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# Quantum Mechanics ## Why Quantum Mechanics? Classical mechanics is useful, but it fails to explain some phenomena, such as: * Blackbody radiation * The photoelectric effect * The Compton effect * Atomic spectra ## Blackbody Radiation A blackbody is an object that absorbs all electroma...

# Quantum Mechanics ## Why Quantum Mechanics? Classical mechanics is useful, but it fails to explain some phenomena, such as: * Blackbody radiation * The photoelectric effect * The Compton effect * Atomic spectra ## Blackbody Radiation A blackbody is an object that absorbs all electromagnetic radiation that falls on it. When heated, a blackbody emits radiation. The spectrum of this radiation depends only on the temperature of the object, not on its material or surface properties. Classical physics predicts that the intensity of blackbody radiation should increase without bound as the frequency increases (the ultraviolet catastrophe). However, experimental results show that the intensity reaches a maximum at a certain frequency and then decreases. **Planck's solution:** Energy is quantized, $E = nhf$, where $n$ is an integer, $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J s}$), and $f$ is the frequency. ## Photoelectric Effect When light shines on a metal surface, electrons can be emitted. This is called the photoelectric effect. Classical physics predicts that the kinetic energy of the emitted electrons should increase with the intensity of the light. However, experimental results show that the kinetic energy depends on the frequency of the light, not on the intensity. Also, there is a threshold frequency below which no electrons are emitted, regardless of the intensity. **Einstein's solution:** Light is made up of particles called photons, each with energy $E = hf$. When a photon strikes the metal surface, it can transfer its energy to an electron. If the energy of the photon is greater than the work function $\phi$ of the metal, the electron can escape with kinetic energy $KE = hf - \phi$. ## Compton Effect When X-rays are scattered by electrons, the scattered X-rays have a longer wavelength than the incident X-rays. This is called the Compton effect. Classical physics predicts that the wavelength of the scattered X-rays should be the same as the wavelength of the incident X-rays. **Compton's solution:** The X-rays are made up of photons, which collide with the electrons like billiard balls. The photon loses some of its energy and momentum in the collision, which results in an increase in its wavelength. $\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos{\theta})$ ## Atomic Spectra When atoms are excited, they emit light at specific wavelengths. This is called atomic spectra. Classical physics predicts that atoms should be able to emit light at any wavelength. **Bohr's solution:** Electrons can only exist in certain orbits around the nucleus, with specific energies. When an electron jumps from one orbit to another, it emits or absorbs a photon with energy equal to the difference in energy between the two orbits. $E = -13.6 \text{ eV } \frac{Z^2}{n^2}$ $E = hf$ ## Wave-Particle Duality Light and matter have both wave-like and particle-like properties. * Light can behave as a wave (interference, diffraction) or as a particle (photoelectric effect, Compton effect). * Matter can behave as a particle (it has mass and momentum) or as a wave (de Broglie wavelength). de Broglie wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$ ## The Uncertainty Principle It is impossible to know both the position and momentum of a particle with perfect accuracy. $\Delta x \Delta p \ge \frac{\hbar}{2}$ It is impossible to know both the energy and time with perfect accuracy. $\Delta E \Delta t \ge \frac{\hbar}{2}$ where $\hbar = \frac{h}{2\pi}$ ## The Schrödinger Equation The Schrödinger equation is the fundamental equation of quantum mechanics. It describes how the wave function of a particle evolves in time. Time-dependent Schrödinger equation: $i\hbar \frac{\partial}{\partial t} \Psi(x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi(x,t) + V(x) \Psi(x,t)$ Time-independent Schrödinger equation: $E\psi(x) = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \psi(x) + V(x) \psi(x)$ Where: * $\Psi(x,t)$ is the wave function, which describes the state of the particle * $V(x)$ is the potential energy of the particle * $E$ is the energy of the particle * $m$ is the mass of the particle ## Particle in a Box A particle in a box is a simple model system that can be used to illustrate some of the basic concepts of quantum mechanics. The particle is confined to a region of space of length $L$, and the potential energy is zero inside the box and infinite outside the box. The solutions to the Schrödinger equation for this system are: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin{\frac{n\pi x}{L}}$ $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$ Where $n$ is a positive integer. ## Quantum Harmonic Oscillator A quantum harmonic oscillator is a model system that describes the vibrations of a molecule. The potential energy is $V(x) = \frac{1}{2} kx^2$, where $k$ is the spring constant. The solutions to the Schrödinger equation for this system are: $E_n = (n + \frac{1}{2}) \hbar \omega$ Where $n$ is a non-negative integer and $\omega = \sqrt{\frac{k}{m}}$.