# Schrodinger equation and free particles

#### kehler

1. The problem statement, all variables and given/known data
Show whether the functions
psi_I = A cos(kx - wt)
psi_II = A sin(kx - wt)
are solutions of Schrodinger equation for a free particle

2. Relevant equations
Schrodinger equation

3. 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

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#### lanedance

Homework Helper
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...?

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