Simple Finite Square Well Problem help *Ignore, made stupid mistake*

[Irishdoug]

Homework Statement

Self studying Quantum Mechanics. I've a finite square well. Inside the well the V(x) = 0. Thus the problem becomes that of the free particle. I'm aware the solution to the SE is Asinkx + Bcoskx however I can't figure out how.

Relevant Equations

$\psi$'' = $-k^2 \psi$ were k = $(2mE)^{0.5}$ / $\hbar$

I've tried to carry out the solution to this as a normal 2nd order Differential Equation
$\psi$'' - $-k^2 \psi$ = 0
Assume solution has form $e^{\gamma x}$
sub this in form $\psi$ and get
$\gamma ^2$ $e^{\gamma x}$ + $k^2 e^{\gamma x}$ = 0
Solution is $\gamma$ = 0 or $k^2$
Now have two real roots that are not equal thus have
c1$e^{\gamma_1 x}$ + c2$e^{\gamma_2 x}$
I presume this is wrong, as I cannot figure out how to turn it into c1sinkx +c2coskx.
I'm aware of Eulers equation but I've no imaginary number and just using the real part means I've no sin term.

 

[George Jones]

$\gamma ^2$ $e^{\gamma x}$ + $k^2 e^{\gamma x}$ = 0
Solution is $\gamma$ = 0 or $k^2$

Check your solution to this equation.

 

[Irishdoug]

When I did it I used $\psi$'' +$k^2 \psi$' instead of simply $+k^2 \psi$ hence my wrong answer. Only realized when I'd written all that out the mistake I'd made. Tis the problem with studying after a full days work!

Cheers for the reply though.