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A particle is subjected to a potential

\[ V(x) = \begin{cases} \infty, & x \le 0 \\ V_{0}, & a \le x \le b \\ 0, & \text{elsewhere} \end{cases} \]

Here, \(a \gt 0\) and \(b \gt a\). If the energy of the particle \(E \lt V_{0}\), which one of the following schematics is a valid quantum mechanical wavefunction \((\Psi)\) for the system?

  • A.Wavefunction Psi with a barrier V0 between a and b: a bump inside the barrier and small oscillations beyond b
  • B.Wavefunction Psi with a barrier V0 between a and b: zero at x = 0, smooth decay through the barrier, small oscillations beyond b
  • C.Wavefunction Psi with a barrier V0 between a and b: oscillation inside the barrier and beyond b
  • D.Wavefunction Psi with a barrier V0 between a and b: non-zero at x = 0, smooth decay through the barrier, small oscillations beyond b

Answer: B

The valid wavefunction is (B).

The requirements are:

  • Ψ = 0 at x = 0, because the potential is infinite for x ≤ 0;
  • between 0 and a, where E > V = 0, Ψ oscillates (here it rises from zero to a maximum);
  • inside the barrier, where E < V0, Ψ is a combination of rising and falling exponentials, so it cannot oscillate or develop a local peak;
  • beyond b, Ψ oscillates again; Ψ and its slope are continuous at a and b.

Sketch (B) meets all four. Sketch (A) has a bump inside the barrier, which an exponential cannot produce. Sketch (C) oscillates inside the barrier, as if E were above V0. Sketch (D) looks like (B) but is not zero at x = 0, violating the condition at the infinite wall.

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