Position and momentum operators on Euclidean L2

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix $\mathrm{\hslash }>0$ and an integer $d\ge 1$, and work in the complex Hilbert space $L^2({\mathbb{R}}^d;\mathbb{C})$ with inner product $⟨f,g⟩={\int }_{{\mathbb{R}}^d}f\bar{g}$ linear in the first argument (Complex Lp classes and Euclidean test-function conventions, The space $L^p(\mu )$ as the quotient by null functions, $L^2$ with the integral pairing is a Hilbert space).

Weak derivatives. For $f\in L^2({\mathbb{R}}^d;\mathbb{C})$ and $j\in \{1,\dots ,d\}$, say that the $j$-th weak derivative of $f$ exists in $L^2$ when there is a class $g\in L^2({\mathbb{R}}^d;\mathbb{C})$ with ${\int }_{{\mathbb{R}}^d}f{D}_j\phi =(-1){\int }_{{\mathbb{R}}^d}g\phi \text{for every }\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d),$ where $D_j\phi$ is the classical partial derivative and the pairing is bilinear without conjugation (Weak derivative of a locally integrable function). Such a $g$ is unique as an $L^2$ class (Uniqueness of a weak derivative as an almost-everywhere class); write $D_{j}f:=g$. The Sobolev space $W^{1,2}({\mathbb{R}}^d;\mathbb{C})$ is the set of classes $f\in L^2$ for which $D_{1}f,\dots ,D_{d}f$ all exist in $L^2$, and $W^{2,2}({\mathbb{R}}^d;\mathbb{C})$ is the set of $f\in W^{1,2}$ with $D_{j}f\in W^{1,2}$ for every $j$, with $D_j{D}_{k}f$ written for the iterated derivative (Integer-order Sobolev spaces and their norms).

(a) Position operators. For $j\in \{1,\dots ,d\}$ define $D(Q_j):=\{f\in L^2({\mathbb{R}}^d;\mathbb{C}):x\mapsto x_{j}f(x)\in L^2({\mathbb{R}}^d;\mathbb{C})\},(Q_{j}f)(x):=x_{j}f(x)\mathrm{.}$ Thus $Q_{j}f$ is the class of the pointwise product with the $j$-th coordinate function.

(b) Momentum operators. For $j\in \{1,\dots ,d\}$ define $D(P_j):=\{f\in L^2({\mathbb{R}}^d;\mathbb{C}):D_{j}f\text{ exists in }{L}^2\},P_{j}f:=-i\mathrm{\hslash }{D}_{j}f\mathrm{.}$

Both $Q_j$ and $P_j$ are densely defined $\mathbb{C}$-linear operators in $L^2({\mathbb{R}}^d;\mathbb{C})$: for $Q_j$ the compactly supported smooth classes lie in the domain because $x_j$ is bounded there, and for $P_j$ they lie in the domain because their classical derivatives are weak derivatives in $L^2$ (Integer-order Sobolev spaces and their norms).

For density, the classes of $C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})$ are dense in $W^{0,2}({\mathbb{R}}^d;\mathbb{C})=L^2({\mathbb{R}}^d;\mathbb{C})$ by Compactly supported smooth functions are dense in W^{k,p}(R^n) and Integer-order Sobolev spaces and their norms. Since these classes lie in each domain as above, both operator domains are dense.

For a class $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ one has $Q_{j}f=x_{j}f$ and $P_{j}f=-i\mathrm{\hslash }{\mathrm{\partial }}_{j}f$ pointwise. The self-adjointness of these operators and their Fourier representation are proved in the companion self-adjointness lemma on this page, not asserted here; for $d=1$ the domain $D(P)$ of the single momentum operator coincides with $W^{1,2}(\mathbb{R};\mathbb{C})$.