Quantum Mechanics: Wave Functions PDF

Modern Physics Quantum Mechanics: Wave Functions Jester Mark Guden Department of Teacher Education Visayas State University Tolosa  Quantum mechanics, is the key to understanding the behavior of matter on the molecular, atomic, and nuclear scales. Quantum mechanics replaces the classical sc...

Modern Physics Quantum Mechanics: Wave Functions Jester Mark Guden Department of Teacher Education Visayas State University Tolosa  Quantum mechanics, is the key to understanding the behavior of matter on the molecular, atomic, and nuclear scales. Quantum mechanics replaces the classical scheme of describing the state of a particle by its coordinates and velocity components. Wave Functions and the One- dimensional Schrӧdinger Equation Schrödinger equation is the fundamental equation that describes the dynamics of matter waves. Wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position, and spin. The customary symbol for this wave function is the Greek letter psi, Ψ or ψ Uppercase Ψ to denote a function of all the space coordinates and time, and a lowercase ψ for a function of the space coordinates only—not of time. Wave Functions: Particle Waves Considering a free particle, one that experiences no force at all as it moves along the x-axis. For such a particle the potential energy Ux has the same value for all x: U=0 Wave Functions: Particle Waves Angular Frequency Wave Number  One-Dimensional Schrӧdinger Equation Developed in 1926 by the Austrian physicist Erwin Schrödinger Solving the Schrodinger equation means finding the ​quantum mechanical wave function​ that satisfies it for a particular situation. The presence of the imaginary number i means that the solutions to the Schrödinger equation are complex quantities, with a real part and an imaginary part. Interpreting Wave Functions The complex nature of the wave function for a free particle makes this function challenging to interpret. Ψ 𝒙, 𝒕 describes the distribution of a particle in space The square of the absolute value of the wave function Ψ 𝟐 of a particle at each point tells us about the probability of finding the particle around that point. Ψ(𝒙, 𝒕) 𝟐 dx is the probability that the particle will be found at time t at a coordinate in the range from x to x + dx. First made by the German physicist Max Born One-dimensional Schrödinger Equation with Potential energy Time-independent One- dimensional Schrödinger Equation Particle in a Box The idea of the particle in a box is to determine, for a given potential energy function U(x), the possible stationary state wave functions, and what are the corresponding energies E. This model might represent an electron that is free to move within a long, straight molecule or along a very thin wire. Particle in a Box Energy Levels for a Particle in a Box Wave Function for a Particle in a Box Sample Problem Find the first two energy levels for an electron confined to a one-dimensional box 5.0 x 10-10 m across (about the diameter of an atom). Potential Wells A potential well is a potential-energy function U(x) that has a minimum. Square-well potential as a simple model of an electron within a metallic sheet with thickness L, moving perpendicular to the surfaces of the sheet.  Potential Barriers A potential barrier is the opposite of a potential well; it is a potential energy function with a maximum. Tunneling If a quantum-mechanical particle encounters a barrier, even if it has energy less, it may appear on the other side in a phenomenon called tunneling.  Tunneling A particle that is initially to the left of the barrier has some probability of being found to the right of the barrier. The tunneling probability T that the particle gets through the barrier is proportional to the square of the ratio of the amplitudes of the sinusoidal wave functions on the two sides of the barrier. Tunneling Sample Problem A 2.0-eV electron encounters a barrier 5.0 eV high. What is the probability that it will tunnel through the barrier if the barrier width is (a) 1.00 nm and (b) 0.50 nm? Applications and Importance of Tunneling Tunnel diode Josephson junction Scanning tunneling microscope (STM) Electron tunneling in enzymes Fusion reaction eNd