# Heisenberg’s Uncertainty Principle

The uncertainty relation is closely related to the quantum wave nature of things. It summarizes the impossibility for two complementary entities, such as position and momentum to be defined with arbitrary precision in the same experimental setting. This is summarized in the inequality:

$$\Delta x \cdot \Delta p_x \ge \frac{\hbar}{2}$$

• $$\Delta x$$ is the accuracy (standard deviation) to which the particle position is defined in the x-direction.
• $$\Delta p_x$$ is the accuracy (standard deviation) to which the particle momentum is defined in the x-direction.
• $$\hbar \simeq 1.1 \times 10^{-34} J s$$ is the reduced Planck quantum of action.

The prefactor of $$\hbar$$  depends on how you define the accuracy of position and momentum. But it is of the order of one.

We can demonstrate the effect and importance of the uncertainty relation in an experiment.

## Experimental Challenge: Molecules at a single slit

Go to the lab and follow the instructions. When you are done, return to this page and continue.

When you reduce the slit width to determine the particle’s position with increasing accuracy you expect to reduce the effective molecular beam width, first. However, below a certain width the increasing momentum uncertainty and with it the directional uncertainty increases and translates into a large position uncertainty further downstream. This is a key mechanism exploited for the preparation of transverse coherence in many of the following experiments: molecules pass through this narrow slit are first delocalized in their momentum and as a consequence of that also delocalized in position, further downstream.

## Uncertainty relation

Test your knowledge!

The following graph shows the width of a molecular beam behind a slit as a function of its width. If the slit width decreases below the uncertainty threshold the molecular beam expands