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Consider 2 cases from intro QM:

Infinite square well

Potential barrier with E > V

_{0}

For the infinite square well, the Schrodinger eqn gives

d2ψ/dx2 + k

^{2}ψ = 0

Since k

^{2}> 0, this gives oscillating solutions (some combination of sines and cosines). Correct?

For the potential barrier with E > V0, the Schrodinger eqn gives (with the potential jumping from 0 to V

_{0}at x = 0)

d

^{2}ψ/dx

^{2}+ k

^{2}

_{I}E = 0 (where k

_{I}= √2mE/(hbar

^{2})

prior to the potential step (i.e. for x < 0) and

d

^{2}ψ/dx

^{2}+ k

^{2}

_{II}E = 0 (where k

_{II}= √2m(E-V0)/(hbar

^{2})

after the potential step (for x > 0)

My problem is this: As far as I can tell, both k

_{I}and k

_{II}are positive numbers (since E > V

_{0}> 0). Why, then, do they give non-oscillating solutions? Whereas, for the infinite square well, a positive k gave oscillating solutions?

Am I making a sign error here or just forgetting something from differential equations? What is different about these equations that one of them gives oscillating, and the other non-oscillating, solutions?

Thanks for any help!