frac{y}{x^2}=0 [/itex] and also satisfies [itex] y(0)=y'(0)=0 [/itex]. Yeah, you may say F has a singularity but you may instead consider the trivial ODE [itex]y''=6x [/itex]!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A property of solutions
  1. Laplacian property (Replies: 0)

Loading...