Question about an equation from anapole measurement paper

In summary, the conversation discusses the use of perturbation theory in calculating the effects of hyperfine and spin-rotational terms in a paper. The speaker agrees with using perturbation theory for most terms, but questions the validity of using it for the term ##c(I\cdot n)(S\cdot n)##. They provide an example using level crossing states and an operator expansion to support their argument. However, they realize that the term in question must be zero as it would connect states of opposite parity. They seek clarification on why the math may suggest otherwise.
  • #1
BillKet
312
29
Hello! I have a question about this paper. They claim that the hyperfine and spin-rotational terms can be treated perturbatively (they do perform a full diagonalization, too, but they claim that perturbation theory is good to get an estimate of the effect). I agree with that for most of the terms, except for ##c(I\cdot n)(S\cdot n)##. For example, let's assume we use as the level crossing the states (using the notation in order N, S, I):

$$\psi_+ = |0,0>|1/2,1/2>|1/2,1/2>$$
and
$$\psi_- = |1,1>|1/2,-1/2>|1/2,1/2>$$

The operator ##c(I\cdot n)(S\cdot n)## can be expanded (I will ignore some constants, I will write just the operators) as ##c(I_zY_1^0+I_+Y_1^{-1}+I_-Y_1^1)(S_zY_1^0+S_+Y_1^{-1}+S_-Y_1^1)## and among these, the term ##cI_zY_1^0S_-Y_1^1## doesn't seem to vanish when calculated between ##\psi_+## and ##\psi_-## i.e.

$$<\psi_-|cI_zY_1^0S_-Y_1^1|\psi_+> \neq 0$$
which is about equal to c. But when doing perturbation theory, the effect of this term would be about ##\frac{c}{E_+-E_-}## and while c is very small, ##E_+-E_-## is much smaller (ideally as small as the parity violation effect), so I don't see how this can be treated pertubatively. Am I missing something? Is that matrix element actually vanishing? Thank you!
 
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  • #2
BillKet said:
Hello! I have a question about this paper. They claim that the hyperfine and spin-rotational terms can be treated perturbatively (they do perform a full diagonalization, too, but they claim that perturbation theory is good to get an estimate of the effect). I agree with that for most of the terms, except for ##c(I\cdot n)(S\cdot n)##. For example, let's assume we use as the level crossing the states (using the notation in order N, S, I):

$$\psi_+ = |0,0>|1/2,1/2>|1/2,1/2>$$
and
$$\psi_- = |1,1>|1/2,-1/2>|1/2,1/2>$$

The operator ##c(I\cdot n)(S\cdot n)## can be expanded (I will ignore some constants, I will write just the operators) as ##c(I_zY_1^0+I_+Y_1^{-1}+I_-Y_1^1)(S_zY_1^0+S_+Y_1^{-1}+S_-Y_1^1)## and among these, the term ##cI_zY_1^0S_-Y_1^1## doesn't seem to vanish when calculated between ##\psi_+## and ##\psi_-## i.e.

$$<\psi_-|cI_zY_1^0S_-Y_1^1|\psi_+> \neq 0$$
which is about equal to c. But when doing perturbation theory, the effect of this term would be about ##\frac{c}{E_+-E_-}## and while c is very small, ##E_+-E_-## is much smaller (ideally as small as the parity violation effect), so I don't see how this can be treated pertubatively. Am I missing something? Is that matrix element actually vanishing? Thank you!
I realized I am dumb, of course that term has to be zero, as it would otherwise connect 2 terms of opposite parity. But I would appreciate if someone can help me figure out why does the math looks like it is not zero?
 

Related to Question about an equation from anapole measurement paper

1. What is an anapole measurement paper?

An anapole measurement paper is a scientific paper that discusses a type of measurement technique used in physics research. The term "anapole" refers to a type of electromagnetic field that is produced by certain particles.

2. What is the equation used in the anapole measurement paper?

The specific equation used in an anapole measurement paper may vary depending on the research being conducted. However, it typically involves mathematical representations of the anapole field and its interactions with other particles.

3. How is the anapole measurement equation derived?

The anapole measurement equation is derived through a combination of theoretical physics principles and experimental data. Scientists use mathematical models and equations to explain and predict the behavior of the anapole field.

4. What is the significance of the anapole measurement equation?

The anapole measurement equation is significant because it allows scientists to accurately measure and understand the behavior of the anapole field. This can lead to a better understanding of fundamental particles and their interactions, and potentially advance our understanding of the universe.

5. Can the anapole measurement equation be applied to other fields of study?

While the anapole measurement equation is primarily used in physics research, it may also have applications in other fields such as engineering or materials science. However, its applicability may depend on the specific context and research being conducted.

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