Question

The speed of an electron is measured to be between 4.3x10^6 m/s and 4.5x10^6 m/s. What is the fractional uncertainty of its speed relative to the lower limit? What is the smallest possible uncertainity in its position? Explain everything in detail.

Answer 1

The upper and lower limits for the velocity measurements are given, so the value of velocity ranges as,

Fractional uncertainty is given be the ratio of the absolute certainty to the best value of the measurement. Absolute uncertainties are not errors but rather are the inherent uncertainty present in all instruments. The best value here is the center value oof 4.4 E+6 m/s.

Hence the fractional uncertainty of our experiment is,

Now, the uncertainty principle is a strange little thing in quantum mechanics. It states that the uncertainty of position and momentum are related to each other such that their product cannot be lower than some constant value.

Which means that if I want to make a very very accurate measurement of the electron's position in some box such that , then the uncertainty in the measurement of the momentum will have to be ATLEAST,

As the uncertainty in position tends to 0, the uncertainty in momentum measurement blows up to crazy high values. The official equation of the principle is,

This means the properties of position and momentum of a wavefunction are inherently related. And indeed they are, but that's me digressing.

Now, momentum of the electron can be written as a product of its mass and velocity,

Taking just the uncertainty, .

Substituting this value in the Uncertainty principle, we get,

So, we can measure the value of the electron's position within 0.5769 nanometers uncertainty.