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Uncertainty principle

We define the uncertainty of a measurement xx as its standard deviation σx\sigma_x, where

σx2=⟨(x−⟨x⟩)2⟩=⟨x2−2x⟨x⟩+⟨x⟩2⟩=⟨x2⟩−2⟨x⟩⟨x⟩+⟨x⟩2=⟨x2⟩−⟨x⟩2. \begin{align*} \sigma_x^2 &= \avg{(x-\avg{x})^2} = \avg{x^2 - 2x\avg x + \avg{x}^2} \\ &= \avg{x^2} - 2\avg{x}\avg{x} + \avg{x}^2 \\ &= \avg{x^2} - \avg{x}^2. \end{align*}

Now consider some observables AA and BB of a wave function ∣Ψ⟩\ket \Psi. By definition,

σA2=⟨(A^−⟨A^⟩)2⟩=⟨Ψ∣(A^−⟨A^⟩)2Ψ⟩. \begin{align*} \sigma_A^2 = \avg{(\hat A - \avg{\hat A})^2} = \braket{\Psi | (\hat A - \avg{\hat A})^2 \Psi}. \end{align*}

Since AA is observable, A^\hat A is Hermitian; so we can say

σA2=⟨(A^−⟨A^⟩)Ψ∣(A^−⟨A^⟩)2Ψ⟩=⟨f∣f⟩. \begin{align*} \sigma_A^2 = \braket{(\hat A - \avg{\hat A}) \Psi | (\hat A - \avg{\hat A})^2 \Psi} = \braket{f|f}. \end{align*}

Similarly,

σB2=⟨g∣g⟩∣g⟩=(B^−⟨B^⟩)∣Ψ⟩. \begin{align*} \sigma_B^2 &= \braket{g|g} \\ \ket g &= (\hat B - \avg{\hat B}) \ket\Psi. \end{align*}

Then, from the Cauchy-Schwarz inequality

σA2σB2=⟨f∣f⟩⟨g∣g⟩≥∣⟨f∣g⟩∣2. \begin{align*} \sigma_A^2 \sigma_B^2 = \braket{f|f} \braket{g|g} \ge |\braket{f|g}|^2. \end{align*}

For any complex number zz,

∣z∣2=Re(z)2+Im(z)2≥Im(z)2=(12i(z−z∗))2. \begin{align*} |z|^2 = \mathrm{Re}(z)^2 + \mathrm{Im}(z)^2 \ge \mathrm{Im}(z)^2 = \left(\frac{1}{2i}(z-z^*)\right)^2. \end{align*}

Which gives us

σA2σB2≥(12i(⟨f∣g⟩−⟨g∣f⟩))2. \begin{align*} \sigma_A^2 \sigma_B^2 \ge \left(\frac{1}{2i}(\braket{f|g} - \braket{g|f})\right)^2. \end{align*}

From some algebra (omitted) we see that ⟨f∣g⟩−⟨g∣f⟩=⟨[A^,B^]⟩\braket{f|g} - \braket{g|f} = \avg{[\hat A, \hat B]}, which finally gives us the generalized uncertainty principle

σAσB≥12i⟨[A^,B^]⟩. \sigma_A\sigma_B \ge \frac{1}{2i} \avg{[\hat A,\hat B]}.

In the special case of xx and pp, where [x^,p^]=iℏ[\hat x, \hat p] = i\hbar, we get the Heisenberg uncertainty principle σxσp≥ℏ2\sigma_x \sigma_p \ge \frac\hbar2.

Geometric interpretation

Consider a state ∣ψ⟩\ket \psi and an observable AA. Define a projector Pψ=∣ψ⟩⟨ψ∣P_\psi = \ket \psi \bra \psi. Then the projection of the measurement onto the state is

PψA∣ψ⟩=∣ψ⟩⟨ψ∣A∣ψ⟩=⟨A⟩∣ψ⟩. \begin{align*} P_\psi A \ket \psi &= \ket \psi \braket{\psi|A|\psi} = \braket{A} \ket \psi. \end{align*}

From which we compute the perpendicular component ∣ψ⊥⟩\ket{\psi_\perp}

∣ψ⊥⟩=(1^−Pψ)A∣ψ⟩=(A−⟨A⟩)∣ψ⟩⟨ψ⊥∣ψ⊥⟩=⟨ψ∣(A−⟨A⟩)2∣ψ⟩=σA2. \begin{align*} \ket{\psi_\perp} &= (\hat 1 - P_\psi) A \ket \psi = (A - \braket{A}) \ket \psi \\ \braket{\psi_\perp | \psi_\perp} &= \braket{\psi | (A - \braket{A})^2 | \psi} = \sigma_A^2. \end{align*}

Geometrically, we see that the uncertainty is the length of the component of A∣ψ⟩A\ket\psi perpendicular to ∣ψ⟩\ket \psi. Intuitively you can think of this as “how much does measuring AA change the state”.