A particle has V(x)= for x0, V(x)=V 0 for a x b, and zero elsewhere, with b>a>0. For E<V 0, which schematic wavefunction is valid?

GATE 2024 PH – Question 22

Quantum Mechanics · one dimensional potentials: step potential, finite rectangular well, tunnelling from a potential barrier, particle in 1,2,3-dimensional box, particle in single and double delta function potentials, 1,2,3 dimensional harmonic oscillator: concept of degeneracy · 1 mark · Multiple choice

A particle has $V(x)=\infty$ for $x\le0$, $V(x)=V_0$ for $a\le x\le b$, and zero elsewhere, with $b>a>0$. For $E<V_0$, which schematic wavefunction is valid?

Finite barrier after a hard wall and four candidate wavefunctions.

Option A in the figure
Option B in the figure
Option C in the figure
Option D in the figure

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) Option B in the figure

Explanation

The wavefunction must vanish at the infinite wall. At each finite barrier boundary both $\psi$ and $d\psi/dx$ are continuous. In the region $E<V_0$, it has exponential rather than sinusoidal character, with $\psi^{\prime\prime}=2m(V_0-E)\psi/\hbar^2$. Outside the barrier it oscillates. The drawing satisfying all these conditions is B.

Official GATE 2024 answer key: https://gate2026.iitg.ac.in/doc/download/2024/PHFinalAnswerKey.pdf