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: In a classically forbidden region the potential energy exceeds the total energy, \(V > E\), so the time-independent Schrodinger equation reads
\[\frac{d^2\psi}{dx^2} = \frac{2m(V-E)}{\hbar^2}\,\psi = \kappa^2 \psi,\]
with \(\kappa^2 = \dfrac{2m(V-E)}{\hbar^2} > 0\).

Step 2: The general solution is \(\psi(x) = A e^{-\kappa x} + B e^{+\kappa x}\), a combination of real exponentials rather than oscillatory sines or cosines.

Step 3: For the wavefunction to remain finite (normalisable) as \(x \to \infty\), the growing term must be discarded, \(B = 0\).

Step 4: Thus the surviving solution is the decaying (negative) exponential \(\psi(x) = A e^{-\kappa x}\), describing the evanescent penetration of the particle into the barrier.
\[\boxed{\psi(x) = A\,e^{-\kappa x}\ \text{(a negative exponential)}}\]