Question

A proton has momentum 10 -20 N s and the uncertainty in the position of the proton is 10 -10 m. What is the minimum fractional uncertainty in the momentum of this proton?

Answer 1

According to Heisenberg's uncertainty principle,

ΔxΔp≥

4π

h

Where Δx&Δp are the uncertainty in position and momentum respectively and h is Planck’s Constant.

Now, the uncertainty in position here is simply the region where your electron is confined. Here, it is 10

−10

∴Δp≥

4π×Δx

h

⇒Δp≥

4×3.14×10

−10

6.63×10

−34

⇒Δp≥5.27×10

−25

gms

−1

Hence, the uncertainty in the momentum is 5.27×10

−25

gms

−1

Answer 2

Concept:

• Heisenberg's uncertainty principle
• Properties of protons
• Uncertainty calculations

Given:

• The momentum p = 10^-20 Ns
• The uncertainty in the position of proton Δx = 10^-10 m
• Value of Planck's constant h = 6.626*10^-34

Find:

• The uncertainty in the momentum of the proton

Solution:

The product of the uncertainties of momentum and position is equal to or greater than the ratio of Planck's constant to 4 π

For minimum uncertainty,

ΔpΔx = h/4π

Δp = h/Δx4π

Δp = 6.626*10^-34/4π* 10^-10

Δp = 5.27 * 10^-25

Fractional uncertainty = Δp/p

Δp/p = 5.27 * 10^-25/10^-20 = 5.27 * 10^-5

The fractional uncertainty in the momentum of the proton is 5.27 * 10^-5

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