Reviewer - Modern Physics Midterm PDF

**[UNIT III- THE WAVE NATURE OF PARTICLES]** **[PART 1]** **DE BROGLIE WAVES** - Prince Louis-Victor de Broglie- (1892--1987) - According to the proposal of Louis de Broglie, electrons and other particles have wavelengths that are inversely proportional to their momentum. **B. ELECT...

**[UNIT III- THE WAVE NATURE OF PARTICLES]** **[PART 1]** **DE BROGLIE WAVES** - Prince Louis-Victor de Broglie- (1892--1987) - According to the proposal of Louis de Broglie, electrons and other particles have wavelengths that are inversely proportional to their momentum. **B. ELECTRON DIFFRACTION** - is the interference of electron beams with atoms, revealing their wave-like nature and crystal structure. - When electrons interact with crystalline materials, the resultant is a pattern of rings accompanied by spots, which characterize the sample. - On a TEM (transmission electron microscope), electron diffraction is performed by focusing the beam down to the point that may be focused at a single particle or the edge of a giant crystal using the magnetic lenses of the beam column. - is a method for determining the crystal structure of materials. **TYPE OF ELECTRON DIFFRACTION PATTERN** The single-crystal material exhibits a spot pattern or Kikuchi line or combination of both Kikuchi and spot pattern, while the PO, crystalline material shows the ring pattern. 1. **Ring Pattern** - These patterns are made due to ultrafine grains of polycrystalline materials. Phases in various polycrystalline materials are determined by interpreting their ring patterns. Polycrystalline specimens are pure aluminum or pure gold. 2. **Spot Pattern** - There are two parameters in the spot diffraction pattern used to index and interpret such types of electron diffraction. 3. **Kikuchi Pattern** - Kikuchi line patterns will happen when the material's thickness is more than expected and almost perfect. - These patterns are made by electrons scattered inelastically in small angles with a slight energy loss. **GED** - It is required for a thorough grasp of structural chemistry. **C. PROBABILITY AND UNCERTAINTY** - There is a certain probability of finding the particle at a given location, and the overall pattern is called a probability distribution. **HEISENBERG UNCERTAINTY** - The uncertainty principle applies to microscopic particles - Electron clouds or orbitals are regions of negative charge surrounding an atomic nucleus that are associated with an atomic orbital. **[UNIT III- THE WAVE NATURE OF PARTICLES]** **[PART 2]** **THE ELECTRON MICROSCOPE** ELECTRON MICROSCOPE - It is a powerful scientific instrument used for imaging and studying objects at the nanoscale level. 2\) Scanning Electron Microscope (SEM) -- used to visualize the surface of objects. **TRANSMISSION ELECTRON MICROSCOPE (TEM)** - The **first TEM** was built by Max Knoll and Ernst Ruska in **1931.** - The TEM was first made available in the market in 1939. - Transmission Electron Microscope (TEM) is a microscopy technique where a beam of electrons is transmitted through an ultra thin specimen. **SCANNING ELECTRON MICROSCOPE (SEM)** - A scanning electron microscope (SEM) is a type of electron microscope that produces images of a sample by scanning it with a focused beam of electrons. - The electrons interact with atoms in the sample, producing various signals that contain information about the sample\'s surface topography and composition. **THE WAVE FUNCTION** **THE SCHRODINGER EQUATION** - **Schrödinger equation**, the fundamental equation of the science of submicroscopic phenomena known as quantum mechanics. - The equation, developed (1926) by the Austrian physicist Erwin Schrödinger. - allow us to calculate the probability distribution of a particle's position or other physical properties in a quantum system. - It is a fundamental tool in quantum mechanics that helps us understand and predict the behavior of quantum systems. **APPLICATION OF SCHRODINGER EQUATION** - used in chemistry and physics to solve problems based on the atomic structure of matter. - helps in studying the structure of the atom [UNIT IV] [QUANTUM MECHANICS] [PART 1] A. **PARTICLE IN A BOX** - The "particle in a box" is a simplified quantum mechanical model used to understand the behavior of particles, such as electrons within a one-dimensional confined space. - In this model, the particle is assumed to be constrained to move within a box or well-defined region, and its behavior is governed by the principles of quantum mechanics. - The solutions to the Schrödinger equation for this system result in quantized energy levels, meaning that the particle can only have certain discrete energy values, which are represented by quantum numbers. - This model is used to explain the concept energy quantization and wave functions in quantum mechanics. **APPLE IN A BOX THEORY** - The intriguing "apple in a box" concepts suggests if an apple were to be left untouched in a box for a billion of years, its fundamental particles would theoretically traverse every possible state they could, eventually leading the apple to return to its original state. **USES OF SCHRÖDINGER EQUATION** 1. **Predicting the behavior of quantum system** - Schrödinger's equation allows us to calculate the probability distribution of a particle's position or other physical properties in a quantum system. It's essential for understanding and predicting the behavior of particles at the quantum level. 2. **Determining energy levels** - This is crucial for understanding the quantization of energy in quantum mechanics. 3. **Wave function analysis** - By analyzing this wave function, we can gain insights into various properties of the system, such as momentum, angular momentum, and spin. 4. **Exploring quantum phenomena** - Schrödinger equation is used to study a wide range of quantum phenomena, from the behavior of electrons in atoms and molecules to the wave-like nature of particles and the quantum entanglement of particles. 