Solving Schrodinger's Equation Homework

[oddiseas]

Homework Statement

At t=0 a particle is described by the eigenfunction:

\Psi= iM e^{\frac{-x}{2}} x \geq 0
0 if x \prec 0

a) Write an expression for the corresponding wave function

b) find the epression for the eigenfunctions.

Homework Equations

The Attempt at a Solution

Does the wavefunction always approach zero as x approaches infinity?

if so this gives me:
f(x)=Be^ikx+Ce^-ikx
f(0)=Aie^(-x/2)
f(\infty)=0 then B=0
f(x)=Aie^(-x/2)e^-ikx

f(x)=Aie^-x(ik+1/2)

then normalising this solution gives A=\sqrt{2}

f_{n}(x)=\sqrt{2}ie^-x(ik+1/2)

then normalising the initial condition give M=1.

\Psi= \sum A*\sqrt{2}ie^-x(ik+1/2)*g(t)

This is as far as i could get;

 

[jdwood983]

Homework Statement

At t=0 a particle is described by the eigenfunction:

\Psi= i M \exp\left({\frac{-x}{2}}\right) x\geq 0
0 if x< 0

The Attempt at a Solution

Does the wavefunction always approach zero as x approaches infinity?

With this eigenfunction, yes. \lim_{x\rightarrow\infty}\exp(-x)=0