Question

We are considering two spin 1/2 particles, whose Hamiltonian is given by The Hamil-ton operator for a hydrogen atom in a weak, homogeneous magnetic field B is (ignoring the electron's spin) $H = frac{p^2}{2m_e} - frac{e^2}{4piepsilon_0r} + frac{2}{m_e}mathbf{B} cdot mathbf{L}$, (38) where $m_e$ is the electron mass. Using the fact that for $B = 0$ the hydrogen wave functions $|nlm angle$ are simultaneous eigenvectors of $H$, $L^2$, and $L_z$ with energy $E_{nlm} = E_1/n^2$ (here $E_1 = -13.6 ext{eV}$ is the ground-state energy) and angular momentum quantum numbers $langle nlm|L^2|nlm angle = hbar^2l(l+1)$ and $langle nlm|L_z|nlm angle = hbar m$. determine the eigen-functions and the corresponding energy eigenvalues for $B eq 0$ without solving the Schrödinger equation from scratch. Sketch the energy levels for $n = 1, 2$ and all possible $l$, $m$ values, assuming that $|ehbar B/2m_e| < |E_1/4|$. Which of these levels are degenerate?

          We are considering two spin 1/2 particles, whose Hamiltonian is given by The Hamil-ton operator for a hydrogen atom in a weak, homogeneous magnetic field B is (ignoring the electron's spin)
$H = frac{p^2}{2m_e} - frac{e^2}{4piepsilon_0r} + frac{2}{m_e}mathbf{B} cdot mathbf{L}$,
(38)
where $m_e$ is the electron mass. Using the fact that for $B = 0$ the hydrogen wave functions $|nlm
angle$ are simultaneous eigenvectors of $H$, $L^2$, and $L_z$ with energy $E_{nlm} = E_1/n^2$ (here $E_1 = -13.6	ext{eV}$ is the ground-state energy) and angular momentum quantum numbers $langle nlm|L^2|nlm
angle = hbar^2l(l+1)$ and $langle nlm|L_z|nlm
angle = hbar m$. determine the eigen-functions and the corresponding energy eigenvalues for $B 
eq 0$ without solving the Schrödinger equation from scratch. Sketch the energy levels for $n = 1, 2$ and all possible $l$, $m$ values, assuming that $|ehbar B/2m_e| < |E_1/4|$. Which of these levels are degenerate?

$We are considering two spin 1/2 particles, whose Hamiltonian is given by The Hamil-ton operator for a hydrogen atom in a weak, homogeneous magnetic field B is (ignoring the electron's spin) H = fracp^22me - frace^24piepsilon0r + frac2memathbfB cdot mathbfL, (38) where me is the electron mass. Using the fact that for B = 0 the hydrogen wave functions |nlm angle are simultaneous eigenvectors of H, L^2, and Lz with energy Enlm = E1/n^2 (here E1 = -13.6 exteV is the ground-state energy) and angular momentum quantum numbers langle nlm|L^2|nlm angle = hbar^2l(l+1) and langle nlm|Lz|nlm angle = hbar m. determine the eigen-functions and the corresponding energy eigenvalues for B eq 0 without solving the Schrödinger equation from scratch. Sketch the energy levels for n = 1, 2 and all possible l, m values, assuming that |ehbar B/2me| < |E1/4|. Which of these levels are degenerate?$

Added by Rodney S.

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