Solving 1-D Potential TISE Homework: Get Solutions & Calculate Energies

[8614smith]

Homework Statement

A particle of mass m and energy E<V₀ is trapped in the 1-D potential well defined by,
V=$$\infty$$, x<0; V=0, 0$$\leq$$x$$\leq$$L; V=V₀, x>L.

(b) Obtain solutions to the time independent Schrödinger equation for the three regions and by appropriate matching at the boundaries, show that the allowed energies are given by the transcendental equation, $$k cot(kL)=-\kappa$$, $$where$$$${k^2}=\frac{2mE}{\hbar^2},{\kappa^2}=\frac{2m({V_0}-E)}{\hbar^2}$$.

Homework Equations

TISE

The Attempt at a Solution

For region 2,

$$\frac{{\partial^2}\psi}{\partial{x^2}}+\left(\frac{2mE}{\hbar^2}\right)\psi=0$$

$$\frac{{\partial^2}\psi}{\partial{x^2}}+{k^2}\psi=0$$

$$\psi(x)=Asin(kx)+Bcos(kx)$$

I don't understand how to get from line two to line 3, and where does the potential term disappear to?

 

[diazona]

For region 2,

$$\frac{{\partial^2}\psi}{\partial{x^2}}+\left(\frac{2mE}{\hbar^2}\right)\psi=0$$

$$\frac{{\partial^2}\psi}{\partial{x^2}}+{k^2}\psi=0$$

$$\psi(x)=Asin(kx)+Bcos(kx)$$

I don't understand how to get from line two to line 3, and where does the potential term disappear to?

Remember what the value of the potential is in region 2...

As for how to get from line 2 to line 3: first of all, I'll tell you directly that
$$\psi(x)=Asin(kx)+Bcos(kx)$$
is the most general possible solution to the differential equation
$$\frac{{\partial^2}\psi}{\partial{x^2}}+{k^2}\psi=0$$
This is a very useful thing for a physicist to know, because that equation comes up very frequently in physics. You're not going to want to go through the whole procedure of solving the equation every time you see it; it's awfully convenient to just say "the solution of [that equation] is..."

Since I gather that you're new to this equation, you should convince yourself that the solution is valid. The simple way is to just plug $\psi(x)$ into the equation and simplify it; you should find that everything cancels out, so the equation is satisfied. Showing that there are no other solutions is a little more complicated, and to learn about that I'd suggest taking a class on differential equations, if you have the opportunity. Otherwise, I'm sure there are good resources somewhere out there on the internet - a Google search or something should get you started.