Schrodinger equation solution when U>E

[td21]

Homework Statement

see attachment

Homework Equations

TISE

The Attempt at a Solution

a)When U>E
$$-\frac{\hbar^2}{2m}\frac{d^2\Psi(x,t)}{d x^2}+U(x)\Psi(x)=E\Psi(x)$$
leads to blahblahblah...(there is only transmittion no reflection as x to infty)
$$\Psi(x)=Be^{-\alpha x}$$, where $$\alpha = \sqrt\frac{2m[U(x)-E]}{\hbar^2}$$This is the solution i work out, but how to "demostrate" it is a solution?

b)
Actually i don't know 2nd order ODE, so i try here:
when U<E, there will be reflection and incident part:
so $$\Psi(x)=Ae^{+i\beta x}+Be^{-i\beta x}$$,where$$\beta =\sqrt\frac{2m[E-U(x)]}{\hbar^2}$$or$$\beta =\sqrt\frac{2m[U(x)-E]}{\hbar^2}$$?
but in any case, the exponential is "complex", yielding the Cosine function,(but where does sine go?)?So U<E there?

c)from part b, $$\beta=10^{9}$$
but i don't know whether $$\alpha =\beta$$ ??
but i just assume so and i calculate: U-E=5.56^-21 J
I would appreciate it if you could help me out esp. the question mark part. Thx!

 

[dingo_d]

Well put it in the equation and see it it will give you something useful (usually if you put the solution in an equation, the condition set on it will be satisfied...).

See the i in the exponentials. If you rewrite them you'll get something real and something imaginary, right? And what does an imaginary solution means in the real world? :D

See what the conditions must be satisfied when solving Schrödinger. And then try to see how they affect your solutions :)

I'm sure that exist in Griffiths or similar books...