Heisenberg’s Uncertainty Principle
Commutators and Heisenberg’s Uncertainty Principle
The commutator of two operators \(\hat{A}\) and \(\hat{B}\) is defined as
\[ \left[ \hat{A} , \hat{B} \right]= \hat{A}\hat{B} - \hat{B}\hat{A}. \]
Key Properties of Commutators
- Two observables \(\hat{A}\) and \(\hat{B}\) have a complete set of common eigenstates
if and only if their commutator vanishes: \[ \left[ \hat{A} , \hat{B} \right]= \hat{0}. \] - If \(\left[ \hat{A} , \hat{B} \right]\neq \hat{0}\), the observables are called incompatible.
They cannot, in general, be measured sharply at the same time.
- This incompatibility is made quantitative by the Heisenberg uncertainty principle: \[ \sigma_A \, \sigma_B \;\geq\; \tfrac{1}{2}\,\big|\langle \left[ \hat{A} , \hat{B} \right]\rangle\big|, \] where \(\sigma_A\) and \(\sigma_B\) denote the standard deviations of observables \(A\) and \(B\).
Let \(\hat{A}\) and \(\hat{B}\) be Hermitian operators.
Define centered operators \[
\Delta \hat{A} = \hat{A} - \langle \hat{A} \rangle, \qquad
\Delta \hat{B} = \hat{B} - \langle \hat{B} \rangle.
\]
For any real number \(\lambda\), consider the inner product \[ \langle \psi | (\Delta \hat{A} + i \lambda \Delta \hat{B})^\dagger (\Delta \hat{A} + i \lambda \Delta \hat{B}) | \psi \rangle \;\geq\; 0. \]
Remark. This is just \(\| (\Delta \hat{A} + i \lambda \Delta \hat{B}) |\psi\rangle \|^2\),
the squared norm of a vector in Hilbert space.
Since norms are always nonnegative, the whole expression is \(\geq 0\).
Expanding gives \[ \langle (\Delta \hat{A})^2 \rangle + \lambda^2 \langle (\Delta \hat{B})^2 \rangle + i \lambda \langle [\Delta \hat{A}, \Delta \hat{B}] \rangle \;\geq\; 0. \]
The expression \[ f(\lambda) = \langle (\Delta \hat{A})^2 \rangle + \lambda^2 \langle (\Delta \hat{B})^2 \rangle + i \lambda \langle [\Delta \hat{A}, \Delta \hat{B}] \rangle \] is a quadratic function of the real variable \(\lambda\).
Remark. The coefficient of \(\lambda^2\) is \(\langle (\Delta \hat{B})^2 \rangle = \sigma_B^2\),
which is strictly positive unless the state is an exact eigenstate of \(\hat{B}\).
Therefore \(f(\lambda)\) is a parabola that opens upwards, ensuring that it has a unique global minimum.
Its minimum value occurs at \[ \lambda_{\min} = -\, \frac{i \langle [\Delta \hat{A}, \Delta \hat{B}] \rangle}{2 \langle (\Delta \hat{B})^2 \rangle}. \]
Substituting back: \[ f(\lambda_{\min}) = \langle (\Delta \hat{A})^2 \rangle - \frac{1}{4 \langle (\Delta \hat{B})^2 \rangle} \, \big| \langle [\Delta \hat{A}, \Delta \hat{B}] \rangle \big|^2 \;\geq\; 0. \]
Rearranging gives \[ \langle (\Delta \hat{A})^2 \rangle \, \langle (\Delta \hat{B})^2 \rangle \;\geq\; \tfrac{1}{4} \big| \langle [\hat{A}, \hat{B}] \rangle \big|^2. \]
Since \(\sigma_A^2 = \langle (\Delta \hat{A})^2 \rangle\) and
\(\sigma_B^2 = \langle (\Delta \hat{B})^2 \rangle\), we finally obtain \[
\sigma_A \, \sigma_B \;\geq\; \tfrac{1}{2} \, \big| \langle [\hat{A}, \hat{B}] \rangle \big|.
\]
This is the Heisenberg uncertainty principle.
Important Examples
Position–Momentum Commutator
\[
\left[ \hat{r}_i , \hat{p}_j \right]= i\hbar \, \delta_{ij},
\] where \(i,j\) label Cartesian components.
This leads directly to the familiar uncertainty relation \[
\sigma_x \, \sigma_{p_x} \geq \tfrac{\hbar}{2}.
\]
Angular Momentum Commutators
The components of angular momentum form a Lie algebra: \[ \left[ \hat{L}_x , \hat{L}_y \right]= i\hbar \hat{L}_z, \qquad \left[ \hat{L}_y , \hat{L}_z \right]= i\hbar \hat{L}_x, \qquad \left[ \hat{L}_z , \hat{L}_x \right]= i\hbar \hat{L}_y. \]
Commuting Quantities
Some pairs of operators do commute, which means they can be simultaneously diagonalized: \[ \left[ \hat{\vec L}^2 , \hat{L}_z \right]= \hat{0}, \qquad \left[ \hat{\vec L}^2 , \hat{H} \right]= \hat{0}. \]
The uncertainty principle is not about experimental imperfections.
It is a fundamental feature of quantum mechanics: non-commuting observables simply do not admit sharp simultaneous values.The stronger the commutator, the stronger the tradeoff between precisions of measurement.