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1d wave equation

  1. Oct 16, 2012 #1
    I was studying a book on QM and found out that the wave function for a free particle of completely undetermined position travelling in positive x direction is given by

    e^(2(pi)i(kx - nt)) where n is the frequency

    i have been trying a lot to derive it but till now i can't . Can any one help me
  2. jcsd
  3. Oct 16, 2012 #2
    It comes from Schrödinger Equation as you know.

    When write time-independent Schrödinger Eq, we get.

    [tex]\frac{d^2\psi }{dx^2}=-k^2\psi [/tex] where [itex]k = \frac{\sqrt{2mE}}{\hbar}[/itex]

    From second order differential equations, it turns out to be:

    [tex]\psi(x)=Ae^{ikx} + Be^{-ikx}[/tex] It is just like the free particle in an infinite square box. But in that case, we had boundary conditions. This time, we have none so time-independent solution is this.

    Since we know that time dependence term is [itex]e^{-\frac{iEt}{\hbar}}[/itex] and [itex]E=\hbar \omega[/itex]

    Our general time-dependent solution becomes

    [itex]\Psi (x,t)=Ae^{i(kx - wt)} + Be^{-i(kx + wt)}[/itex]

    I hope this helps you.
  4. Oct 16, 2012 #3
    thanks coki . But in the book they are saying it comes for maxwell's electromagnetic theory .
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