notes
home
math
algebra and trigonometry
differential equations
linear algebra
multivariable calculus
probability and statistics
physics
classical mechanics
electricity and magnetism
vibrations and waves
quantum mechanics
relativity
thermodynamics and stat. mech.
computer science
math for computer science
languages
french

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-Schwartz 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.