Expressions for \gamma and \theta in terms of \alpha and \beta?

[Slepton]

Homework Statement

For a free particle, i have two expressions.

$$\varphi$$(x) = $$\alpha$$e^ikx + $$\beta$$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 $$\beta$$.

Homework Equations

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

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

The Attempt at a Solution

I replaced sin and cosine in the second equation.

 

[gabbagabbahey]

The Attempt at a Solution

I replaced sin and cosine in the second equation.

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

 

[Slepton]

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

I have,

2(Ae^ikx + Be^-ikx) = -Me^ikx + Me^-ikx + Ne^ix + Ne^-ix

 

[gabbagabbahey]

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

I have,

2(Ae^ikx + Be^-ikx) = -Me^ikx + Me^-ikx + Ne^ix + Ne^-ix

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?

 

[Slepton]

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

 

[gabbagabbahey]

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