Question

The wave function of a particle in a region classically forbidden region is __________.

• A. A sine function
• B. A cosine function
• C. A positive exponential
• D. A negative exponential

Hint:

With \(V>E\), \(\psi''=+\kappa^2\psi\); the normalisable solution is the decaying exponential \(e^{-\kappa x}\).

Answer 1

The Correct Option is D

Step 1: Compare the two regions. Where $E > V$ (allowed), the Schrodinger equation gives $\psi'' = -k^2\psi$ with $k^2 > 0$, whose solutions are oscillatory $\sin$ and $\cos$. Where $E < V$ (forbidden), the sign flips.

Step 2: With $V > E$ we get $\psi'' = +\kappa^2\psi$, $\kappa = \sqrt{2m(V-E)}/\hbar$. A positive coefficient on the right means non-oscillatory, exponential behaviour.

Step 3: The two independent exponentials are $e^{+\kappa x}$ (grows without bound) and $e^{-\kappa x}$ (decays). A physically acceptable, square-integrable wavefunction cannot blow up, so only the decaying branch survives deep in the barrier.

Step 4: Hence in the classically forbidden region the wavefunction is a decaying (negative) exponential, giving the familiar exponentially damped tunnelling amplitude.
\[\boxed{\psi \propto e^{-\kappa x}\ \text{(a negative exponential)}}\]