Discussion
A one-dimensional harmonic oscillator in ground state is subjected to time dependent perturbation \( V(t) = 0 \) for \( t \lt 0 \); \( V(t) = xe^{-at} \) for \( t \gt 0 \). The probability that the system is in third excited state at \( t = \infty \) is :
Answer: C
The probability is zero.
In first-order time-dependent perturbation theory, the amplitude to reach state \( |n\rangle \) is proportional to the matrix element \( \langle n|x|0\rangle \). Since \( x = \sqrt{\frac{\hbar}{2m\omega}}(a + a^{+}) \), the perturbation connects the ground state only to \( |1\rangle \):
\[ \begin{aligned} \langle n|x|0\rangle &= \sqrt{\frac{\hbar}{2m\omega}}\, \langle n|1\rangle \\ &= \sqrt{\frac{\hbar}{2m\omega}}\, \delta_{n,1} \end{aligned} \]
So the system can be found in the first excited state, but the transition to the third excited state (\( n = 3 \)) is forbidden in this order; this selection rule is the reading the key relies on. The time dependence \( e^{-at} \) affects only the size of the transition to \( n = 1 \), so 1/3, 1 and \( e^{-a/3} \) have no basis.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development