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

[Slepton]

Homework Statement

For a free particle, i have two expressions.

[tex]\varphi[/tex](x) = [tex]\alpha[/tex]e^ikx + [tex]\beta[/tex]e^-ikx
and


[tex]\varphi[/tex](x) = [tex]\gamma[/tex]sin(kx) + [tex]\theta[/tex]cos(kx)

I have to determine expressions for [tex]\gamma[/tex] and [tex]\theta[/tex] in terms of [tex]\alpha[/tex] and [tex]\beta[/tex].

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:

[itex]2(\alpha e^{ikx}+\beta e^{-ikx})=(\theta-\gamma)e^{ikx}+(\gamma+\theta)e^{-ikx}[/itex]

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 [itex]2\alpha=\theta-\gamma[/itex] and [itex]2\beta=\theta+\gamma[/itex]...can't you?