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Chemistry · Ch 2 — Structure of Atom

Significance of Uncertainty Principle

2.5.2a

Significance of Uncertainty Principle

Significance of the Uncertainty Principle

Trajectories Do Not Exist for Electrons

One of the most profound implications of the uncertainty principle is that it rules out the existence of definite paths or trajectories for electrons and other subatomic particles.

The trajectory of an object is determined by its location and velocity at various moments. If you know where a body is at a particular instant, and you also know its velocity and the forces acting on it, you can predict where it will be at any later time. Position and velocity together fix the trajectory.

For a subatomic object like an electron, it is impossible to simultaneously determine position and velocity at any given instant to an arbitrary degree of precision. Therefore, it is not possible to speak of the trajectory of an electron. The classical idea of a well-defined path simply does not apply.

Why Macroscopic Objects Are Unaffected

The uncertainty principle is significant only for microscopic objects. For macroscopic objects, the uncertainties are so tiny that they are completely negligible.

Consider an object of mass about one milligram (10−610^{-6} kg). The product Δv⋅Δx\Delta v \cdot \Delta x obtained from the uncertainty principle is extremely small — far below any measurable threshold. For milligram-sized or heavier objects, the associated uncertainties are of no practical consequence.

Now consider an electron with mass 9.11×10−319.11 \times 10^{-31} kg. The product Δv⋅Δx\Delta v \cdot \Delta x is much larger. If we try to locate the electron to within an uncertainty of only 10−810^{-8} m in position, then:

Δv≥h4πmΔx\Delta v \ge \frac{h}{4\pi m \Delta x}

Δv≥6.626×10−344×3.1416×9.11×10−31×10−8\Delta v \ge \frac{6.626 \times 10^{-34}}{4 \times 3.1416 \times 9.11 \times 10^{-31} \times 10^{-8}}

Δv≥6.626×10−341.145×10−37\Delta v \ge \frac{6.626 \times 10^{-34}}{1.145 \times 10^{-37}}

Δv≥5.79×103 m s−1  (≈104 m s−1)\Delta v \ge 5.79 \times 10^3 \text{ m s}^{-1} \; (\approx 10^4 \text{ m s}^{-1})

This uncertainty in velocity is enormous by everyday standards — several kilometres per second. Such a large uncertainty means that the classical picture of electrons moving in fixed, well-defined orbits (like Bohr's orbits) cannot hold. (Locating the electron still more precisely makes it worse: for Δx=0.1\Delta x = 0.1 Å the uncertainty grows to 5.79×1065.79 \times 10^6 m s−1^{-1} — the chapter's Problem 2.15.) …

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