5. **Developing quantum models** - It serves as the foundation for creating quantum models of physical systems, which are vital in fields like chemistry, solid-state physics, and quantum computing. **B. POTENTIAL WELLS** - A potential well is the energy necessary to be in a specific location and the strong nuclear force makes it nearly impossible for the protons and neutrons to escape within the nucleus, but once they are just out of range, they can leave very quickly. - Energy captured in a potential well is unable to convert to another type of energy because it is captured in the local minimum of a potential well. - In a finite Potential the potential energy outside the box is constant **Example of Potential Well:** If you have a hole and a ball in it certain, the ball will require a certain amount of energy to climb up and out of the hole. However, if it lacks that energy, the ball becomes stuck at the bottom of the hole. C. Barriers and Tunneling Barriers - In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave mechanical reflection. **Entropy** is a measure of randomness or disorder of a system. **Quantum mechanics** is a branch of physics that deals with the behaviour of matter and light on a subatomic and atomic level. - In quantum physics, **wave function** is represented by the Greek letter Ψ, a mathematical description of the quantum state of an isolated quantum system. **Quantum Measurement Problem** - The quantum measurement problem refers to the challenges and questions surrounding the act of measurement in quantum mechanics. It arises because the process of measuring a quantum system can affect the system itself, leading to questions about the nature of reality and the role of observation in quantum mechanics. **Wave-Particle Duality** - Wave-particle duality is a fundamental concept in quantum mechanics. - It suggests that particles, such as electrons and photons, can exhibit both wave-like and particle-like characteristic depending on the experimental conditions. **Quantum Entanglement** - Quantum entanglement is a phenomenon where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the state of the others, no matter how far apart they are. **Quantum Uncertainty** - often referred to as **Heisenberg's uncertainty principle** - Quantum tunneling is defined as a quantum mechanical process where wavefunctions can penetrate through a potential barrier. **[Applications of Quantum Tunneling ]** 1. **Scanning Tunneling Microscope** - It is a type of microscope that helps to observe objects at atomic levels. - It functions by utilizing the connection between quantum tunneling with distance. 2. **Nuclear Fusion** - Quantum tunneling is crucial part of nuclear fusion. - The average temperature of a star's core is usually not sufficient for atomic nuclei to overcome the Coulomb barrier and kick start thermonuclear fusion. - The tunneling increases the chances if infiltrating this barrier. - Though the probability is still low, the huge number of nuclei in the stellar core is enough to drive a steady fusion reaction. 3. **Electronics** - Tunneling is a frequent source of current leakage in very large-scale integration (VLSI) electronics. **IMPORTANCE OF QUANTUM MECHANICS** - Explain phenomena found in nature. - Develop technologies that rely upon quantum effects, like integrated circuits and lasers - Understand how individual atoms are joined by covalent bonds to form molecules **[UNIT IV]** **[QUANTUM MECHANICS]** **[PART 2]** **HARMONIC OSCILLATOR** - It is a fundamental concept in physics, describing systems oscillating around a stable equilibrium. - It is a system executing harmonic motion - The **HARMONIC OSCILLATOR** is a great approximation of a molecular vibration, but has key limitations; **SIMPLE HARMONIC OSCILLATOR** - A simple harmonic oscillator is a type of oscillator that is either damped or driven. **HISTORY: The Harmonic Oscillator** - The **Harmonic Oscillator** played a leading role in the development of quantum mechanics. - **In 1900 , PLANCK** made the bold assumption that atoms acted like oscillators with quantized energy when they emitted and absorbed radiation. - **In 1905 , EINSTEIN** assumed that electromagnetic radiation acted like electromagnetic harmonic oscillators with quantized energy. **APPLICATION** 1. **Clocks -** Many traditional clocks use a pendulum to keep time. 2. **Musical Instruments -** The vibrations of these components can be analyzed as simple harmonic motion. 3. **Electrical Systems -** Circuits with capacitors and inductors can exhibit oscillatory behavior, and their 4. **Biological Systems -** Some biological systems, like the motion of your leg while walking or the beating 5. **Molecular Vibrations -** Even at the microscopic level, molecules vibrate and oscillate. **DISCRETE ENERGY STATE - It distinguishes the Quantum Harmonic Oscillator from the Classical Harmonic Oscillator** **CLASSICAL HARMONIC OSCILLATOR** - It generally consists of a mass' m', where a lone force 'F' pulls the mass in the trajectory of the point x = 0, and relies only on the position 'x' of the body and a constant k. **QUANTUM HARMONIC OSCILLATOR** - The quantum harmonic oscillator is the **subatomic analogue version** of the conventional harmonic oscillator. - It is one of the most relevant model systems in quantum physics. - A random smooth potential can generally be estimated as a harmonic potential at the locale of a stable equilibrium point. **Models of Harmonic Oscillator** The Following are some examples to illustrate the scope of its applications from classical to quantum physics : 1. **Hooke\'s Law** - This is the classical example involving mass motion by spring. - **According to this law, within the elastic limit, stress is proportional to the strain.** 2. **Coupled Oscillators** - This system introduces the concept of eigenvalues and eigenvectors at the level of classical physics. - The normal modes describe coherent motion of atoms in molecules and in crystal lattice**.** 3. **Quantum Field Theory** - Imitating a system of free harmonic oscillators to provide the basic concept for the quantization of a field. - The three-dimensional quantum mechanical model is like the VIP section of quantum physics, where particles aren\'t just bouncing back and forth in one dimension, they\'re free to roam in three. - In this model, you\'re dealing with the position of a particle in three-dimensional space. - Instead of a simple one-dimensional wave function, you now have a three-dimensional wave function Ψ(x,y,z)) that describes the probability amplitude of finding a particle at a particular position.