Heisenberg Uncertainty Principle

Overview

• A principle of quantum mechanics concerning the precision of measuring certain pairs of complementary variables, such as position and momentum.
• For a particle with mass m and velocity v, its linear momentum p is given by:
  • p = m * v
• According to the principle, exact position and momentum cannot be known simultaneously.
  • If position is known accurately, momentum is uncertain and vice versa.

Mathematical Expression

• The product of the uncertainties in position (Δx) and momentum (Δp) is compared to a constant:
  • Δx * Δp ≥ h / (4π)
  • h is Planck’s constant.

Inverse Proportionality

• Uncertainties in position and momentum are inversely proportional:
  • Increasing accuracy in position results in increased uncertainty in momentum.
  • Example calculations to illustrate:
    • Δx = 2, Δp = 2 gives Δx * Δp = 4
    • Decrease Δx to 1, Δp increases to 4 to maintain the product.

Application to Bohr Model of Hydrogen Atom

• Bohr model describes an electron orbiting a nucleus similar to a planet around the sun.
• Radius of electron in ground state: 5.3 x 10^-11 meters
  • Diameter: 1.06 x 10^-10 meters
• Calculating velocity and momentum:
  • Velocity: 2.2 x 10^6 meters/second
  • p = m * v with 10% uncertainty in velocity
  • Calculating uncertainty in momentum: 2.0 x 10^-25 kg m/s

Applying Uncertainty Principle

• Using momentum uncertainty to find position uncertainty:
  • Δx * 2.0 x 10^-25 ≥ h / (4π)
  • Solving gives: Δx ≥ 2.6 x 10^-10 meters
• Conclusion for Bohr model:
  • Uncertainty in position exceeds diameter of hydrogen atom, indicating limitations of the Bohr model.

Intuition and Macroscopic Scale

• Everyday experience doesn’t align with the uncertainty principle due to larger scales and masses.
  • For larger particles, uncertainties become negligible.
• Quantum mechanics and its principles like the uncertainty principle are not intuitive.

Further Topics

• Future discussions will delve deeper into quantum mechanics and its effects on electrons and atoms.