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Richard Feynman is a great explainer, and I got through the first 2 volumes without problems. The 3rd one, treating the subject of quantum mechanics, gets very complex and he doesn't always provide enough explanations to my liking.

My questions may be very stupid, but keep in mind that I'm trying to teach myself, and I don't have a teacher to answer questions for me. So I'd like to know if what I think I understand is good or not. So, this is like a "true or false" multiple questions post. Please tell me if I am wrong about anything, and what is the proper answer. I searched quite a bit on this site, and have lurked on it for quite a while before joining today, and I finally decided to ask a question, because I couldn't find an answer to a lot of things.

[I should mention that I have posted on vBulletin-based forums before, but stackexchange is quite different. If my question needs re-wording or if it doesn't conform to the required format, I'm sorry: I checked all the FAQs and it didn't seem to forbid asking multiple questions in the same post.]

So am I correct to say:
- the H11 and H22 Hamiltonian matrices represent the time evolution of a state from state 1 to state 1, and from state 2 to state 2. The H12 and H21 represent the time evolution to go from state 1 to 2, and 2 to 1, respectively.
- The ammonia molecule represents a 2 state system. An electron spin-half as a free particle, without an EM field, would correspond to the 2-state system of an ammonia molecule without an EM field.
- The probability amplitude of the ammonia molecule varies sinusoidally with time, between its 2 states, without an EM field. In a varying EM field, it can experience resonance at a given frequency that corresponds to the energy difference between the two states, divided by Planck's constant.

Does that mean the probability amplitude electron's z-spin (which he treats a few chapters later, but in an EM field, with Pauli spin matrices) also varies sinusoidally with time, even without any EM field?

Since the probability amplitude goes to 0 and back to 1 for both state 1 and 2, I guess that means the state itself must constantly be changing back and forth? If this is correct, what is the frequency at which the electron would oscillate between spin up and down?

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    $\begingroup$ You point out perhaps one of the greatest values of the PSE, the ability to serve as a community teacher for the self learner. I also self studied the Feynmans lectures, agree he had a remarkable ability to explain things, but the 3rd volume is difficult. Quantum physics in general difficult, so you need to compliment your study with other books, and questions here. Bravo! $\endgroup$
    – docscience
    Oct 23, 2016 at 15:24
  • $\begingroup$ Thank you for the warm welcome. I also have Dirac's "Principle of Quantum Mechanics" and Griffiths' "Introduction to Quantum Mechanics". I've also found some course notes on the Standard Model, chromodynamics, etc., but I definitely felt like I needed to clarify this before reading more. I'll need to re-read a few chapters of Feynman's book now that I understand. $\endgroup$
    – Simon
    Oct 23, 2016 at 15:36
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    $\begingroup$ You are welcome. Another good book, quantum physics by Stephen Gasiorowicz $\endgroup$
    – docscience
    Oct 23, 2016 at 17:17

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The ammonia molecule, in the absence of an external driving field, oscillates between Feynmann's two states if you assume it starts off entirely in one of the states. That's because the basis states he chooses aren't eigenstates...they are the obvious up/down geometrical states. The exact solution of the ammonia atom shows the true eigenstates are the symmetric and anti-symmetric superpositions of these two geometric states. So if you start in one of the pure geometric states, it means you really are in a superposition of the two eigenstates. Since those two states have slightly different frequencies, there is an oscillation at the difference frequency.

And it goes on forever. Except...it doesn't really. Because this analysis ignores radiation. In fact, there is an oscillating electric dipole moment associated with this superposition, and it radiates exactly according to antenna theory as described by Maxwell's equations. The energy radiated away from the system comes at the expense of the higher (anti-symmetric) eigenstate, and the end result is the ammonia molecule ends up in the ground state...the symmetrical superposition of the two geometrical states. Feymann doesn't exactly mention that.

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  • $\begingroup$ OK. This makes a lot of sense: there is a difference between geometrical states and the actual eigenvector/eigenstate of the Hamiltonian matrix. Is it the same for the electron's spin (let's choose the z-spin)? In other words, are the electron spins up and down eigenstates? Do they have different energy in the absence of an EM field? $\endgroup$
    – Simon
    Oct 23, 2016 at 1:50
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    $\begingroup$ I think they're degenerate in free space. But I think Feynmann analyzes the case of the hydrogen atom, with the two eigenstates of the electron relative to the spin of the proton. In that case they are indeed eigenstates with different energies. They can be driven to oscillate by a field of the right frequency...that's the famous 21 cm line of hydrogen, I think. $\endgroup$ Oct 23, 2016 at 2:15
  • $\begingroup$ Ok, with those concepts in mind, I'll re-read chapters 5 to 11. I think I finally understand what I was missing: the difference between identifiable/geometrical states and actual eigenvectors/eigenstates of a time evolution matrix . I finally understand why he uses states 1 and 2, versus states I and II. Thank you so much. I'll accept your answer now. $\endgroup$
    – Simon
    Oct 23, 2016 at 2:21

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