- #1
td21
Gold Member
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- 8
Homework Statement
see attachment
Homework Equations
TISE
The Attempt at a Solution
a)When U>E
[tex]-\frac{\hbar^2}{2m}\frac{d^2\Psi(x,t)}{d x^2}+U(x)\Psi(x)=E\Psi(x)[/tex]
leads to blahblahblah...(there is only transmittion no reflection as x to infty)
[tex]\Psi(x)=Be^{-\alpha x}[/tex], where [tex]\alpha = \sqrt\frac{2m[U(x)-E]}{\hbar^2}[/tex]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 [tex]\Psi(x)=Ae^{+i\beta x}+Be^{-i\beta x}[/tex],where[tex]\beta =\sqrt\frac{2m[E-U(x)]}{\hbar^2}[/tex]or[tex]\beta =\sqrt\frac{2m[U(x)-E]}{\hbar^2}[/tex]?
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, [tex]\beta=10^{9}[/tex]
but i don't know whether [tex]\alpha =\beta [/tex] ??
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!