Determine the wave function of the particle

[rmfw]

[STRIKE][/STRIKE]

Homework Statement

A particle coming from +∞ with energy E colides with a potential of the form:

V = ∞ , x<0 (III)
V = -V₀ , 0<x<a (II)
V = 0, x>a (I)

a) Determine the wave function of the particle considering that the amplitude of the incident wave is A. Writting the amplitude of the reflected wave at x=a in the form

$\frac{B}{A} = e^{i\delta}$

determine $\delta$ . What is the value of $\delta$ in the limit where V₀ = 0 ?

b) Determine the probability density current in x>a

The Attempt at a Solution

a)

For region I :

$\Psi_{I} (x) = Ae^{-ikx} + Be^{ikx} , k=\frac{\sqrt{2mE}}{\hbar}$

II:

$\Psi_{II} (x) = Csin(lx) + Dcos(lx) , l=\frac{\sqrt{2(mE+V_{0}}}{\hbar}$

III:

$\Psi_{III} (x) = 0$Boundary conditions:

at x=0:

$0 = Csin(lx) + Dcos(lx) (=) D = 0$

So $\Psi_{II} (x) = Csin(lx)$

at x=a:

$Ae^{-ika} + Be^{ika} = Csin(la)$ (1)

and

$-ikAe^{-ika} + ikBe^{ika} = lCcos(la)$ (2)

Dividing (1) for (2) I got:

$\frac{B}{A} = e^{-2ika} \frac{(-\frac{1}{l}tan(la)ik-1)}{(1-\frac{1}{l}tan(la)ik)}$

How can I get rid of that constant that is multiplying by the exponential? Is this even right?

 

[Simon Bridge]

B/A appears to be a complex number - is that right?

Note: did you try writing $\psi_{II}$ as a sum of complex exponentials?

 

[rmfw]

B/A appears to be a complex number - is that right?

Note: did you try writing $\psi_{II}$ as a sum of complex exponentials?

Hi Simon Bridge, thank you for your reply.

I did write psiII as a sum of complex exponentials and I got the following:

$\frac{B}{A} = \frac{\frac{k}{l}e^(ika-ila)-\frac{k}{l}e^(-ika + ila) - e^(-ika -ila) - e^(-ika + ila)}{\frac{k}{l}e^(ika-ila) - \frac{k}{l}e^(ika+ila) +e^(ika+ila) + e^(ika-ila)}$

is there any way to simplify this ? This problem is killing me, because I'm not even sure what "writing the amplitude of the reflected wave at x=a in the form $\frac{B}{A}$ " means.

I mean, I assumed the amplitude of the reflected wave at x=a is B, if I write B/A it isn't the amplitude of the reflected wave anymore.

 

[Simon Bridge]

B/A is the proportion of the incident amplitude that is reflected.
If you multiplied it by 100, you's be able to say, "100B/A percent got reflected".

You basically have to look for common factors and cancel things off a bit at a time.
It is not going to be easy. But that, I am afraid, is the exercise.

Note: $e^{a-b}+e^{-a-b}=(e^a+e^{-a})e^{-b}$ ... stuff like that. maybe $e^{-b}$ cancels something in the denominator? If not - look for something else.

Of course, since they are all constant terms, it may be possible to add them up geometrically.
Separate the real and imaginary components.