Contingent of the Theorem of Campbell-Baker-Hausdorff-Dynkin with Dirac Matrices

According to Gottfried’s spin representation¹ under a rotating transverse field ψ(t) = e^-iI_zωt e^{-i(I_z(ω₀−ω)+I_xω₁)t}ψ(0),² the second matrix exponential operator prevents one to obtain the exact answer attributed to the mix of diagonal and off-diagonal parts in the exponent I_z(ω₀ − ω) + I_xω₁. The CBHD(Campbell-Baker-Hausdorff-Dynkin) formula³ for general Lie group u, v ∈ G expands the exponential functions to infinite order in the Lie series e^ue^v = e^{u+v+1/2[u,v]+···}, that means, expanding Gottfried’s solution with any finite number of high-order terms is not exact. The above difficulty leads to the probability evaluation only for the single state m to single state m′ transition.

Here with Dirac matrices I_z, I_x, I_y for arbitrary spin, we obtain the exact answer ψ(t) = e^-iI_zωte^-iI_yθe^-iI_zΩte^iI_yθψ(0) by making use of the third element of the Lie group I_y to diagonalize the off-diagonal exponent by a unitary rotation, and ψ(t) satisfies the time-dependent Schrödinger equation. Our expression has a single operator for each matrix exponent which makes it convenient to fully evaluate the probability density matrix, in other words, the general state to general state transition probability can be found. The operation can be utilized for NMR and EPR precise measurements of atomic magnetic and electric moments.

¹ K. Gottfried, “Quantum Mechanics Volume I: Fundamentals”, Benjamin (1966)

² Where ω₀, ω₁, ω are Larmor frequencies of homogeneous, inhomogeneous fields and transverse rotation.

³ A. Bonfiglioli and R. Fulci, “Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin”, Springer (2012)

* h/2π = 1 in our expression.