Spin angular momentum operators

In summary: 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)? From (2) to (3) they've passed through time-averaging and setting the spin operators product to 1 (unit operator, probably).
  • #1
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 [Broken]

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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  • #2
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.
 
  • #3
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)?
 
  • #4
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.
 
  • #5
OK. I could see that if the spin operators are unitary. However, I am also confused about why there is an [tex]H^D[/tex] 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.
 
  • #6
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?
 

1. What is spin angular momentum?

Spin angular momentum is a physical quantity that describes the intrinsic angular momentum of a particle. It is a fundamental property of particles, similar to their mass and charge.

2. How is spin angular momentum different from orbital angular momentum?

Orbital angular momentum describes the rotational motion of a particle around a point, while spin angular momentum describes the internal rotational motion of a particle. They are fundamentally different types of angular momentum.

3. What are spin angular momentum operators?

Spin angular momentum operators are mathematical operators used in quantum mechanics to describe the behavior of particles with spin. They are represented by matrices and act on the wave function of a particle to determine its spin state.

4. How are spin angular momentum operators related to the Pauli matrices?

The Pauli matrices are a set of three spin angular momentum operators that are commonly used in quantum mechanics. They represent the x, y, and z components of spin angular momentum and are essential in describing the behavior of spin-1/2 particles, such as electrons.

5. What is the significance of spin angular momentum in quantum mechanics?

Spin angular momentum is a crucial concept in quantum mechanics as it explains many fundamental properties of particles, such as their magnetic moment and the structure of atoms. It also plays a significant role in quantum computing and other advanced technologies.

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