Finding \Psi(x,t) From Schrodinger Equation

[puneet.988]

hi
is there any way we can find $$\Psi$$(x,t) for a given $$\psi$$(x,0) ?
i got stuck with schrodinger equation...

 

[SpectraCat]

hi
is there any way we can find $$\Psi$$(x,t) for a given $$\psi$$(x,0) ?
i got stuck with schrodinger equation...

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

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

There are more complicated versions that work for time-dependent Hamiltonians.

 

[puneet.988]

actually I've got this wave function $$\psi$$(x,0)=A sin 2$$\Pi$$x cos $$\Pi$$x

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

we need to find $$\Psi$$(x,t) at a later time t...

how should i go for it?
i tried to normalize it but got stuck...

 

[xepma]

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 $\psi_1(x)$ is $e^{iE_1t/\hbar} \psi_1(x)$
4/ Just replace each eigenstate by it's time-dependen version, $\psi_1\rightarrow e^{iE_1t/\hbar} \psi_1(x)$ and you're done!

 

[puneet.988]

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.

 

[puneet.988]

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.

psi(x,0)=A sin 2x cos x