Uncertainty Relations

The goal of this section is to show that the uncertainties $\Delta A = \sqrt{<(\hat{A}-<\hat{A}>)^2>}$ and $\Delta B = \sqrt{<(\hat{B}-<\hat{B}>)^2>}$, of any pair of hermitian operators $\hat{A}$ and $\hat{B}$, satisfy the uncertainty relation:R3(437)

$$(\Delta A)^2(\Delta B)^2 \geq \frac{1}{4} < i [A, B] > ^2.$$ (7)

In particular, when $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}$, we obtain the Heisenberg uncertainty relation:

$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}.$$ (8)

Proof:

$\hat{U} \equiv \hat{A} - <A>,$          $\phi(\lambda, x) \equiv (\hat{U} + i \lambda \hat{V}) \Phi (x),$

$\hat{V} \equiv \hat{B} - <B>,$          $I(\lambda) \equiv \int{dx \phi^* (\lambda, x) \phi (\lambda, x)} \geq 0,$

$$I(\lambda) = \int{dx [(\hat{A} -<A>) \Phi(x) + i \lambda(\hat{B}-<B>) \Phi(x)]^*[(\hat{A}-<A>)\Phi(x)+ i \lambda( \hat{B} -<B>) \Phi(x)]},$$

$$I(\lambda) = <\hat{U}\Phi\vert\hat{U}\Phi> + \lambda^2 <V\Phi\vert V\Phi> - i \lambda (<V\Phi\vert U\Phi> - <U\Phi\vert V\Phi>),$$

$$I(\lambda) = <\Phi\vert U^2\vert\Phi> + \lambda^2 <\Phi\vert V^2\vert\Phi> - i \lambda <\Phi\vert UV - VU\vert\Phi> \geq 0,$$ (9)

The minimum value of $I(\lambda)$, as a function of $\lambda$, is reached when $\partial{I}/ \partial{\lambda} = \partial{I}/ \partial{\lambda^*} = 0.$

This condition implies that

$$2\lambda (\Delta B)^2 = i <[A,B]>, \qquad => \qquad \lambda= \frac{i<[A,B]>}{2(\Delta B)^2}.$$

Substituting this expression for $\lambda$ into Eq. (9), we obtain:

$$(\Delta A)^2 + \frac{i^2 <A,B>^2}{4(\Delta B)^2} - \frac{i^2 <A,B>^2}{2(\Delta B)^2} \geq 0,$$

$$(\Delta A)^2 (\Delta B)^2 \geq \frac{i^2 <A,B>^2}{4}.$$

Exercise 8: Compute $<X>$, $<P>$, $\Delta X$ and $\Delta P$ for the particle in the box in its minimum energy state and verify that $\Delta X$ and $\Delta P$ satisfy the uncertainty relation given by Eq. (8)?

With the exception of a few concepts (e.g., the Exclusion Principle that is introduced later in these lectures), the previous sections have already introduced most of Quantum Theory. Furthermore, we have shown how to solve the equations introduced by Quantum Theory for the simplest possible problem, which is the particle in the box. There are a few other problems that can also be solved analytically (e.g., the harmonic-oscillator and the rigid-rotor described later in these lectures). However, most of the problems of interest in Chemistry have equations that are too complicated to be solved analytically. This observation has been stated by Paul Dirac as follows: The underlying physical laws necessary for the mathematical theory of a large part of Physics and the whole of Chemistry are thus completed and the difficulty is only that exact application of these laws leads to the equations much too complicated to be soluble. It is, therefore, essential, to introduce approximate methods (e.g., perturbation methods and variational methods).