# Time dependent Pauli matrices

For an operator $\sigma$ (which can be written in terms of Pauli matrices $\sigma_{x/y/z}$), the time evolution can be given by standard quantum rule $\sigma(t) = U^\dagger(t) \sigma U(t)$, where $U(t)=exp(-iHt)$, for some Hamiltonian $H$. I have come across the representation in which $\sigma(t_k)=n_k.\sigma$, where $n_i$ is the unit vector. It seems that the time dependence is shifted from the vector $\sigma$ to the unit vector $n_k$. How is this justified?

• If you have a $2 \times 2$ matrix that is a linear combination (with real coefficients) of the 3 traceless and hermitian pauli matrices, you end up with a hermitian and traceless operator. In fact, with arbitrary real coefficients you get the most general hermitian and traceless $2 \times 2$ matrix. When you time-evolve them, you use a unitary transformation which preserves the trace and hermiticity. The result can then again be written as a linear combination (with real, time-dependent coefficients) of the pauli matrices. – secavara Mar 23 '18 at 20:36
• Thanks, @secavara. Could you kindly suggest some references? – W. Voltera Mar 23 '18 at 20:52
• I think you can find most of these properties scattered in different parts of the standard quantum mechanics books or in wikipedia. I cannot really point to a specific section where I could find all of them together... section 3.2 in Sakurai's book, Modern Quantum Mechanics, mentions several properties of the pauli matrices and then in the first chapter it discusses properties of unitary transformations. – secavara Mar 23 '18 at 21:18
• @secavara you can write that as an answer! – glS Mar 26 '18 at 10:08

## 1 Answer

If you have a 2$\times$2 matrix that is a linear combination (with real coefficients) of the 3 traceless and hermitian Pauli matrices, you end up with a hermitian and traceless operator. In fact, with arbitrary real coefficients you get the most general hermitian and traceless 2$\times$2 matrix. When you time-evolve them, you use a unitary transformation which preserves the trace and hermiticity. The result can then again be written as a linear combination (with real, time-dependent coefficients) of the Pauli matrices.

• Yes, this is adjoint action on a Pauli vector--formally the Rodrigues rotation formula for the direction of the original orientation of σ. – Cosmas Zachos Nov 8 '18 at 22:45