How Does a Free Particle's Wave Function Evolve Over Time?

[mac_guy_ver]

Homework Statement

free particle of mass m moving in 1d
state: $$\Psi(x,0) = sin(k_{0}x)$$

Homework Equations

$$\Psi(x,t) = \stackrel{1}{\overline{\sqrt{2\pi}}}\overline{}$$$$\int^{\infty}_{-\infty}b(k)e^{i(kx-\omega t)}$$

The Attempt at a Solution

b(k)=$$\stackrel{1}{\overline{\sqrt{2\pi}}}\overline{}$$$$\int^{\infty}_{-\infty}sin(k_{0}x)e^{-ikx}$$

 

[jambaugh]

The problem here is that you have sine instead of $$e^{ikx}$$ factor.
Use the fact that:
$$sin(kx) = \frac{1}{2i}(e^{ikx}-e^{-ikx})$$

Second your answer should be:
$$\Psi(x,t) = ...$$
not
$$b(k) = ...$$
Look up the source of your "relevant equation" to see what role $$b(k)$$ plays...also you should give the variable of integration which appears to be k.

Your attempted solution is I suppose integrated over x? The integral you give has a known solution in terms of Dirac delta functions (which relates to my point above)

Remember ultimately you are looking for a solution to the Schrödinger equation which a.) is properly normalized and b.) satisfies the initial condition given.