Q7(b)(i). Uncertainty in momentum and speed of an electron, Δ x≈50 pm

Given: Δ x=50 pm=50×10⁻¹² m; h=6.626×10⁻³⁴ J s; mₑ=9.109×10⁻³¹ kg; 1 J=1 kg m² s⁻².

1. The smallest uncertainty in momentum is the one that just satisfies the relation: Δ pₓ=(h)/(Δ x)
2. Δ pₓ=6.626×10⁻³⁴ kg m² s⁻¹50×10⁻¹² m, Δ pₓ=1.33×10⁻²³ kg m s⁻¹
3. The mass of the electron is known exactly, so Δ pₓ=mₑ Δ v: 3pt] Δ v=(Δ pₓ)/(mₑ)=1.3252×10⁻²³9.109×10⁻³¹, Δ v=1.45×10⁷ m s⁻¹

Comment 1: the uncertainty is larger than the quantity itself

• The electron in the first Bohr orbit of hydrogen moves at 2.19×10⁶ m s⁻¹, so its momentum is mₑ v=2.0×10⁻²⁴ kg m s⁻¹. The uncertainty in speed is 1.45×10⁷/2.19×10⁶=6.6 times the speed itself.
• Once the electron is placed within the size of an atom, its speed, and with it the next position, is unknown. A definite orbit (a path with position and speed fixed together) has no meaning. The electron is described instead by an orbital, which gives only the probability of finding it in a region.

Comment 2: the effect is large only for small masses

• Δ v is 1.45×10⁷/2.998×10⁸=4.9 % of the speed of light: an enormous spread for a particle bound in an atom. Δ v∝1/(m Δ x), so the small mass of the electron and the small Δ x together make it large.
• A body of 1 g located to the same 50 pm has Δ v=1.3×10⁻²⁰ m s⁻¹, far below anything measurable. The principle matters only for electrons, nuclei and other particles of atomic mass.

The exact form: with Δ pₓ Δ x≥ħ/2=h/4π, the limits are 4π times smaller: Δ pₓ=1.05×10⁻²⁴ kg m s⁻¹ and Δ v=1.16×10⁶ m s⁻¹, still about half the Bohr speed. Both conclusions remain.

Application. the energy–time form of the principle sets the width of spectral lines. A state of lifetime τ has an energy spread δ E≈ħ/τ (lifetime broadening): a lifetime of 1 ps gives a line about 5.3 cm⁻¹ wide, so short-lived excited states give broad lines.