Spin angular momentum operators

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Discussion Overview

The discussion revolves around the transition between two equations in a paper related to spin angular momentum operators and their relation to the Hamiltonian operator in quantum mechanics. Participants are exploring the implications of these equations in the context of quantum chemistry, particularly focusing on the removal of operators and the interpretation of expectation values.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the transition from equation (2) to equation (3) in the paper, questioning whether this involves applying both sides to a wavefunction.
  • Another participant suggests that if equation (2) was mediated, the spin part could produce 1, implying a potential disconnect between the two equations.
  • A different participant challenges the assertion of no connection, noting the similarity between the two equations and seeking clarification on the removal of operators.
  • One participant proposes that the transition involves time averaging and setting the spin operators to a unit operator, though they express uncertainty about their interpretation.
  • Another participant raises a question regarding the presence of the Hamiltonian operator on the left side of (2) and the absence of it in (3), specifically inquiring about the transition from an operator to a coupling value.
  • A later reply suggests that equation (3) represents the expectation value of the Hamiltonian from equation (2), questioning how the spin angular momentum operators are accounted for in this context and whether unitary operators disappear in expectation value calculations.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple competing views regarding the connection between equations (2) and (3), the role of spin operators, and the nature of unitary operators in expectation values.

Contextual Notes

There are unresolved assumptions regarding the nature of the operators involved, the definitions of time averaging, and the specific conditions under which operators may be removed or simplified in the equations.

ehrenfest
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The context of this question is chemistry but I think that it contains enough quantum mechanics to warrent posting it here instead of in the chemistry forum.

Go to section 2.1.1 at the following site:

http://tesla.ccrc.uga.edu/publications/papers/qrevbiophys_v33p371.pdf

I am confused about how you go from equation (2) in 2.1.1 which contains the Hamiltonian operator on the left side and the spin angular momentum operators on the right side to equation (3) in 2.1.2 without any operators in the equation at all.

Is this just the result of applying both sides to a wavefunction or something?
 
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Apparently if (2) was mediated, the spin part produces 1 by mediation, that is, of course of there's any connection between (2) and (3) at all. Actually the spin part is time independent and, since it doesn't appear in (3), i guess there's no connection between the 2 formulae.
 
What do you mean there is no connection between (2) and (3)?

They are nearly identical.

I just want to know how you get rid of the operators when you go from (2) to (3)?
 
They mean a time average, the way they've written it. But the spin part, from (2) appears to be time-independent, so if the 2 formulae are connected one to another, it follows that from (2) to (3) they've passed through time-averaging and setting the spin operators product to 1 (unit operator, probably).

Perrhaps I'm not seeing something; i hope someone else can join us in this thread.
 
OK. I could see that if the spin operators are unitary. However, I am also confused about why there is an H^D operator on the LHS of (2) and only D on the LHS of (3).

D is the coupling value so I am confused about how you go from an operator to a value.
 
I reread it and I think equation (3) is the expectation value for the Hamiltonian operator in equation (2). So, now my question is how did they arrive at this expectation value? What happened to the spin angular momentum operators?? Do unitary operators always disappear in an expectation value calculation? Are they even unitary operators?
 

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