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- Homework Statement
- What is the explicit expression of the propagator for a particle in a magnetic field along with z-axis

- Relevant Equations
- ## H = \omega J_z ##

##J^2 |jm> = \hbar^2 j(j+1) |jm>##

##J_z |jm> = \hbar m |jm > ##

To show that when ##[J^2, H]=0 ## the propagator vanishes unless ##j_1 = j_2## , I did (##\hbar =1##)

$$ K(j_1, m_1, j_2 m_2; t) = [jm, e^{-iHt}]= e^{iHt} (e^{iHt} jm e^{-iHt}) - e^{-iHt} jm $$

$$ = e^{iHt}[jm_H - jm] $$

So we have

$$ \langle j_1 m_1 | [jm, e^{-iHt} ] | j_2 m_2 \rangle $$

$$ = (j_1 m_1 - j_2 m_2) \langle j_1 m_1 | e^{-iHt} | j_2 m_2 \rangle $$

$$ = \langle j_1 m_1 | e^{-iHt} [ jm_H -jm] | j_2 m_2 \rangle $$

$$ \frac{d jm_H}{dt} = \frac{i}{\hbar} [ H, jm_H] $$

Thus

$$ jm_H = jm $$

And

$$ (j_1 m_1 - j_2 m_2) \langle j_1 m_1 | e^{-iHt} | j_2 m_2 \rangle = \langle j_1 m_1 | e^{-iHt} | j_2 m_2 \rangle $$

$$ (j_1 m_1 - j_2 m_2) K(j_1 m_1, j_2 m_2; t) =0 $$

So the propagator vanishes unless ##j_1 = j_2##

b) Splitting the time evolution

$$ \langle j m_1 | e^{-iHt} | j m_2 \rangle = \langle | e^{-iH(t-t_1)} e^{-iHt_1} | j m_2 \rangle $$

In the continuous (usual) case we have

$$ K(x,t,x', 0) = \int dx'' K(x,t, x'', t_1) K(x'', t_1, x', 0) $$

In the discrete case this is simply just a sum

$$ K(x,t,x', 0) = \sum K(x,t, x'', t_1) K(x'', t_1, x', 0) $$

$$ = K (t-t_1) K(t_1) $$

I am stuck on part c (see attached file) where I need to show that when

$$ H= \omega J_z $$

for a particle in a magnetic field, what is the propagator?

$$ K(j_1, m_1, j_2 m_2; t) = [jm, e^{-iHt}]= e^{iHt} (e^{iHt} jm e^{-iHt}) - e^{-iHt} jm $$

$$ = e^{iHt}[jm_H - jm] $$

So we have

$$ \langle j_1 m_1 | [jm, e^{-iHt} ] | j_2 m_2 \rangle $$

$$ = (j_1 m_1 - j_2 m_2) \langle j_1 m_1 | e^{-iHt} | j_2 m_2 \rangle $$

$$ = \langle j_1 m_1 | e^{-iHt} [ jm_H -jm] | j_2 m_2 \rangle $$

$$ \frac{d jm_H}{dt} = \frac{i}{\hbar} [ H, jm_H] $$

Thus

$$ jm_H = jm $$

And

$$ (j_1 m_1 - j_2 m_2) \langle j_1 m_1 | e^{-iHt} | j_2 m_2 \rangle = \langle j_1 m_1 | e^{-iHt} | j_2 m_2 \rangle $$

$$ (j_1 m_1 - j_2 m_2) K(j_1 m_1, j_2 m_2; t) =0 $$

So the propagator vanishes unless ##j_1 = j_2##

b) Splitting the time evolution

$$ \langle j m_1 | e^{-iHt} | j m_2 \rangle = \langle | e^{-iH(t-t_1)} e^{-iHt_1} | j m_2 \rangle $$

In the continuous (usual) case we have

$$ K(x,t,x', 0) = \int dx'' K(x,t, x'', t_1) K(x'', t_1, x', 0) $$

In the discrete case this is simply just a sum

$$ K(x,t,x', 0) = \sum K(x,t, x'', t_1) K(x'', t_1, x', 0) $$

$$ = K (t-t_1) K(t_1) $$

I am stuck on part c (see attached file) where I need to show that when

$$ H= \omega J_z $$

for a particle in a magnetic field, what is the propagator?

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