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PROBLEM 2 (25 points) The Hamiltonian for an electron in a hydrogen atom subject to a constant magnetic field \( \mathbf{B} \) is given by \[ H=\frac{\mathbf{p}^{2}}{2 m_{e}}-\frac{e^{2}}{4 \pi \varepsilon_{0} r}+\frac{e}{2 m_{e}}(\mathbf{L}+2 \mathbf{S}) \cdot \mathbf{B} \] where \( \mathbf{L} \) and \( \mathbf{S} \) are the angular momentum and spin operators respectivelyr. Assume the magnetic field points in the z -direction. (a) How many distinct energy levels will the \( n=3 \) state have? (b) Consider the line corresponding to the transition \( (n=3, l=2) \rightarrow(n=2, l=1) \). Find the energy of the emitted photons ( \( \hbar \omega=\Delta E \), with \( \Delta E= \) the energy difference between initial and final state), assuming the possible transitions are constrained by the selection rule \( \Delta m=0, \pm 1 \).

          PROBLEM 2 (25 points)
The Hamiltonian for an electron in a hydrogen atom subject to a constant magnetic field \( \mathbf{B} \) is given by
\[
H=\frac{\mathbf{p}^{2}}{2 m_{e}}-\frac{e^{2}}{4 \pi \varepsilon_{0} r}+\frac{e}{2 m_{e}}(\mathbf{L}+2 \mathbf{S}) \cdot \mathbf{B}
\]
where \( \mathbf{L} \) and \( \mathbf{S} \) are the angular momentum and spin operators respectivelyr. Assume the magnetic field points in the z -direction.
(a) How many distinct energy levels will the \( n=3 \) state have?
(b) Consider the line corresponding to the transition \( (n=3, l=2) \rightarrow(n=2, l=1) \). Find the energy of the emitted photons ( \( \hbar \omega=\Delta E \), with \( \Delta E= \) the energy difference between initial and final state), assuming the possible transitions are constrained by the selection rule \( \Delta m=0, \pm 1 \).

PROBLEM 2 (25 points)
The Hamiltonian for an electron in a hydrogen atom subject to a constant magnetic field 𝐁 is given by

H=(𝐩^2)/(2 me)-(e^2)/(4 πε0 r)+(e)/(2 me)(𝐋+2 𝐒) ·𝐁

where 𝐋 and 𝐒 are the angular momentum and spin operators respectivelyr. Assume the magnetic field points in the z -direction.
(a) How many distinct energy levels will the n=3 state have?
(b) Consider the line corresponding to the transition (n=3, l=2) →(n=2, l=1). Find the energy of the emitted photons ( ħω=Δ E, with Δ E= the energy difference between initial and final state), assuming the possible transitions are constrained by the selection rule Δ m=0, ± 1.

Added by Stephen C.

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