uncertainty in position of electron

Question

The mass of an electron is 9.11x10^-31 kg  What is the incertainty in the position of an electron moving at 4.00x10^6 m/s with an uncertainty of change in v = 0.01x10^6 m/s?

Answer 1

Take Planck's Constant as h = 6.62606876E-34 J•s.

Take (Light Velocity, Electron Mass) as (c, m_e) = (299792458 m/s, 9.109389754E-31kg).

Relativistic Mass m_r is calculated by way of

9.109389754E-31kg ÷ √[1 − (4E6/2.99792458E8)² ] equal to 9.110200707E-31 kg.

Here m_e & m_r have only a tiny difference; either can be used.

Choosing m_r, guide on the formula Δx ≥ h / 4πm_rΔv.

That is to say, Δx is at least 6.62606876E-34 J•s divided by

(4π • 9.110200707E-31 kg • 1E4 m/s).

This last simplifies to Δx ≥ 5.787861488E-9 meters, roughly

5.788 nanometers (or 5.788 billionths of a meter).