GATE 2024 PH – Question 47

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 · 2 marks · Multiple select

$V(x)=V_0+\frac12m\omega_0^2x^2$ for x > 0 and infinity for x ≤ 0. Figures P,Q,R,S specify offsets and frequencies. With $E_j^{(X)}$ the j-th level of X, which statement(s) is/are true?

Half-oscillator potentials P, Q, R, S with different offsets and frequencies.

$E_0^{(P)}=E_0^{(Q)}$
$E_0^{(Q)}=E_0^{(S)}$
$E_0^{(P)}=E_1^{(R)}$
$E_0^{(R)}\ne E_0^{(Q)}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $E_0^{(Q)}=E_0^{(S)}$; (C) $E_0^{(P)}=E_1^{(R)}$; (D) $E_0^{(R)}\ne E_0^{(Q)}$

Explanation

The half oscillator has $E_j=V_0+(2j+3/2)\hbar\omega_0$. In units of $\hbar$, the ground energies P,Q,R,S are 18,21,10,21 and the first excited R energy is 18.