The Hamiltonian for an electron in a hydrogen atom subject...

The Hamiltonian for an electron in a hydrogen atom subject to a constant magnetic field B is (neglecting spin): H = p^2/(2m_e) - e^2/(4??_0r) + e/(2m_e) L · B where L is the angular momentum operator. Assume the magnetic field points in the z-direction. (a) Write down the values of the first 5 distinct energy levels. (b) Consider the spectral line corresponding to the transition (n = 4, l = 3) ? (n = 3, l = 2). Find the energy of the emitted photons. Consider only cases for which the transitions are constrained by the selection rule ?m = 0, ±1.

Transcript

00:01 In this question, basically, you look at hydrogen atom, right? hydrogen is subject to a constant magnet field directly around z direction.

00:09 So the hamiltonian, according to the question is given by, was p squared over 2m, me, minus e squared over 4 pi, absent zero arrow, and price e over 2me, the angular momentum, z times b, right? the decomponent it's very clear that if you call this h0, we see that this part, let's call it h prime, and then very clearly that h0 and h prime commute, right? so the eigenstates of h0 is also the eigenstate of h prime, right? so the against state of h0, we normally write it as n -a -o -m, right? so it's very clear that this must also be the eigenstate of h -o -m.

00:59 H prime.

01:00 So if you look at the energy, if you act homony on this date, you'll simply get h0, nm plus h prime, and lm, right? and h node, act on nm, of course, is giving, you know, you know the answer, which is actually minus 13 .60 lighten volts over n squared at times n -lm.

01:26 How about h prime? h prime, of you will find this to be e over 2me times.

01:33 It depends only on the l -z, so it depends only on the m, right? so it will be m times m -l -m, right? so that would be, so it means that the energy of the system now is given by, it depends on both and m, right? so it's given by e -m over 2 -m -e, minus 13 .6 legion intervals over n squared.

02:04 So that would be the energy levels we will have.

02:07 And you ask you to write down the values for the first distinct energy leverage.

02:11 First you look at, you can look at the ground state, right? so if you look at the ground state, that's n equals 1, right? and then of course, a equals zero, then m equals zero, right? so that's first energy state.

02:24 And the second one, and of course, for this energy you can easily see, it's exactly the same as hydrogen atom, right? so the n0 and that will be just minus 0 .6 electron volts like this.

02:35 And similarly, the next level is n equals 2.

02:38 And the air, you can, l can be 0 and it could be 1 somewhere...

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