Translation operator unitarily equivalent to multiplication by exponential

Question

This is part of a problem from Hall's book "Quantum Theory for Mathematicians".

Determine the unitary operator $U:L^2(\mathbb{R^n})\to L^2(\mathbb{R^n})$ (unique up to a constant) such that
$$Ue^{itP_j}U^{-1}=e^{itX_j}$$
$$Ue^{-itX_j}U^{-1}=e^{itP_j}.$$

For those unfamiliar, $X_j$ and $P_j$ are the position and momentum operators defined as multiplication by $x_j$ and $-i\hbar\frac{\partial}{\partial x_j}$ respectively. From here, one can show that $e^{itX_j}$ is just multiplication by $e^{itx_j}$ and $e^{itP_j}$ is just translation to the left by $t\hbar\mathbf{e_j}$. So we are looking for a unitary operator $U$ such that
$$UT^j_{t\hbar}U^{-1}\psi(\mathbf{x})=e^{itx_j}\psi(\mathbf{x})$$
$$Ue^{-itX_j}U^{-1}\psi(\mathbf{x})=T^j_{t\hbar}\psi(\mathbf{x})=\psi(\mathbf{x}+t\hbar \mathbf{e_j}).$$

Something that might help is that $P_j$ and $X_j$ satisfy the exponentiated commutation relation:
$$e^{X_j}e^{P_j}=e^{X_j+P_j+\frac12[X_j,P_j]}=e^{X_j+P_j+\frac12i\hbar I}.$$

Answer 1

This is but the π/2 rotation in phase space, a canonical transformation generated by the quantum harmonic oscillator Hamiltonian $H=(X^2+P^2)/2\hbar$. I am skipping the superfluous subscripts j. The structure actually originates in classical mechanics.

That is to say, given $$ [H,P]= i X, \qquad [H,X]= -iP ~. $$

It then follows that, from the Hadamard Lemma (adjoint action), $$ e^{-i\pi H/2}P e^{i\pi H/2}=P-i{\pi\over 2} [H,P] -{\pi^2\over 2!~~2^2 }[H,[H,P]]+...= X,\\ e^{-i\pi H/2}X e^{i\pi H/2}=X-i{\pi\over 2} [H,X] -{\pi^2\over 2!~~2^2 }[H,[H,X]]+...= -P, $$ a right-angle rotation. Intercalating this similarity transformation, further transforms all functions $f(P)\mapsto f(X)$, and $g(X)\mapsto g(-P)$, whence your exponential desiderata relations!

This was first noticed in print by Condon 1937, and serves as the formal underpinning of the celebrated Fractional Fourier Transform.