Peeter
Dec14-10, 10:20 AM
1. The problem statement, all variables and given/known data
An initial state is given:
{\lvert {\psi(0)} \rangle} = \frac{1}{{\sqrt{3}}} \left( {\lvert {100} \rangle} + {\lvert {210} \rangle} + {\lvert {211} \rangle}\right)
An L_z measurement is performed with outcome 0 at time t_0. What is the appropriate form for the ket {\lvert {\psi_{\text{after}}(t_0)} \rangle} right after the measurement?
2. Relevant equations
t_0 = \frac{4 \pi \hbar }{E_I}
E_n = - E_I/n^2
3. The attempt at a solution
For the evolved state at this time t_0 (specially picked so that the numbers work out nicely) I get:
{\lvert {\psi(t_0)} \rangle} = \frac{1}{{\sqrt{3}}} \left( e^{-i2\pi i/3} {\lvert {100} \rangle} + e^{i\pi i/3} ({\lvert {210} \rangle} + {\lvert {211} \rangle} )\right).
A zero measurement means that we've measured an L_z eigenvalue of m \hbar = 0, or m = 0, so we can only have states
{\lvert {\psi_{\text{after}}(t_0)} \rangle} = a {\lvert {100} \rangle} + b {\lvert {210} \rangle}
There are multiple m=0 states in the t_0 pre-L_z ket, so I don't think that we can know any more about the distribution of those states after the L_z measurement. Other than {\left\lvert{a}\right\rvert}^2 + {\left\lvert{b}\right\rvert}^2 = 1, can we say anything more about this post L_z measurement state?
An initial state is given:
{\lvert {\psi(0)} \rangle} = \frac{1}{{\sqrt{3}}} \left( {\lvert {100} \rangle} + {\lvert {210} \rangle} + {\lvert {211} \rangle}\right)
An L_z measurement is performed with outcome 0 at time t_0. What is the appropriate form for the ket {\lvert {\psi_{\text{after}}(t_0)} \rangle} right after the measurement?
2. Relevant equations
t_0 = \frac{4 \pi \hbar }{E_I}
E_n = - E_I/n^2
3. The attempt at a solution
For the evolved state at this time t_0 (specially picked so that the numbers work out nicely) I get:
{\lvert {\psi(t_0)} \rangle} = \frac{1}{{\sqrt{3}}} \left( e^{-i2\pi i/3} {\lvert {100} \rangle} + e^{i\pi i/3} ({\lvert {210} \rangle} + {\lvert {211} \rangle} )\right).
A zero measurement means that we've measured an L_z eigenvalue of m \hbar = 0, or m = 0, so we can only have states
{\lvert {\psi_{\text{after}}(t_0)} \rangle} = a {\lvert {100} \rangle} + b {\lvert {210} \rangle}
There are multiple m=0 states in the t_0 pre-L_z ket, so I don't think that we can know any more about the distribution of those states after the L_z measurement. Other than {\left\lvert{a}\right\rvert}^2 + {\left\lvert{b}\right\rvert}^2 = 1, can we say anything more about this post L_z measurement state?