## Heisenberg Uncertainty Principle

Werner Heisenberg considered the limits of how accurately we can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurately we measure the momentum of a particle, the less accurately we can determine its position at that time, and vice versa. This is summed up in what we now call the **Heisenberg uncertainty principle**: *It is fundamentally impossible to determine simultaneously and exactly both the momentum and the position of a particle*. For a particle of mass *m* moving with velocity *v _{x}* in the

*x*direction (or equivalently with momentum

*p*

_{x}), the product of the uncertainty in the position, Δ

*x*, and the uncertainty in the momentum, Δ

*p*, must be greater than or equal to \(\phantom{\rule{0.2em}{0ex}}\frac{\hslash }{2}\) (recall that ,\(\hslash \phantom{\rule{0.2em}{0ex}}\text{}=\phantom{\rule{0.2em}{0ex}}\frac{h}{2\pi },\) the value of Planck’s constant divided by 2

_{x}*π*).

\(\text{Δ}x\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{Δ}{p}_{x}=\left(\text{Δ}x\right)\left(m\text{Δ}v\right)\ge \phantom{\rule{0.2em}{0ex}}\cfrac{\hslash }{2}\)

This equation allows us to calculate the limit to how precisely we can know both the simultaneous position of an object and its momentum. For example, if we improve our measurement of an electron’s position so that the uncertainty in the position (Δ*x*) has a value of, say, 1 pm (10^{–12} m, about 1% of the diameter of a hydrogen atom), then our determination of its momentum must have an uncertainty with a value of at least

\(\left[\text{Δ}p=m\text{Δ}v=\phantom{\rule{0.2em}{0ex}}\cfrac{h}{\left(2\text{Δ}x\right)}\phantom{\rule{0.2em}{0ex}}\right]=\phantom{\rule{0.2em}{0ex}}\cfrac{\left(1.055\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-34}{\phantom{\rule{0.2em}{0ex}}\text{kg m}}^{\text{2}}\text{/s}\right)}{\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1{0}^{-12}\phantom{\rule{0.2em}{0ex}}\text{m}\right)}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1{0}^{-23}\phantom{\rule{0.2em}{0ex}}\text{kg m/s}.\)

The value of *ħ* is not large, so the uncertainty in the position or momentum of a macroscopic object like a baseball is too insignificant to observe. However, the mass of a microscopic object such as an electron is small enough that the uncertainty can be large and significant.

It should be noted that Heisenberg’s uncertainty principle is not just limited to uncertainties in position and momentum, but it also links other dynamical variables. For example, when an atom absorbs a photon and makes a transition from one energy state to another, the uncertainty in the energy and the uncertainty in the time required for the transition are similarly related, as Δ*E* Δ*t* ≥ .\(\frac{\hslash }{2}.\) As will be discussed later, even the vector components of angular momentum cannot all be specified exactly simultaneously.

Heisenberg’s principle imposes ultimate limits on what is knowable in science. The uncertainty principle can be shown to be a consequence of wave–particle duality, which lies at the heart of what distinguishes modern quantum theory from classical mechanics. Recall that the equations of motion obtained from classical mechanics are trajectories where, at any given instant in time, both the position and the momentum of a particle can be determined exactly.

Heisenberg’s uncertainty principle implies that such a view is untenable in the microscopic domain and that there are fundamental limitations governing the motion of quantum particles. This does not mean that microscopic particles do not move in trajectories, it is just that measurements of trajectories are limited in their precision. In the realm of quantum mechanics, measurements introduce changes into the system that is being observed.

Read this article that describes a recent macroscopic demonstration of the uncertainty principle applied to microscopic objects.