The Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states that it is not possible for us to know both the exact momentum and exact location of an electron in space. Heisenberg related the uncertainty in the position, Δx and the uncertainty in momentum, Δ(mv) as follows:

\(\displaystyle \Delta x⋅\Delta(mv)\;\geq\;\frac{h}{4\pi}\;\;\;(Equation\;1)\)

It is not possible to know both the position and the momentum of a moving particle, with small masses, like that of an electron, at the same time. The more accurately we know the speed, the less accurately we know the position, and vice versa. Suppose we have an electron with a mass of 9.11 x 10-31 kg moving at a speed of 3.0 x 105 m/s. If we know the speed to an uncertainty of 1% we can calculate the uncertainty in the position of the electron. The uncertainty is 0.01 x (3.0 x 105 m/s) = 3.0 x 103 m/s.

\(\displaystyle \Delta x\;\geq\;\frac{h}{4\pi m\Delta\nu}\;=\;\frac{6.626\times\;10^{-34}\frac{kg⋅s^2}{m^2}⋅s}{4\pi(9.11\times\;10^{-31}kg)\times\;(3.0\times\;10^3\;m/s)}\;=\;1.9\times\;10^{-8}\;m\)

A hydrogen atom has a diameter of about 1 x 10-10 m. The uncertainty is much greater than the size of the atom, therefore, we cannot know the location of the electron in the atom. In other words, we cannot describe the electron as moving in a definite orbit.

If we have a golf ball with a mass of 0.0459 kg moving at a speed of 72.6 m/s. The uncertainty in the speed is 1% and is 0.01 x 72.6 m/s = 0.726 m/s

\(\displaystyle \Delta x\;\geq\;\frac{h}{4\pi m\Delta\nu}\;=\;\frac{6.626\times\;10^{-34}\frac{kg⋅s^2}{m^2}⋅s}{4\pi(0.0459\;kg\times\;0.726\;m/s)}\;=\;1.5\times\;10^{-33}\;m\)

This value is much too small to measure, and does not mean anything.

Thanks to de Broglie and Heisenberg, a new model of atomic structure was developed. The electron is described by its wave characteristics and predicts the energy of the electron as well the electron’s location in terms of probabilities.

Exercises

Exercise 1. Calculate the uncertainty in the position of a proton moving at a speed of (2.80 ± 0.01) x 104 m/s.

Exercise 2. Calculate the uncertainty in the position of a neutron moving at a speed of (1.80 ± 0.01) x 105 m/s.

Check Solutions/Answers to Exercises

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