Planck's Math PDF

# Planck's Math ## Probability - Wave vs. Particle * Probability -> wave-like * Quantized energy $E=nhf$ -> particle-like * Blended wave and particle energy ## Wave-like Properties of Light * Stated that small things exhibit properties of both a wave and a particle * Test -> wave = double slit...

# Planck's Math ## Probability - Wave vs. Particle * Probability -> wave-like * Quantized energy $E=nhf$ -> particle-like * Blended wave and particle energy ## Wave-like Properties of Light * Stated that small things exhibit properties of both a wave and a particle * Test -> wave = double slit interference ## Particle-like Properties of Light * Particle -> quantized or discrete values of energy * To test particle nature of light, photoelectric effect was complete. ### The Photoelectric Effect * Metal * Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ .... -> require a specific amount of energy to eject (remove) an electron from metal. * Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ ... -> free to move * $E_k=hf$ * red -> less energy * blue -> more energy * more intense red (more energy) -> more intense wave * Even with the most intense red light, no electrons were ejected ### Testing the Particle Nature of Light * By changing the color or frequency of light, electrons were ejected. * This confirms the particle nature of light. ## Energy of Light - Particle-like * Light has energy like a particle -> so "collides" with electrons like particles to inelastically transfer energy. * 1.77 eV -> reflects * 2.25 eV -> ejected * 3.1 eV -> ejected and faster * 0.25 eV -> kinetic energy * 1.1 eV -> kinetic energy * Potassium -> 2.0 eV to eject electron * Work function is the amount of energy needed to eject an electron from a metal. * $W = E = hf$ * $f_0$ is the threshold frequency (minimum f) to eject an electron. A tiny frequency of light used will eject electrons faster. * $E_k$ (for electron) $= E$ (of light) $- W$ * $E_k = hf - hf_0$, or * $E_k = \frac{hf}{mc}+\frac{h f_0}{mc}$ * $E =hf$ * $hf = (6.63x10^{-34}J\cdot s) (1Hz)$ * $\frac{1}{1.6x10^{-19}J} = 1Hz$, or * $Ek = 1.6x10^{-19}J$ * $E_k = hf-W$ * $E_k = (6.63x10^{-34}J\cdot s)(8.0x10^{14} Hz) - 3.7x10^{-19}J$ * $E_k \approx 2.0x10^{-19}J$ * Light is quantum, and has wave-particle duality. * We call this quantum object a PHOTON. ## Photons * Photons -> light waves carrying "packets" of energy like a particle. * $E_{photon} = hf$ * ↑f means ↑$E_{photon}$ ### Photoelectric Effect * Used to eject electrons from just outside the visible spectrum. * Photon collides inelastically with an electron. * Photon's energy is transferred to the electron, causing it to move. ### Compton Effect * Continue to test collisions with photons using higher energy (x-rays). * X-ray has higher energy and momentum. * X-ray collides with a photon. * Photon has less energy, and has a lower frequency, but it's moving faster. * $E = hf$ * Photon can still transfer its energy to the electron, though the photon itself is now moving faster. ## Momentum * Collision demonstrates that photons have momentum. * $P_{particle} = mv$ * $P_{photon}$ -> photons (light) has no mass -> no momentum? * If a photon has energy, we can associate it with mass. * $P_{photon} = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}$ * ↑f or ↑λ -> ↑P * ↓f or ↓λ -> ↓P * **Example:** * Determine the momentum of an x-ray with λ = 1.0x10^-10 m. And determine the energy in eV. * $P_{x-ray} = \tfrac{h}{\lambda} = \tfrac{6.63x10^{-34}J\cdot s}{1.0x10^{-10}m}$ * $P_{x-ray} = 6.63x10^{-24}J\cdot s$ * $E = hf$ or $P_{photon}c$ * $E = Pc$ * $E = (6.63x10^{-34}J\cdot s)(3.00x10^{8}m/s)$ * $E = 2.0x10^{-15}J$ * $\frac{1 eV}{1.6x10^{-19}J} = 12500 eV$ ## Wave-Particle Duality * Light is a wave, and has particle properties. * Particles - Can they act as waves? (Double slit test). * Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ -> interference patter * Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ -> a particle is created. * Based on probability, we see regions where the electron is more likely (or less likely) to hit.. ## Particle-Like Properties of Particles * Particles acting like waves: * After doing the double-slit experiment with electrons, protons, neutrons, atoms, and some molecules, an interference pattern was observed above the interference. * Those particles also have wave-particle duality. * Some math is used to understand variables. * So, a particle has a λ associated with it. * λ (electron) = 3.0x10^-15 m * λ (green light) = 532 nm = 5.32x10^-7 m * λ (electron) <<< λ (visible light) * Louis de Broglie found a wavelength for particles: * $P_{photon} = \frac{h}{\lambda} ->\lambda = \frac{h}{p} = \frac{h}{m}$ * This can be used to predict the behavior of electrons using the double slit experiment. * ↑m and ↑p of particles -> the associated λ decreases. * This creates a limitation in using this method. * The slit width is related to λ, and has to be greater than λ. * Evenually, the size of the particle is too large to fit through the opening. * Still, it is able to be tested. ### Measuring the Interference Pattern * Δx is inversely proportional to λ. * If λ is small, then Δx is small, and particles are more likely to be detected. * If λ is large, then Δx is large, and interference patterns cannot be observed (particles are more likely to miss the slits). ### Quantum Measurement * When measuring objects in the quantum world, objects do not change as they do in the classical world. * When measuring particles in the quantum world, we are using photons, which can disturb the particle. * So we are not able to completely understand all the details about small quantum objects. #### Analogy: * Imagine you are using a basketball to measure a baseball. The baseball will not be measured accurately and its behavior can be affected. * Example: Watching electrons pass through a double-slit experiment resulted in no interference pattern. * Even though we don't understand why the math works, it still does. ## Probability and Uncertainty * Electrons act as waves, but they do not have a localized position. * If we try to measure the position of an electron (as a particle), it will disturb it and change the system. * An electron can be measured across a space, like a wave -> wave function. * Not only a particle, but a wave function. * The wave function of the electrons creates an interference pattern. * Because we have probabilities, the exact values of an electron (a quantum object) is not perfect. * There is an error (uncertainty) associated with every measurement. * The uncertainty is not zero. * 1.0 m -> distance - Δd * 0.9 m - 1.1m * Δd = > 0.1m (net Δd) * Uncertainty in Δd = > 0.1 m * Short Δt * Long Δt * Δf Δx = constant * Heisenberg's Uncertainty Principle: Measurement results are not exact or change for quantum. ## Heisenberg Uncertainty Principle * Δx -> uncertain in position. * Δp -> uncertainty in momentum/motion (speed and direction). * Δx Δp = constant (never 0) * Δx Δp = h/4π ### Diffraction of Light * Large opening -> large Δx * Small opening -> small Δx * The screen is hit in a small blob. * The range of direction is minimal. * If Δp is small (the spread of momentum is small) then the spread of position is small. * The Δp is small because photon's wavelengths are small. * If Δx is small, then the photon is more likely to spread out and produce an interference pattern. * The small Δx contributes to a small Δp. ## Quantum Theory * Quantum (quanta) -> a discrete physical property. * All measurements are related to the smallest value. * There is no continuous range of measurement. * **The Number Line** * → -2 -1 0 1 2...