21.8 Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle is one of the most profound results of quantum mechanics. It states that there are fundamental limits to how precisely certain pairs of physical quantities can be known simultaneously — not because of instrument limitations, but because of the wave-particle nature of matter itself.

---

21.8.1 Position–Momentum Uncertainty

Statement: It is impossible to simultaneously measure both the position ($x$) and the linear momentum ($p$) of a particle with perfect precision. The product of their uncertainties satisfies:

$\Delta x\cdot \Delta p\ge \hslash$

where $\hslash =\frac{h}{2\pi }\approx 1.054\times 10^{-34}$ J s is the reduced Planck's constant.

• $\Delta x$ = uncertainty in position
• $\Delta p$ = uncertainty in momentum

If $\Delta x$ is made smaller (position measured more precisely), then $\Delta p$ must become larger (momentum becomes less certain), and vice versa.

---

21.8.2 Energy–Time Uncertainty

A second form of the uncertainty principle relates energy and time:

$\Delta E\cdot \Delta t\ge \hslash$

where:

• $\Delta E$ = uncertainty in the energy of a quantum state
• $\Delta t$ = lifetime of that state

Interpretation: A quantum state that exists for a very short time ($\Delta t$ small) has a large uncertainty in its energy ($\Delta E$ large). Conversely, a long-lived state has a very well-defined energy.

Example: An electron in an excited state with lifetime $\Delta t=10^{-8}$ s has a minimum energy uncertainty: $\Delta E\ge \frac{\hslash }{\Delta t}=\frac{1.054\times 10^{-34}}{10^{-8}}\approx 1.05\times 10^{-26}\text{ J}$

---

21.8.3 Physical Origin: Wave-Particle Duality

The uncertainty principle is not a result of clumsy measurement — it is a fundamental consequence of wave-particle duality.

A particle is described by a wave packet — a superposition of many waves of different wavelengths. The spatial extent of the wave packet determines $\Delta x$, while the spread in wavelengths determines $\Delta p$ (since $p=h/\lambda$):

• To localize a particle (small $\Delta x$): many different wavelengths must be superposed → large spread in $\lambda$ → large $\Delta p$.
• A single wavelength (precise $p$, small $\Delta p$): gives a wave spread over all space → large $\Delta x$.

This trade-off is mathematically unavoidable.

---

21.8.4 Why the Principle is Not Noticeable in Everyday Life

For macroscopic objects, $\hslash \approx 1.054\times 10^{-34}$ J s is so incredibly small that the resulting uncertainties in position and momentum are far below the sensitivity of any measuring instrument. For example, a 1 kg ball moving at 1 m/s has a momentum of 1 kg m/s; the corresponding position uncertainty is of order $10^{-34}$ m — completely undetectable.

The principle only becomes significant at the atomic and subatomic scale, where masses and momenta are of order $10^{-30}$ to $10^{-27}$ kg m/s.

---

21.8.5 Using the Uncertainty Principle to Explain Measurement Uncertainty

The uncertainty principle has direct consequences for measurement:

1. Perfect position measurement ($\Delta x\to 0$) implies $\Delta p\to \infty$ — the momentum becomes completely undefined.
2. Perfect momentum measurement ($\Delta p\to 0$) implies $\Delta x\to \infty$ — the particle's location is completely unknown.
3. These are not technological limitations — they are inherent properties of quantum systems.

This principle also explains the natural linewidth of spectral lines: because an excited atomic state has a finite lifetime $\Delta t$, its energy has an inherent spread $\Delta E\ge \hslash /\Delta t$, causing the emitted photon to have a small range of frequencies rather than a perfectly sharp line.

---

Summary Table