How to create good quantum states from good quantum numbers?

In summary, to create "good" quantum states from "good" quantum numbers, one must ensure that the perturbation H' commutes with the operators L^2, S^2, J^2, and Jz. Then, for the strong field Zeeman effect, the "good" state can be represented as | n l ml s ms > and for the fine structure correction, it can be represented as | n l s j mj >. To go from these representations to the sum of states, the Clebsch-Gordan coefficients are used to account for the addition of angular momentum.
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mattlorig
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how to create "good" quantum states from "good" quantum numbers?

I think I am finally understanding what the "good" quantum numbers are in degenerate perturbation theroy. Basically, given a perturbation H', if
[H', L^2] = [H', S^2] = [H', J^2] = [H', Jz] = 0, then
l, s, j, and mj are the "good" quantum numbers.

But, I'm a little confused as to how one goes about creating a "good" quantum state from the "good" quantum numbers.

In particular, I'm wondering about how to create a "good" states for the strong field Zeeman effect ("good" state = | n l ml s ms > ) and the fine structure correction ("good state = | n l s j mj > ).

i.e. how do I go from: | n l s j mj > to SUM( c R_nl Y_lm |s ms> )?
 
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FAQ: How to create good quantum states from good quantum numbers?

How do I determine the "good" quantum numbers for a given quantum state?

The "good" quantum numbers for a quantum state can be determined by first identifying the quantum system and its relevant properties, such as spin, orbital angular momentum, and total angular momentum. These properties will correspond to the allowed quantum numbers for the system and can be determined through mathematical calculations or experimental observations.

What makes a quantum state "good"?

A "good" quantum state is one that is well-defined and has a high degree of stability and coherence. This means that the state has a low level of uncertainty and can be accurately measured and controlled. Additionally, a good quantum state should have a long lifetime and be resistant to external disturbances.

How can I create a good quantum state from good quantum numbers?

To create a good quantum state from good quantum numbers, one must first prepare the system in a way that allows for precise control over the quantum numbers. This can be achieved through techniques such as cooling, trapping, and manipulating the system using electromagnetic fields. By carefully controlling these parameters, a good quantum state can be created from the good quantum numbers of the system.

Can any quantum state be created from good quantum numbers?

No, not all quantum states can be created from good quantum numbers. A quantum state must satisfy certain conditions, such as obeying the laws of quantum mechanics and having a well-defined set of quantum numbers, in order to be considered "good". Additionally, the ability to create a particular quantum state from good quantum numbers may depend on the specific experimental setup and techniques available.

How important are good quantum states and quantum numbers in quantum technologies?

Good quantum states and quantum numbers are essential in quantum technologies. These technologies rely on the precise control and manipulation of quantum systems, which can only be achieved through good quantum states and their associated quantum numbers. Without them, it would be impossible to accurately measure and control quantum systems, hindering the development and progress of quantum technologies.

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