Baker-Campbell-Hausdorff Formula

hfwang132@gmail.com (hfwang132) — Thu, 07 Mar 2024 16:13:09 +0800

Chinese version here

Baker-Campbell-Hausdorff Formula can be used to compute operator evolution in the Heisenberg picture:

$e^X Y e^{-X}=Y+[X,Y]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots$

This formula is actually just a younger sibling of the BCH formula.

Because the evolution rule of operators in the Heisenberg picture is $A\rightarrow UAU^{\dag}$, where $U$ is a unitary evolution operator.

If $U$ is generated by $H$, then it becomes $A\rightarrow e^{\frac{t}{i\hbar}H}Ae^{-\frac{t}{i\hbar}H}$.

Example 1: Phase Shifter The Hamiltonian is $H=\varphi n$, and the annihilation operator $a$ evolves as: $\begin{aligned} e^{-i\varphi n} a e^{i\varphi n}&= a + i\varphi [n, a] - \frac{\varphi}{2!} [n,[n,a]] - \cdots \\ &= a (1+i\varphi -\frac{\varphi^2}{2!} - \cdots)\\ &= e^{i\varphi} a \end{aligned}$

What is the relationship between Lie derivative and covariant derivative?

hfwang132@gmail.com (hfwang132) — Sat, 15 Jul 2023 16:52:03 +0800

I. Differences and Similarities in Properties

Lie derivative $\mathcal{L}_V$ and covariant derivative $\nabla_V$ share many common points:

Both $\mathcal{L}_V$ and $\nabla_V$ preserve the type of tensors, mapping $\mathcal{T}^p_q(M)$ to $\mathcal{T}^p_q(M)$. $\mathcal{T}^p_q(M)$ represents the set of all smooth tensor fields of type (p, q) on $M$.

Particularly, for (0,0) type tensor fields, i.e., scalar fields $f\in \mathcal{F}(M)$, we have $\mathcal{L}_V f=\nabla_V f=Vf$.

Both satisfy linearity and the Leibniz rule:

$ \begin{aligned} \mathcal{L}_V(\mu A + \lambda B) &= \mu \mathcal{L}_V A + \lambda \mathcal{L}_V B, \\ \mathcal{L}_V (A \otimes B) &= (\mathcal{L}_V A)\otimes B + A \otimes (\mathcal{L}_V B) \end{aligned} $

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Baker-Campbell-Hausdorff Formula

What is the relationship between Lie derivative and covariant derivative?

I. Differences and Similarities in Properties