Exploring Free Particle Scattering and Tunneling with Trigonometric Functions

[splitringtail]

[SOLVED] Making the math easier

Homework Statement

I am doing a free particle scattering/tunneling by a well and a barrier combination. There are four regions of interest where the wave numbers are different. I have seen the solution inside the well be expressed as a linear combination of exponents or sine and cosine functions.

Now I have only seen free particles in these problems as a linear combination of exponents... can I rewrite them as a linear combination of sine and cosine. I am finding these exponent forms very very cumbersome b/c I get two terms when I match the wave functions @ x=0, if they were in a sine and cosine combination, then it would sine would be zero @ x=0

Homework Equations

The Attempt at a Solution

This is more of a discussion and there is too much work to present here.

 

[CompuChip]

You can use the identity
[tex]e^{i\theta} = \cos\theta + i \sin\theta[/itex].
So, for example
$$\sum_k a_k e^{ikx} = \sum_k a_k (\cos(kx) + i \sin(kx))$$
and then you can try to rewrite this (for example, if the sum runs from -infinity to +infinity, you can use that sin(-kx) = -sin(kx) and cos(-kx) = cos(kx) to simplify).
If the exponent is real, you can use
$$e^{\theta} = e^{i(-i\theta)} = \cos(-i\theta) + i sin(-i\theta)$$
and subsequently
$$\sinh x = -i \sin(i\theta), \cosh x = \cos(i\theta)$$
to write them in hyperbolic sines and cosines.

Conversely, if you have a combination of sines and cosines, you can always write them in exponentials. All of this is the reason we usually write a plane wave as something like exp(i(kx - wt)) with k the wavevector and w the frequency: you can write it out in sines and cosines to say that it really "waves"