Heisenberg's Uncertainty Principle

Theoretical Foundations and the Transition from Bohr's Model

• Bohr's Theory Context: Considerations within Bohr's theory treat the electron as a material particle. In this framework, both the position and momentum of the electron can be determined with accuracy.

• The de-Broglie Influence: The understanding of the electron shifted when it was considered in the form of a wave, as suggested by de-Broglie.

• Wave Nature Implications: Because a wave necessarily extends throughout a region of space, it becomes impossible to ascertain simultaneously the exact position and velocity of the electron more precisely at any given instant.

Heisenberg's Uncertainty Principle

• Historical Context: In 1927, Werner Heisenberg presented a fundamental principle known as the Heisenberg Uncertainty Principle.

• Definition: The principle states: "It is impossible to measure simultaneously the exact position and exact momentum of a body as small as an electron."

Mathematical Formulations

• Variable Definitions:

  • $\Delta x$: Uncertainty in the measurement of position.

  • $\Delta p$: Uncertainty in the measurement of momentum.

  • $m$: Mass of the particle.

  • $\Delta v$: Uncertainty in the measurement of velocity.

  • $h$: Planck's constant.

• Position-Momentum Relationship:   The uncertainty in position and momentum are related according to the following inequality:   $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

• Position-Velocity Relationship:   By substituting the definition of momentum ($p = m \cdot v$) into the uncertainty relation, the following expressions are derived:   $\Delta x \cdot m \Delta v \geq \frac{h}{4\pi}$   $\Delta x \cdot \Delta v \geq \frac{h}{4\pi m}$

• Energy-Time Relationship and Derivation:   The uncertainty principle can also be expressed in terms of energy ($\Delta E$) and time ($\Delta t$) through the following derivation steps:   $\Delta t \times \Delta x \times \frac{\Delta p}{\Delta t} \geq \frac{h}{4\pi}$   $F \times \Delta t \times \Delta x \geq \frac{h}{4\pi}$   $\Delta E \times \Delta t \geq \frac{h}{4\pi}$

Physical Implications and Limits of Precision

• Reciprocal Relationship: The uncertainties are inversely related; increasing the precision of one measurement necessarily decreases the precision of the other.

• Extreme Case - Infinite Uncertainty in Velocity:   When position is known with perfect accuracy ($\Delta x = 0$), the uncertainty in velocity becomes infinite:   $\Delta v = \infty$

• Extreme Case - Infinite Uncertainty in Position:   When velocity is known with perfect accuracy ($\Delta v = 0$), the uncertainty in position becomes infinite:   $\Delta x = \infty$

• General Rule: If the position is determined quite accurately (i.e., $\Delta x$ is very small), then the error or uncertainty in velocity ($\Delta v$) becomes very large, and vice versa.