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 Schrodinger equation which a.) is properly normalized and b.) satisfies the initial condition given.