Free particle (spatial part)

1. Oct 16, 2009

Slepton

1. The problem statement, all variables and given/known data

For a free particle, i have two expressions.

$$\varphi$$(x) = $$\alpha$$eikx + $$\beta$$e-ikx
and

$$\varphi$$(x) = $$\gamma$$sin(kx) + $$\theta$$cos(kx)

I have to determine expressions for $$\gamma$$ and $$\theta$$ in terms of $$\alpha$$ and $$\beta$$.

2. Relevant equations

sin(kx) = (eikx - e-ikx)/2i

cos(kx) = (eix + e-ix)/2

3. The attempt at a solution

I replaced sin and cosine in the second equation.

2. Oct 16, 2009

gabbagabbahey

That's a good start.....what did you end up with?

3. Oct 16, 2009

Slepton

replacing
alpha with A
beta with B
gamma with M
theta with N

I have,

2(Aeikx + Be-ikx) = -Meikx + Me-ikx + Neix + Ne-ix

4. Oct 16, 2009

gabbagabbahey

Good, now just group like terms together:

$2(\alpha e^{ikx}+\beta e^{-ikx})=(\theta-\gamma)e^{ikx}+(\gamma+\theta)e^{-ikx}$

Surely you can see where to go from here?

5. Oct 16, 2009

Slepton

actually that's where i'm stuck at. I know its should be something simpler but my system has lasted on me...

6. Oct 16, 2009

gabbagabbahey

Surely you can see that $2\alpha=\theta-\gamma$ and $2\beta=\theta+\gamma$....can't you?