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Physlib.QuantumMechanics.OneDimension.Operators.Commutation

Commutation relations

The commutation relations between different operators.

Commutation relation between position and momentum operators

3 declarations

theorem

Canonical Commutation Relation [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar for Schwartz functions

For any Schwartz function ψ∈S(R,C)\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C}), the position operator x^\hat{x} and the momentum operator p^\hat{p} satisfy the following commutation relation: x^(p^ψ)−p^(x^ψ)=iℏψ\hat{x}(\hat{p}\psi) - \hat{p}(\hat{x}\psi) = i\hbar\psi where ii is the imaginary unit and ℏ\hbar is the reduced Planck constant. Here, the position operator x^\hat{x} acts as (x^ψ)(x)=xψ(x)(\hat{x}\psi)(x) = x\psi(x) and the momentum operator p^\hat{p} acts as (p^ψ)(x)=−iℏdψdx(x)(\hat{p}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x).

theorem

x^p^ψ=p^x^ψ+iℏψ\hat{x}\hat{p}\psi = \hat{p}\hat{x}\psi + i\hbar\psi for Schwartz functions

For any Schwartz function ψ∈S(R,C)\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C}), the position operator x^\hat{x} and the momentum operator p^\hat{p} satisfy the relation: x^(p^ψ)=p^(x^ψ)+iℏψ\hat{x}(\hat{p}\psi) = \hat{p}(\hat{x}\psi) + i\hbar\psi where ii is the imaginary unit and ℏ\hbar is the reduced Planck constant.

theorem

p^x^=x^p^−iℏ\hat{p}\hat{x} = \hat{x}\hat{p} - i\hbar for Schwartz functions

For any Schwartz function ψ∈S(R,C)\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C}), the composition of the momentum operator p^\hat{p} and the position operator x^\hat{x} satisfies the relation: p^(x^ψ)=x^(p^ψ)−iℏψ\hat{p}(\hat{x}\psi) = \hat{x}(\hat{p}\psi) - i\hbar\psi where ii is the imaginary unit, ℏ\hbar is the reduced Planck constant, the position operator is defined by (x^ψ)(x)=xψ(x)(\hat{x}\psi)(x) = x\psi(x), and the momentum operator is defined by (p^ψ)(x)=−iℏdψdx(\hat{p}\psi)(x) = -i\hbar \frac{d\psi}{dx}.