What is the Heisenberg Uncertainty Principle?

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What is the Heisenberg Uncertainty Principle?

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Answer 1 · 2014-12-19T11:15:54

It says that certain factors of a phenomenon are complementary: if you know a lot about one of the factors, you know little about the others.

Heisenberg talked about this in the context of a particle with a certain speed and location. If you know the speed very precisely, you wont know much about the location of the particle. It also works the other way around: if you know the location of a particle precisely, you won't be able to accurately describe the velocity of the particle.

(Source: what I remember from chemistry class. I'm not entirely sure if this is correct.)

Answer 2 · 2015-10-16T23:44:18

For a quantum mechanical (itty-bitty/subatomic) particle such as an electron, the Heisenberg Uncertainty Principle applies in a significant manner so as to assert that:

$σ_{x} σ_{p} \geq \frac{h}{4 π}$ $color(blue)(sigma_xsigma_p >= h/(4pi))$

What this says is that the product of the position's standard deviation $σ_{x}$ $sigma_x$ and the momentum's standard deviation $σ_{p}$ $sigma_p$ is too significantly large to make confident judgments about the statistics of the electron when the product $σ_{x} σ_{p}$ $sigma_xsigma_p$ is above $\frac{6.626 \times 10^{- 34} J \cdot s}{4 π}$ $(6.626xx10^(-34) "J"*"s")/(4pi)$ .

This is the main statement---that the more precisely you know the position of an electron, the less precisely you know its momentum, and vice versa.

Or, you could say that you can't observe both at the same time with good certainty.

Alone, they may be under $\frac{h}{4 π}$ $h/(4pi)$ , but multiplied, they may not be. This describes why you can observe one at a time.

For the electron using a "Particle in a Box" model (electron/particle in a chemical system/box), for example, it has been determined that:

$σ_{x} σ_{p} = \frac{h}{4 π} \sqrt{\frac{n^{2} π^{2}}{3} - 2}$ $color(green)(sigma_xsigma_p = color(blue)(h/(4pi))sqrt((n^2pi^2)/3 - 2))$

where $n$ $n$ is the principle quantum number you learned in General Chemistry, and $h$ $h$ is Planck's constant as usual.

You can tell that with the lowest value using $n = 1$ $n = 1$ (the lowest energy level and the best possible quantum mechanical condition), we still satisfy the condition:

$[σ_{x} σ_{p} = \frac{h}{4 π} \sqrt{\frac{π^{2}}{3} - 2}] \geq \frac{h}{4 π}$ $[color(blue)(sigma_xsigma_p) = h/(4pi)sqrt((pi^2)/3 - 2)] color(blue)(>= h/(4pi))$

since:

$[\sqrt{\frac{π^{2}}{3} - 2} \approx 1.136] > 1$ $[sqrt((pi^2)/3 - 2) ~~1.136] > 1$

In contrast, the uncertainties for normal objects like baseballs and basketballs are so low that we can say with good certainty what their positions and momenta are, mainly due to their size, giving them negligible wave characteristics.

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What is the Heisenberg Uncertainty Principle?

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