→ * → -1.6x10^-19 0 1.6 x10^-19...→ * Charge -1.6x10^-19, 0, 1.6x10^-19 * q = Ne * **Particle** * Discrete quantities * Localized position * There is no interference pattern in the double slit experiment. * **Wave** * Carriers energy continuously in space. * Has interference patterns. * EM wave, vibration f, v=cλ (speed of light) ## Classical Physics * Classical physics distinguishes between a wave and a particle. ## Blackbody Radiation * Blackbody radiation in classical physics did not match the observed spectrum for blackbody radiation. ### Problem * The UV Catastrophe: The smaller waves being emitted from the blackbody suddenly drop off or stop. ## Solution * Max Planck decided to create a function of temperature. * To fit the graph, math was needed that quantized waves. * Put wave and particle equations together. * Quantized energy of a wave: $E = nhf$ * q = Ne * Planck's constant h = 6.63x10^-34J⋅s * The math is based on probability. ### Waves Emitted by a Blackbody * Waves emitted by a blackbody are quantized. * Not all frequencies are generated, but since h is small, it looks like it does. * $E=nhf$. * So at high frequencies (for a low λ), the energy jump of an atom is very large, so the probability of an atom vibrating at this value is low (drops off). ## Relativistic Energy * E = mc² * $E_{total} = E_{kr} + E_{rest}$ * $E_T = \frac{mc^2}{\sqrt{1- \frac{v^2}{c^2}}}$ * $E_{kr} = \frac{1}{2} mv^2$ (classical or proper measurement) * $E_{kr} = E_T - E_{rest}$ * Energy measured in Joules is too small. * A better unit for energy is electron volts (eV) or MeV. * 1 MeV = 10⁶ eV = 1.6 x10^-13 J ### Mass * Mass -> kg or J * E = mc² ### Kinetic Energy Changes * 1 eV = 1.6 x10^-19J * ΔV -> 100 V * Gain E_k -> ΔE_k * ΔE_k = (Δ-E_e) * ΔE_k = (-q ΔV) * ΔE_k = (1.6 x10^-19 C) (100V) * 1 eV = 1.6x10^-19 J * **Example:** A proton in a particle accelerator reaches a speed of 0.70c. Find the relativistic kinetic energy (in Mev) and the classical kinetic energy (in Mev) to see the difference. * $E_{kr} = E_{total}- E_{rest}$ * $E_{kr} = \frac{mc^2}{\sqrt{1- \frac{v^2}{c^2}}}-mc^2$ * $E_{kr} = mc^2 \left(\frac{1}{\sqrt{1- \frac{v^2}{c^2}}}-1\right)$ * $E_{kr}= 1.5x10^{-13}J (0.40)$ * $E_{kr}=6.02x10^{-11}J$ * $E_{kr} = \tfrac{1}{2} mv^2$ * $E_{kr} = \tfrac{1}{2} (1.67 x10^{-27} kg)(0.70(3.00 x10^8 m/s)^2$ * $E_{kr} = 3.68 x10^{-11}J$ * $E_{kr} = \tfrac{3.68 x10^{-11}J}{1.6x10^{-13}J/MeV}$ * $E_{kr} = 230 MeV$ * $E_{kr} = \tfrac{6.02 x10^{-11}J}{1.6 x10^{-13}J/MeV}$ * $E_{kr} = 376 MeV$ * $E_{kr} > E_{kr}$ * $E_{kr}$ is about 1.63 times $E_{kr}$. ## Special Relativity * Δt passes observers differently. * How different these times are depends on the relative motion between the observer and the object. * **Proper Measurement:** No relativistic motion between the observer and what they are measuring. * **Subscripts:** * **S** for Stationary * **P** for Proper * **O** for Original * **Proper Time:** Δt_S * **Proper Distance:** Δd_S * **Proper Time:** Δt_P * **Proper Distance:** Δd_P * **Original Time:** Δt_O * **Original Distance:** Δd_O * No motion = Proper measurement. * Ad = cΔt_p * y = vΔt_O * x = cΔt_O * x² = Δd²+y² * (cΔt_O)² = (cΔt_P)²+(vΔt_O)² * (cΔt_O)² - (vΔt_O)² = (cΔt_P)² * c²(Δt_O)² - v²(Δt_O)² = c²(Δt_P)² * Δt_O²(c²-v²) = c²(Δt_P)² * Δt_O² ( $\frac{c^2-v^2}{c^2}$ ) = Δt_P² * * Δt_O $\sqrt{\frac{(c^2-v^2}{c^2}}$ = Δt_P * Δt_O = $\frac{Δt_P}{\sqrt{1- \frac{v^2}{c^2}}}$ * This equation accounts for motion between the observer and what they are measuring. * **V** is the relative speed of the object. * **Δt (relative time)** is the time measured by an observer at rest. * **Δt_P** is the proper time, which is the time measured by an observer moving with the object (in the object's rest frame). * **Δt_O** is the time measured in the object's rest frame. ## Things to Remember * The only thing that changes in the special relativity equation for time is **V**, the relative speed between the observer and the measurement. * As **V** approaches **c**, the denominator of the equation approaches 0. * This means **ΔtO** approaches ∞. * If **V << c** * $\frac{V^2}{c^2} \approx 0$ * $√(\frac{1-v^2}{c^2}) \approx 1$ * Δt_O = Δt_P * if **0 << c** * $\frac{V^2}{c^2} << 1$ * $0 < √(\frac{1-v^2}{c^2}) < 1$ * Δt_O > Δt_P -> time dilation (bigger time) * If **V ≥ c** * $\frac{V^2}{c^2} ≥ 1$ * 1- $\frac{V^2}{c^2} ≤ 0$ * Δt_O is undefined, or not real. * This does not occur. ## Example * Your favorite song on Earth lasts 3.00 minutes. If someone were to listen to your song on a spacecraft moving at: * a) 0.100 c * b) 0.90 c * **a) 0.100c** * Δt_P = 3.00 min * Δt_O = ? * Δt_O= $\frac{Δt_P}{\sqrt{1- \frac{v^2}{c^2}}}$ * Δt_O = $\frac{3.00 min}{\sqrt{1-(0.10c)^2}}$ * Δt_O ≈ 3.02 minutes * **b) 0.90c** * Δt_P = 3.00 min * Δt_O = ? * Δt_O = $\frac{Δt_P}{\sqrt{1- \frac{v^2}{c^2}}}$ * Δt_O = $\frac{3.00 min}{\sqrt{1-(0.90c)^2}}$ * Δt_O ≈ 6.88 minutes ## Time Dilation * Δt_O = $\frac{Δt_P}{\sqrt{1- \frac{v^2}{c^2}}}$ * Δt_O > Δt_P (at fast speed) * vΔt_O = vΔt_P * vΔt_O = $\frac{vΔt_P}{\sqrt{1- \frac{v^2}{c^2}}}$ * V = $\frac{Δd}{Δt}$ (constant) * vΔt_O = $\frac{Δd}{\sqrt{1- \frac{v^2}{c^2}}}$ * L_P = $\frac{L_V}{\sqrt{1- \frac{v^2}{c^2}}}$ * L_V = L_P $\sqrt{1- \frac{v^2}{c^2}}$ ## Length Contraction * L_V > L_P (at fast speeds) * Length contraction ### Example * A spacecraft has a measured length of 78 m while moving at 0.85c away from Earth. What length (L_P) would the person inside the spaceship measure? * L_V = 78 m * L_P = ? * L_V = L_P $\sqrt{1- \frac{v^2}{c^2}}$ * $\frac{L_V}{\sqrt{1- \frac{v^2}{c^2}}} = L_P$ * $\frac{78m}{√(1-0.85^2)} = L_P$ * $L_P = 148m$ ## Relativity * All measurements have a "relativistic" counterpart. * They are based on speed. * Momentum: p=mv * $P_{r} = \frac{P_P}{\sqrt{1- \frac{v^2}{c^2}}}$ * M and V are changing, * P_P is proper mass. * m is the mass of the object at rest. * P_r > P_P * In order to move an object fast, a lot of work must be done. * This results in the object gaining kinetic energy. * If you supply a lot of energy, it doesn't entirely transfer to kinetic energy. * There is a point where the object reaches a speed where energy transforms to mass (E=mc²). * This means objects just become heavier, not faster. * Proper mass (rest mass) has an associated energy (E=mc²). * Proper mass (as long as it is greater than zero) can never be calculated beyond c. ## Momentum Graph * P (momentum) * **Red line**: Classical momentum * **Blue line**: Relativistic momentum * P=mv * 0.5v (in terms of c) ## Speed of Light * If you were light, v=c. * Δt_O = Δt_P * $\frac{Δt_P}{\sqrt{1- \frac{v^2}{c^2}}} = Δt_O$ * Δt_O would not exist. * Time would not exist for you. * Time is not defined. * L_V = L_P $√(1- \frac{v^2}{c^2})$ * L_V = L_P (0) * Length would be zero (so no space). ## General Relativity * Acceleration -> gravity. * Measurements change based on mass distorting the space-time fabric. * Satellite -> moves faster, 7ys a day. * One right -> General Relativity * 45 Ms a day * Δd = vΔt * v = 3.0x10⁸ * Δt = 3.10^-6 * Δd < vΔt