Question about $e^{\frac{-itA}{\hbar}}(\hat{Q}+ \hat{P})e^{\frac{itA}{\hbar}}$

Question

This arises in the context of trying to rigorously understand quantum dynamics but it's a functional analysis issue. For simplicity suppose we are in dimension $1$.

Let $\hat{Q}$ be the operator that's multiplication by $x$ and $\hat{P} = \frac{h}{i}\frac{d}{dx}$. Suppose $A$ is a denseley defined self-adjoint operator on $L^2(\mathbb{R})$ and that $$e^{\frac{-itA}{\hbar}}(\hat{Q}+ \hat{P})e^{\frac{itA}{\hbar}} = a\hat{Q} + b\hat{P}$$ where $a,b \in\mathbb{R}$

The author then says it follows from "exponentiation" that $$e^{\frac{-itA}{\hbar}}e^{\frac{i}{\hbar}(\hat{Q}+ \hat{P})}e^{\frac{itA}{\hbar}} = e^{\frac{i}{\hbar}(a\hat{Q} + b\hat{P})}$$ I can't think of a rigorous explanation for what exponentiation means and why the first identity implies the second. In particular I'm uncomfortable with the fact that conjugating by a unitary operator goes through the exponentiation. I'm assuming this is a functional calculus thing. I was wondering if anyone knew of a rigorous explanation for this.

Answer 1

I'm not sure this is the level of rigor you need. But in physics this identity is normally motivated as follows (let me set $h=1$ for readability): $$ U = e^{it A}\;\Longrightarrow\; U U^{\dagger} = \mathbb{1}\,. $$ Then $$ \begin{aligned} U^\dagger\, e^{i(\hat{Q}+\hat{P})} U &= \sum_{n=0}^\infty U^\dagger\, \frac{i^n}{n!}(\hat{Q}+\hat{P})^n \,U \\&= \sum_{n=0}^\infty\frac{i^n}{n!}\, U^\dagger\, (\hat{Q}+\hat{P})\,UU^\dagger (\hat{Q}+\hat{P}) \cdots U U^\dagger (\hat{Q}+\hat{P}) \,U \\& =\sum_{n=0}^\infty\frac{i^n}{n!}\, \big(U^\dagger\, (\hat{Q}+\hat{P})\,U\big)^n = \\&= e^{i \,U^\dagger\, (\hat{Q}+\hat{P})\,U}\,. \end{aligned} $$ From which the desired identity follows. In the second line I simply inserted $\mathbb{1}$ in between each factor in the form of $UU^\dagger$.

I guess one should also motivate that the exponential of the operator $\hat{Q} + \hat{P}$ is well defined as a power series. For the operator $\hat{Q}$ alone this is just multiplication by $e^x$ and it clearly matches the sum $\sum_n x^n/n!$. For $\hat{P}$ the convergence of the exponential is the same as the convergence for the Taylor expansion $$ e^{ia\hat{P}} f(x) = \sum_{n=0}^\infty \frac{a^n}{n!}f^{(n)}(x) = f(x+a)\,. $$ Assuming $f \in C^\infty(\mathbb{R}) \cap L^2(\mathbb{R})$. In order to exponentiate $\hat{Q} + \hat{P}$ you may use Baker–Campbell–Hausdorff formula. $$ e^{\hat{Q}}e^{\hat{P}} = e^{\hat{Q}+\hat{P} + \frac12 i}\,, $$ where I used $[\hat{Q},\hat{P}] = i \mathbb{1}$ and the fact that all other terms are commutators with $\mathbb{1}$ which vanish.