# 7.2 The heisenberg uncertainty principle

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By the end of this section, you will be able to:
• Describe the physical meaning of the position-momentum uncertainty relation
• Explain the origins of the uncertainty principle in quantum theory
• Describe the physical meaning of the energy-time uncertainty relation

Heisenberg’s uncertainty principle    is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately.

## Momentum and position

To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x -direction. The particle moves with a constant velocity u and momentum $p=mu$ . According to de Broglie’s relations, $p=\hslash k$ and $E=\hslash \omega$ . As discussed in the previous section, the wave function for this particle is given by

${\psi }_{k}\left(x,t\right)=A\left[\text{cos}\left(\omega \text{ }t-k\text{ }x\right)-i\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\omega \text{ }t-k\text{ }x\right)\right]=A{e}^{\text{−}i\left(\omega \text{ }t-k\text{ }x\right)}=A{e}^{\text{−}i\text{ }\omega \text{ }t}{e}^{i\text{ }k\text{ }x}$

and the probability density $|{\psi }_{k}\left(x,t\right){|}^{\text{ }2}={A}^{2}$ is uniform and independent of time. The particle is equally likely to be found anywhere along the x -axis but has definite values of wavelength and wave number, and therefore momentum. The uncertainty of position is infinite (we are completely uncertain about position) and the uncertainty of the momentum is zero (we are completely certain about momentum). This account of a free particle is consistent with Heisenberg’s uncertainty principle.

Similar statements can be made of localized particles. In quantum theory, a localized particle is modeled by a linear superposition of free-particle (or plane-wave) states called a wave packet    . An example of a wave packet is shown in [link] . A wave packet contains many wavelengths and therefore by de Broglie’s relations many momenta—possible in quantum mechanics! This particle also has many values of position, although the particle is confined mostly to the interval $\text{Δ}x$ . The particle can be better localized $\left(\text{Δ}x$ can be decreased) if more plane-wave states of different wavelengths or momenta are added together in the right way $\left(\text{Δ}p$ is increased). According to Heisenberg, these uncertainties obey the following relation.

## The heisenberg uncertainty principle

The product of the uncertainty in position of a particle and the uncertainty in its momentum can never be less than one-half of the reduced Planck constant:

$\text{Δ}x\text{ }\text{Δ}p\ge \hslash \text{/}2.$

This relation expresses Heisenberg’s uncertainty principle. It places limits on what we can know about a particle from simultaneous measurements of position and momentum. If $\text{Δ}x$ is large, $\phantom{\rule{0.2em}{0ex}}\text{Δ}p$ is small, and vice versa. [link] can be derived in a more advanced course in modern physics. Reflecting on this relation in his work The Physical Principles of the Quantum Theory , Heisenberg wrote “Any use of the words ‘position’ and ‘velocity’ with accuracy exceeding that given by [the relation] is just as meaningless as the use of words whose sense is not defined.”

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