Heisenberg's Uncertainty Principle
Structure of Atom | Quantum Mechanical Model | Class 11
1. Introduction
Proposed by Werner Heisenberg in 1927, this principle is a fundamental consequence of the dual nature of matter and radiation. It places a fundamental limit on the precision with which certain pairs of physical properties can be known.
2. Statement & Formula
Mathematically, the product of uncertainty in position ($\Delta x$) and uncertainty in momentum ($\Delta p$) is always greater than or equal to $\frac{h}{4\pi}$.
Since momentum $p = mv$, and mass ($m$) is constant:
$$ \Delta x \cdot m \Delta v \geq \frac{h}{4\pi} $$ $$ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} $$Where:
- $\Delta x$ = Uncertainty in position
- $\Delta p$ = Uncertainty in momentum
- $\Delta v$ = Uncertainty in velocity
- $m$ = Mass of the particle
- $h$ = Planck's constant ($6.626 \times 10^{-34} J \cdot s$)
3. Physical Significance
A. Microscopic Objects (Electrons)
For an electron ($m \approx 9.1 \times 10^{-31} kg$), the value of $\frac{h}{4\pi m}$ is significant. If we try to measure position accurately ($\Delta x$ is small), the uncertainty in velocity ($\Delta v$) becomes extremely large. This implies that electrons do not follow definite paths or trajectories (like Bohr's orbits).
B. Macroscopic Objects (Cricket Ball)
For a heavy object (e.g., a ball of 1 mg or 1 kg), the mass $m$ is very large. This makes the value of $\frac{h}{4\pi m}$ extremely small (negligible). Therefore, the uncertainties are too small to observe, and classical mechanics holds true. We can define precise trajectories for macroscopic objects.
4. Conclusion
Heisenberg's principle rules out the existence of definite circular paths or trajectories for electrons. Instead, we speak of the probability of finding an electron in a given region (Orbitals).
Practice Quiz
Test your understanding of Uncertainty.
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