GATE 2024 PH – Question 64
An electron in a proton Coulomb field has $\Psi=\psi_{100}/3+\psi_{210}/\sqrt3-\sqrt5\psi_{320}/3$. Energy measured at $t_1$ is $E_2$. Total angular momentum is measured at $t_2>t_1$, and energy again at $t_3>t_2$. If the final probability of $E_2$ is P/9, the integer P is _____
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Correct answer: 9
Explanation
The state is $\Psi=\dfrac13\psi_{100}+\dfrac1{\sqrt3}\psi_{210}-\dfrac{\sqrt5}{3}\psi_{320}$ (the probabilities $\tfrac19+\tfrac13+\tfrac59=1$ ✓).
**First measurement (energy at $t_1$).** The result $E_2$ means $n=2$. The only $n=2$ component is $\psi_{210}$, so the state collapses to $\psi_{210}$.
**Second measurement (total angular momentum, $L^2$, at $t_2$).** $\psi_{210}$ is already an eigenstate of $L^2$ with $l=1$, so the measurement leaves the state unchanged.
**Third measurement (energy at $t_3$).** The state is still $\psi_{210}$, an energy eigenstate with $E_2$, so the probability of finding $E_2$ again is **1**:
$$\frac P9=1\;\Rightarrow\;P=\mathbf{9}.$$
(Between measurements the state evolves with a phase factor only, which does not change the probabilities.)