When analyzing spin dynamics, particularly in the context of magnetic resonance or quantum computing, it's often beneficial to describe the system from a rotating frame of reference. This allows for simplifications and intuitive interpretations of spin behaviour. In a rotating frame, the magnetic field may appear to be time-varying, which can lead to easier mathematical treatment.

Now, let's discuss the calculation of expectation values ⟨Sx⟩, ⟨Sy⟩, and ⟨Sz⟩ from this rotating frame perspective:

Expectation Value ⟨Sx⟩: In the rotating frame, the x-component of the spin operator, Sx, remains unchanged. Thus, the expectation value ⟨Sx⟩ can be calculated as usual, without any additional complexity due to the rotating frame.
Expectation Value ⟨Sy⟩: In the rotating frame, the y-component of the spin operator, Sy, evolves under the rotating frame Hamiltonian. This evolution may introduce additional time dependence in the expectation value ⟨Sy⟩, which needs to be accounted for in the calculations.
Expectation Value ⟨Sz⟩: In the rotating frame, the z-component of the spin operator, Sz, is affected by the rotation. The expectation value ⟨Sz⟩ may exhibit time dependence due to this effect, requiring careful consideration in the analysis.

Unitary matrix for the rotational operator

where 𝜃 is the angle of rotation. J is the total angular momentum operator. If we consider only spin degree of freedom then J =S = ℏ 𝜎/ 2

This matrix is unitary and it's determinant is one. If l f> = (C+ C-) represent the spin wave function as seen from a certain frame of reference then Rn(𝜃) l f> represent the spin wave function from rotated frame of reference.

Overall, while describing spin dynamics in a rotating frame simplifies the Hamiltonian, it introduces additional complexities in the time evolution of spin operators and expectation values. However, this approach provides valuable insights into spin behaviour and facilitates the analysis of spin-related phenomena in various applications, including magnetic resonance spectroscopy, quantum computing, and spintronics.

This note is a part of the Physics Repository.

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