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Wave function

  1. Feb 12, 2010 #1
    hi
    is there any way we can find [tex]\Psi[/tex](x,t) for a given [tex]\psi[/tex](x,0) ?????
    i got stuck with schrodinger equation........
     
    Last edited: Feb 12, 2010
  2. jcsd
  3. Feb 12, 2010 #2

    SpectraCat

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    One way is to apply the propagator [tex]exp\left\{-\frac{i\hat{H}t}{\hbar}\right\}[/tex] to the wavefunction. This works for a time-independent Hamiltonian.

    Then [tex]\Psi\left(x,t\right)=e^{-\frac{i\hat{H}t}{\hbar}}\psi(x,0)[/tex]

    There are more complicated versions that work for time-dependent Hamiltonians.
     
  4. Feb 12, 2010 #3
    actually i've got this wave function [tex]\psi[/tex](x,0)=A sin 2[tex]\Pi[/tex]x cos [tex]\Pi[/tex]x

    this wave function is for a one dimensional box of unit length...A is normalization constant

    we need to find [tex]\Psi[/tex](x,t) at a later time t......

    how should i go for it?????
    i tried to normalize it but got stuck.........
     
  5. Feb 12, 2010 #4
    1/ Find the eigenstates of the system [itex](\psi_1(x), \psi_2(x),\ldots)[/tex]
    2/ Write your wavefunction as a sum over these eigenstates ([itex]\Psi(x,0) = c_1\psi_1 + \ldots[/tex]. Note: it can very well be your wavefunction is identical to an eigenstate.
    3/ The time evolution of one eigenstate is very simple: it is multiplication with a phase factors. So the time evolution of [itex]\psi_1(x)[/itex] is [itex]e^{iE_1t/\hbar} \psi_1(x)[/itex]
    4/ Just replace each eigenstate by it's time-dependen version, [itex]\psi_1\rightarrow e^{iE_1t/\hbar} \psi_1(x)[/itex] and you're done!
     
  6. Feb 12, 2010 #5
    hi
    i want to normalize that wave function but could'nt. can somone help me over this.
    i need to find out the value of A.
     
  7. Feb 12, 2010 #6
    psi(x,0)=A sin 2x cos x
     
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