Paschen back effect and commutator [J^2,Lz]

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In summary, the commutator ##[J^2,L_z]## is equal to ##2i\bar{h}[\hat{L_x}\hat{S_y}-\hat{L_y}\hat{S_x}]##. This relates to the Paschen Back effect by showing that in a weak magnetic field, the spin and orbital angular momentum are coupled, but in a strong magnetic field, they become decoupled and can be measured separately. This is further explained in the article provided.
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Taylor_1989
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Homework Statement


I have been given a question on how the commutator relates to the paschen back effect the exact question is as follows

Calculate the commutator ##[J^2,L_z]## where ##\vec{J}=\vec{L}+\vec{S}## and explain the relevance of this with respect to the paschen back effect

Homework Equations

The Attempt at a Solution


I normally it forum rules to preferable use latex, but it would take me sometime to write all my working out so I have photo in my working for the ##[J^2,L_z]## part
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So my final result for the commutator was ##[J^2,L_z]=2i\bar{h}[\hat{L_x}\hat{S_y}-\hat{L_y}\hat{S_x}]##

So how this relates to paschen back I am not sure exactly I have been looking at, how spin orbit coupling works in weak and strong mag fields and my working theory is as follows.

In a weak mag field the spin/orbit couple form the constant total angular momentum vector or total angular quantum number honestly I am always confused why vectors suddenly become operators I know how to work them out but everything iv watch on you-tube or even read never really explains it in full, maybe someone could expand on this?

Anyway, so whilst in a weak mag field we would see that the commutator ##[J^2,L_z]## would not be commutable as J is partly formed by L so there would be no way to know simultaneously the measurement of ##J## or a component of ##L## in this case ##L_x##.

With respect to the panache effect the magnetic field is strong enough to decouple the spin orbit and so the two act independently precessing around the magnetic field in the direction in which the magnetic field point in, then as there has been a decoupling then there is no total angular moment and the square magnitudes of the angular momentum and intrinsic spin angular momentum become commutable with ##L_z##.

In honestly I am still unsure of my explanation and have only come up with by looking at a diagram of spin/orbit coupling in a magnetic field. Would it be possible if someone maybe could explain, if I am at least on the correct lines or well off the mark.

Any advice would be much appreciated.
 

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  • #2
I don't know what the precise answer to your question is, but try reading this: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/paschen.html It seems to be a pretty good write-up. ## \\ ## Additional item: The energy eigenstates in the weak magnetic field are eigenstates of the ## J_z ## operator, at least to first order, if I understand it correctly. Meanwhile, with a strong magnetic field, the eigenstates switch to being eigenstates of ## L_z ## and ## S_z ##.
 
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Related to Paschen back effect and commutator [J^2,Lz]

1. What is the Paschen-Back effect?

The Paschen-Back effect is a phenomenon in atomic physics where the spectral lines of an atom are split into multiple components under the influence of an external magnetic field. This effect is named after Friedrich Paschen and Alfred Back, who first observed it in 1913.

2. How does the Paschen-Back effect occur?

The Paschen-Back effect occurs due to the interaction of the magnetic dipole moment of an atom with the external magnetic field. This interaction causes the energy levels of the atom to split into different components, leading to the observed spectral line splitting.

3. What is the significance of the Paschen-Back effect?

The Paschen-Back effect is significant in understanding the behavior of atoms in the presence of magnetic fields. It is also used in various fields such as astrophysics, plasma physics, and spectroscopy to study the properties of atoms and molecules.

4. What is a commutator in quantum mechanics?

In quantum mechanics, a commutator is an operation that determines how two physical quantities (usually represented by operators) interact with each other. It is defined as the difference between the product of the two operators in two different orders. The commutator of two operators is used to calculate the uncertainty in the measurement of the corresponding physical quantities.

5. What is the relationship between J^2 and Lz in the Paschen-Back effect?

J^2 and Lz are two operators in quantum mechanics that represent the total angular momentum and the z-component of angular momentum, respectively. In the Paschen-Back effect, these operators are used to calculate the energy levels of an atom in the presence of an external magnetic field. The commutation between J^2 and Lz is important in determining the energy splitting of the spectral lines.

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