Wave-Particle Duality & Uncertainty Principle PDF

# Wave-Particle Duality and Uncertainty Principle ## Wave-Particle Duality ### de Broglie Relation The de Broglie relation describes the wave nature of matter. * For a particle with momentum *p*, the wavelength is: λ = *h* / *p* Where *h* is Planck's constant. ### Derivation of de B...

# Wave-Particle Duality and Uncertainty Principle ## Wave-Particle Duality ### de Broglie Relation The de Broglie relation describes the wave nature of matter. * For a particle with momentum *p*, the wavelength is: λ = *h* / *p* Where *h* is Planck's constant. ### Derivation of de Broglie Relation The de Broglie relation can be derived by considering the energy and momentum of a particle: * Energy: *E* = *mc*² * Momentum: *p* = *mv* 1. **Relating Energy and momentum:** *E* = *pc*²/<br> *dw*/ *dk* = 2π*p*c² / *h* <br> 2. **Introducing Group Velocity:** *dw*/ *dk* = 2πν*dp*/ *h* Where *v* is the group velocity of the wave. 3. **Solving the Differential Equation** Integrating the above equation and assuming the integration constant is zero: *p* = (*h*/2π) *k* 4. **Substituting for k and rewriting the equation:** *λ* = *h*/*p* Which is the de Broglie relation. ## Uncertainty Principle The Uncertainty Principle states that it is impossible to know both the position and momentum of a particle with perfect accuracy. The principle is mathematically expressed as: Δ*x* Δ*p* ≥ *ħ*/2 where Δ*x* is the uncertainty in position, Δ*p* is the uncertainty in momentum and *ħ* is the reduced Planck's constant. ## Experiments Demonstrating Wave-Particle Duality 1. **Davisson and Germer's Experiment (1925)** * This experiment famously confirmed the de Broglie relation. * A beam of electrons was diffracted by a nickel crystal. * The experiment showed that the electron beam behaved as a wave, resulting in a diffraction pattern characteristic of wave interference. * The wavelength of the electrons was measured and found to be consistent with the de Broglie relation. 2. **G. P. Thomson's Experiment** * Similar to Davisson and Germer's experiment. * A beam of electrons was passed through a thin gold foil, showing diffraction patterns. 3. **Two-Slit Interference Experiment** * Demonstrates the wave nature of electrons. * Electrons were allowed to pass through two narrow slits. * An interference pattern was observed on a screen behind the slits, indicating the wave nature of electrons. 4. **Diffraction at a Straight Edge Experiment** * Electrons were diffracted by an edge, exhibiting a diffraction pattern, again supporting the wave nature of matter. ## Experiments to Verify the Uncertainty Principle 1. **Gamma Ray Microscope Experiment** * Imagined to verify the uncertainty principle by Heisenberg. * A microscope is used to observe an electron. * The resolving limit of the microscope (Δ*x*) is related to the wavelength of the light used *(λ*). * The uncertainty in the electron's momentum is related to the Compton effect (Δ*p*). * The experiment shows that the product of the uncertainties in position and momentum is greater than or equal to *ħ*/2, verifying the uncertainty principle. 2. **Diffraction of a Beam of Electrons by a Slit** * Similar to the two-slit experiment. * The uncertainty of the position of an electron as it passes through the slit is related to the slit width. * The uncertainty of the momentum of the electron is related to the diffraction pattern observed. * This experiment illustrates the uncertainty principle as well, as it's impossible to know both the position and momentum of the electron with perfect accuracy. ## Applications of the Uncertainty Principle * **Size of Atoms** * Relates to the uncertainty of an electron’s position. * The uncertainty in position leads to a minimum radius for the electron’s orbit, ultimately defining the atomic size. * **Stability of Atoms** * According to classical physics, an electron orbiting the nucleus should radiate energy and eventually spiral into the nucleus, causing atomic collapse. * The uncertainty principle prevents this by preventing the electron from having a precise position and momentum, meaning it cannot lose all energy and collapse into the nucleus. ## The Nucleus and Electron Limitations * **Electrons Cannot Reside Inside the Nucleus:** * If electrons resided inside the nucleus, the uncertainty in their position would be on the order of the nuclear radius (10-14 m). * Based on the uncertainty principle, this would result in a very large uncertainty in the electron's momentum and a very high energy. * This energy is much greater than the observed energy of electrons emitted from the nucleus, indicating that electrons cannot reside inside the nucleus.