Schrodinger equation and free particles

[kehler]

Homework Statement

Show whether the functions
psi_I = A cos(kx - wt)
psi_II = A sin(kx - wt)
are solutions of Schrödinger equation for a free particle

Homework Equations

Schrödinger equation

The Attempt at a Solution

For psi_I = A cos(kx - wt),
d²psi_I/dx² = -Ak²psi[/SUB]I[/SUB]
dpsi_I/dt = Awsin(kx - wt)
Substituting into S.E,
ih_bar Awsin(kx - wt) = ((h_bar)²/2m) Ak²psi_I + 0 psi (free particle so V=O)
So it doesn't satisfy the S.E.

For psi_II = A sin(kx - wt),
d²psi_II/dx² = -Ak²psi[/SUB]II[/SUB]
dpsi_II/dt = -Awcos(kx - wt)
Substituting into S.E,
ih_bar -Awcos(kx - wt) = ((h_bar)²/2m) Ak²psi_II + 0 psi
So it doesn't satisfy the S.E either.

Is this correct? Or is there some way that I'm supposed to manipulate both sides to equal each other? :S

 

[lanedance]

hi kehler

as you've shown, they will not be a solution of the Time Independent SE for a free particle by themselves...

How about a linear combination of the 2 states? ie
$$\psi = C\psi_1 + D\psi_2$$

what do C & D have to staisfy to be a solution...?