How to treat spin orbit operator directly

In summary: You can then just multiply the matrix elements of the ##j_z## matrix in the ##j## basis by the corresponding states of ##s##.In summary, the spin operator for an electron, S_z, is represented by a 2x2 matrix with spin up and down as its bases. The angular momentum operator, L_z, with l=1 is a 3x3 matrix. To treat the L_z S_z operator directly in matrix form, we use the total angular momentum operator, J_z, which is given by J=L+S. The eigenvalues of J_z are -3/2, -1/2, 1/2, and 3/2, and the matrix operator of J_z is
  • #1
hokhani
483
8
For an electron the spin operator [itex]S_z[/itex]is represented by a [itex]2×2[/itex] matrix, with spin up and down as its bases. Consider the angular momentum operator [itex]L_z[/itex] with [itex]l=1[/itex] which is a [itex]3×3[/itex] matrix. How can we treat the [itex]L_z S_z[/itex] operator directly in matrix form?
 
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  • #2
If with ##L_z S_z## you mean the total angular momentum ##J_z## than you have that ##J=L+S##, so ##j=l+s=\frac{3}{2}##. The eigenvalues of ##J_z## are ##m_j=-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}## (##-j \le m_j \le j##). By choosing a basis where ##J^2## and ##J_z## are diagonal, your matrix operator of ##J_z## is:

##J_z = \hbar
\begin{pmatrix}
\frac{3}{2} & 0 & 0 & 0 \\
0 & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{1}{2} & 0 \\
0 & 0 & 0 & -\frac{3}{2}
\end{pmatrix}
##

P.S.: I'm studying this right now, so maybe i could probably make some mistakes
 
  • #3
In non-relativistic physics (I guess that's all about non-relativistic quantum theory) spin and orbital angular momentum commute, and total angular momentum is given by
$$\vec{J}=\vec{L}+\vec{S}.$$
Then according to representation theory of the rotation group, if the orbital angular angular momentum has definite ##l## and spin ##s##, then the total angular momentum has ##j \in \{|l-s|,|l-s|+1,\ldots,l+s \}##, i.e., in your case you have ##j \in \{1/2,3/2 \}##.

Indeed the total-angular momentum space is ##(2l+1)(2s+1)##, i.e., in your case ##3 \cdot 2=6## dimensional. The possible values for ##j## imply the same dimension, i.e., ##2+4=6##.

For the eigenvalues of ##j_z=l_z+s_z##. You should look this up in a good textbook. Look for Clebsch-Gordan coefficients.
 
  • #4
Thanks, but my question is how to multiply the two matrixes directly without solving the problem this way.
 
  • #6
The usual way to get ##L S## is to use the identities
##J^2 = (L + S)^2 = L^2 + S^2 + 2LS##
##S## is a constant, so the result depends on ##L^2## and ##J^2##. With ##l=1## fixed, you can write out the basis states of ##j## in terms of the basis states of ##s##.
 

1. How does the spin orbit operator affect atomic structure?

The spin orbit operator is responsible for the coupling of an electron's spin and orbital motion. This results in the splitting of energy levels and can affect the overall electronic structure of an atom.

2. Can the spin orbit operator be treated separately from other operators?

Yes, the spin orbit operator can be treated independently as it only involves the interaction between an electron's spin and orbital motion. However, it is often considered alongside other operators in order to fully understand and describe the behavior of electrons.

3. How do we account for the spin orbit operator in quantum mechanics?

In quantum mechanics, the spin orbit operator is included in the Hamiltonian, which is the operator that describes the total energy of a system. This allows for the spin orbit interaction to be incorporated into the equations used to describe the behavior of particles at the atomic level.

4. What is the significance of the spin orbit operator in spectroscopy?

The spin orbit operator plays a crucial role in spectroscopy as it causes the splitting of energy levels, leading to the observation of fine and hyperfine structure in atomic and molecular spectra. This allows for the identification of different elements and the determination of their electronic configurations.

5. How is the spin orbit operator calculated and represented mathematically?

The spin orbit operator is a vector operator that can be calculated using the spin and orbital angular momentum operators. It is represented by the symbol ζ in equations and is typically described in terms of its matrix elements in a given basis set.

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