Heisenberg Uncertainty Principle

Claimed by Nicholas Maldonado 11/27/22

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is a concept that describes uncertainty relationships between a particle's different properties, those being position and momentum, as well as energy and time.The biggest takeaway from this concept is that there is always some measure of uncertainty in at least one of any given particle's properties.

Generally speaking, uncertainty relationships are not unique to modern physics. This is a concept that occurs in classical physics as well. Consider a wave, described by some combination of sines and cosines. Any wave has an amount of uncertainty between its frequency and its position. For example, a wave with a single period would give you a lot of information about its position, but not much information about its frequency. Conversely, a wave of many periods would allow you to precisely determine its frequency, but the position of the wave would be less certain.

A Mathematical Model

The total uncertainty between a particle's position and momentum is given as [math]\displaystyle{ {\Delta p \Delta x} }[/math]. The minimum uncertainty a particle can have is [math]\displaystyle{ {\frac{{\hbar}}{2}} }[/math]. Therefore, we can establish the relationship of:

[math]\displaystyle{ {\Delta p \Delta x \geq \frac{{\hbar}}{2}} }[/math]

Oftentimes, it is also good enough to simply approximate their general relationship as:

[math]\displaystyle{ {\Delta p \Delta x} }[/math] ~ [math]\displaystyle{ {\hbar} }[/math]

NOTE: [math]\displaystyle{ {\hbar = \frac{{h}}{2\pi}} }[/math] where [math]\displaystyle{ {h} }[/math] is Planck's constant.

This principle can also be extended to a relationship between energy and time, where:

[math]\displaystyle{ {\Delta E \Delta t \geq \frac{{\hbar}}{2}} }[/math]

and

[math]\displaystyle{ {\Delta E \Delta t} }[/math] ~ [math]\displaystyle{ {\hbar} }[/math]

A Computational Model

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Connectedness

Heisenburg uncertainty principle later plays into quantum tunneling, where this inherent uncertainty in a particle may sometimes allow it to overcome a potential energy barrier it otherwise would not be able to overcome classically.
As a consequence of Heisenberg uncertainty principle, microelectronics are limited in size because electrons are able to tunnel through transistors that are small enough for uncertainty principle to have an effect.

History

The development of the Heisenberg uncertainty principle is largely accredited to Werner Heisenberg.

Single-slit Diffraction Experiments

Heisenberg uncertainty relationships were first physically demonstrated through single-slit diffraction experiments. The experiment is conducted by launching beam of electrons through a narrow slit. Initially, the electrons have zero momentum in the x-direction. After passing through the slit, information about an electron's x position is acquired, resulting in an increased uncertainty in the particle's x momentum. Making the slit narrower therefore increases the uncertainty of the particle's x momentum while decreasing the uncertainty of its x position, while making the slit wider has the opposite effect. This effectively demonstrates the relationship between the uncertainty of a particle's position and momentum.

The Uncertainty Principle. Uncertainty principle. (n.d.). Retrieved April 20, 2022, from hyperphysics.phy-astr.gsu.edu

Libretexts. (2021, April 23). Heisenberg's Uncertainty Principle. Chemistry LibreTexts. Retrieved April 20, 2022, from chem.libretexts.org

Heisenberg Uncertainty Principle

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