## 3

## The Uncertainty Principle in Optics

### 3.1 Approximate Derivation of the Uncertainty Principle

Heisenberg’s uncertainty principle (Heisenberg 1927) is of fundamental importance to optics and to laser optics in particular. Here, optical arguments are applied to outline an approximate derivation of the uncertainty principle.

#### 3.1.1 The Wave Character of Particles

The quantum energy of a particle is given by the well-known quantum energy equation

$$\mathit{E}\mathrm{=}\mathit{h}\mathrm{v}\left(3.1\right)$$

where:

*h* is Planck’s constant

Equating this to the relativistic energy *E* = *mc*^{2} and using the identity λ = *c*/ν, an expression for the momentum of a particle can be given as

$$\mathit{p}\mathrm{=}\frac{\mathit{h}}{\mathit{\lambda}}\left(3.2\right)$$

which, using the identity

$$\mathit{k}\mathrm{=}\frac{\mathrm{2}\mathit{\pi}}{\mathit{\lambda}}\left(3.3\right)$$

can be restated as

$$\mathit{p}\mathrm{=}\mathit{\u0127}\mathit{k}\left(3.4\right)$$

This momentum equation was applied to particles, ...

Get *Tunable Laser Optics, 2nd Edition* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.