Commutators
The origin of the Heisenberg Uncertainty Principle.
In quantum mechanics, physical observables (such as position, momentum, and energy) are represented by operators. In general, the order in which operators act on a wavefunction matters. To quantify this, we define the commutator of two operators \( \hat{A} \) and \( \hat{B} \) as
\[ [\hat{A},\hat{B}] \equiv \hat{A}\hat{B} - \hat{B}\hat{A}. \]
If the commutator is zero, the operators are said to commute, and the order in which they act does not matter. If the commutator is nonzero, the operators do not commute, and their order of operation changes the result.
Commutators play a central role in quantum mechanics because they determine whether two observables can be known simultaneously with arbitrary precision. In fact, nonzero commutators are directly connected to uncertainty relations.
Worked example: position and momentum in one dimension
In one dimension, the position operator is \( \hat{x}=x \), and the momentum operator is \( \hat{p}_x=-i\hbar\,\frac{d}{dx} \). We evaluate their commutator by letting it act on an arbitrary test wavefunction \( \psi(x) \).
First compute \( \hat{x}\hat{p}_x\psi(x) \):
\[ \hat{x}\hat{p}_x\psi(x) = x\left(-i\hbar\frac{d\psi}{dx}\right) = -i\hbar\,x\,\frac{d\psi}{dx}. \]
Next compute \( \hat{p}_x\hat{x}\psi(x) \). The momentum operator must act on the entire product.
\( x\psi(x) \):
Because this derivative acts on a product of two functions, we must use the product rule:
\[ \frac{d}{dx}\!\left(u v\right) = u\,\frac{dv}{dx} + v\,\frac{du}{dx}. \]
This gives us the result
\[ \hat{p}_x\hat{x}\psi(x) = -i\hbar\,\frac{d}{dx}\!\left(x\psi(x)\right) = -i\hbar\left(\psi(x)+x\frac{d\psi}{dx}\right). \]
The commutator acting on \( \psi(x) \) is therefore
\[ [\hat{x},\hat{p}_x]\psi(x) = \hat{x}\hat{p}_x\psi(x) - \hat{p}_x\hat{x}\psi(x) = i\hbar\,\psi(x). \]
Since this result holds for any wavefunction, we conclude that
\[ [\hat{x},\hat{p}_x] = i\hbar. \]
This nonzero commutator shows that position and momentum do not commute and therefore cannot be simultaneously specified with arbitrary precision. This mathematical result underlies the Heisenberg uncertainty principle.
Useful properties of commutators
From the definition \( [\hat{A},\hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A} \), several important properties follow immediately:
- An operator always commutes with itself: \( [\hat{A},\hat{A}] = 0 \).
- Multiplying an operator by a constant does not change its commutation with itself: \( [k\hat{A},\hat{A}] = 0 \), where \(k\) is a constant.
- Constants factor out of commutators: \( [k\hat{A},\hat{B}] = k[\hat{A},\hat{B}] \).
- Reversing the order of operators changes the sign of the commutator: \( [\hat{A},\hat{B}] = -[\hat{B},\hat{A}] \).
These properties are often used to simplify commutators before explicitly evaluating them.
Big idea: commutators measure the incompatibility of quantum observables and reveal fundamental limits on what can be known at the same time. This is the origin of the Heisenberg Uncertainty Principle.