Schrodinger Equation for Potential Well

[freddy_12345]

Homework Statement

A particle of mass m and total energy E < 0 is confined to a potential given by:

where $$\alpha$$ is some positive constant.

Show that the wavefunction

is a solution of the time independent Schrödinger equation when x > 0. Find the associated energy Eigenvalue E.

Homework Equations

The TISE:

The Attempt at a Solution

The second part of the question is straightforward: you take the equation for $$\psi$$(x) and its second derivative and plug them into the TISE, cancel and rearrange for E. I got:

Which I'm over 50% sure is correct.

But for the first part, what does it mean to show that Nx exp (-$$\alpha$$x) is a solution?

 

[gabbagabbahey]

The second part of the question is straightforward: you take the equation for $$\psi$$(x) and its second derivative and plug them into the TISE, cancel and rearrange for E. I got:

Which I'm over 50% sure is correct.

But for the first part, what does it mean to show that Nx exp (-$$\alpha$$x) is a solution?

You've basically already done the first part. All you need to do is plug it into the left-hand side of the TISE, calculate the derivatives and show that for x>0, you get a constant times the wavefunction for all values of x>0. Doing so, proves that it is a solution to the TISE (If it weren't, then you would not end up with a constant multiple of the wavefunction when you calculate the LHS). The second part simply asks you for the value of that constant (E).