Solving Schrodinger's Equation Homework

[oddiseas]

Homework Statement

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

$$\Psi$$= i$$M$$ $$e^{\frac{-x}{2}}$$ 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. $\lim_{x\rightarrow\infty}\exp(-x)=0$