## 21.8 HEISENBERG'S UNCERTAINTY PRINCIPLE Formulated by Werner Heisenberg in 1927, this principle states that it is impossible to accurately find both the position and speed (and thus momentum) of a particle, such as a photon or electron, simultaneously with perfect accuracy. * It is a fundamental consequence of the **dual nature of matter** (wave-particle duality). * On a microscopic level, any measurement interaction with a particle introduces unavoidable uncertainty. ### Position-Momentum Uncertainty Relation The product of uncertainties in determining the position ($\Delta x$) and corresponding momentum ($\Delta p$) of a particle at the same instant is approximately equal to $\hbar$ (h-cut). **Formula:** $$\Delta x \cdot \Delta p \geq \hbar$$ Where $\hbar = \frac{h}{2\pi} = 1.054 \times 10^{-34} \text{ J s}$ (h is Planck's constant). **Implication:** If the uncertainty in position ($\Delta x$) is zero, then the uncertainty in momentum ($\Delta p$) must be infinite, and vice-versa. ### Wave Description of Particles In quantum mechanics, a particle is described by a wave. * *Position* refers to the point where the wave is concentrated. Uncertainty in position means the position is uncertain to the degree the wave is spread out. * *Momentum* is related to wavelength. Uncertainty in momentum means the momentum is uncertain to the degree that the wavelength is unclear. ### Inherent Uncertainty These uncertainties are inherent in the physical world and are not due to the skill of the observer or limitations of measuring instruments. Due to the very small value of $\hbar$, these uncertainties are not observable in normal everyday (macroscopic) situations. ### Measurement Interaction To achieve high accuracy in position measurement, one must use photons of shorter wavelength. * Short wavelength implies high-frequency, high-energy photons, which transfer more momentum to the target particle, thereby increasing the uncertainty in its momentum. * Using longer wavelength (less energetic) photons reduces momentum transfer but compromises resolution and position accuracy. ### Energy-Time Uncertainty Relation Another form of the uncertainty principle relates the uncertainty in the energy ($\Delta E$) of a quantum state and its lifetime ($\Delta t$). **Formula:** $$\Delta E \cdot \Delta t \geq \hbar$$ **Implication:** A system remaining in a metastable state for a very long time ($\Delta t$ is large) can have a well-defined energy ($\Delta E$ is small). Conversely, if a system remains in a state for only a short time ($\Delta t$ is small), the uncertainty in energy ($\Delta E$) must be correspondingly greater. ## Possible Questions/Answer * **Q:** Calculate the uncertainty in the momentum of an electron if uncertainty in its position is $10^{-10} \text{ m}$. **A:** Given $\Delta x = 10^{-10} \text{ m}$. Using $\Delta p \geq \frac{h}{2\pi\Delta x}$: $$ \Delta p = \frac{6.63 \times 10^{-34} \text{ J s}}{2(3.14)(10^{-10} \text{ m})} = 5.28 \times 10^{-25} \text{ kg m s}^{-1} $$ * **Q:** A ball with a mass of $0.1 \text{ kg}$ moves at a speed of $30 \text{ m s}^{-1}$. The uncertainty in its momentum is $0.000001$ of its actual momentum. Determine the uncertainty in its position. **A:** 1. Calculate actual momentum: $p = mv = (0.1 \text{ kg})(30 \text{ m s}^{-1}) = 3 \text{ kg m s}^{-1}$. 2. Calculate uncertainty in momentum: $\Delta p = 0.000001 \times 3 \text{ kg m s}^{-1} = 3 \times 10^{-6} \text{ kg m s}^{-1}$. 3. Using $\Delta x \geq \frac{h}{2\pi\Delta p}$: $$ \Delta x = \frac{6.63 \times 10^{-34} \text{ J s}}{2(3.14)(3 \times 10^{-6} \text{ kg m s}^{-1})} \approx 3.52 \times 10^{-29} \text{ m} $$