Discussion
A quantum particle is incident on a potential barrier. The particle has incident energy that is less than the height of the potential barrier. When the particle penetrates the barrier, its wave function is :
Answer: B
Inside the barrier the wave function is exponentially decreasing.
Where \(E \lt V_{0}\), the Schrödinger equation becomes \(\psi'' = \kappa^{2}\psi\) with \(\kappa = \sqrt{2m(V_{0} - E)}/\hbar\), whose solutions are real exponentials \(e^{\pm\kappa x}\). For a wave entering from the left, the physical part decays as \(e^{-\kappa x}\) into the barrier; this is quantum tunnelling.
Oscillatory solutions occur only where \(E \gt V\). A growing exponential is not normalisable for a wide barrier, and a constant would need \(E = V_{0}\) exactly.
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