Nanoelectronics Chapter 3 Quantum Mechanics of Electrons

Slides:



Advertisements
Similar presentations
Physical Chemistry 2nd Edition
Advertisements

The Quantum Mechanics of Simple Systems
1. Quantum theory: introduction and principles
1. Quantum theory: introduction and principles
Commutators and the Correspondence Principle Formal Connection Q.M.Classical Mechanics Correspondence between Classical Poisson bracket of And Q.M. Commutator.
Dirac’s Quantum Condition
Quantum One: Lecture 5a. Normalization Conditions for Free Particle Eigenstates.
Wavefunction Quantum mechanics acknowledges the wave-particle duality of matter by supposing that, rather than traveling along a definite path, a particle.
Chap 12 Quantum Theory: techniques and applications Objectives: Solve the Schrödinger equation for: Translational motion (Particle in a box) Vibrational.
PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10)
Overview of QM Translational Motion Rotational Motion Vibrations Cartesian Spherical Polar Centre of Mass Statics Dynamics P. in Box Rigid Rotor Angular.
PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10)
Group work Show that the Sx, Sy and Sz matrices can be written as a linear combination of projection operators. (Projection operators are outer products.
Spin and addition of angular momentum
PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10)
PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10)
Lecture 13 Space quantization and spin (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.
PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10)
Quantum mechanics review. Reading for week of 1/28-2/1 – Chapters 1, 2, and 3.1,3.2 Reading for week of 2/4-2/8 – Chapter 4.
MSEG 803 Equilibria in Material Systems 6: Phase space and microstates Prof. Juejun (JJ) Hu
Angular Momentum. What was Angular Momentum Again? If a particle is confined to going around a sphere: At any instant the particle is on a particular.
1 The Failures of Classical Physics Observations of the following phenomena indicate that systems can take up energy only in discrete amounts (quantization.
Chap 12 Quantum Theory: techniques and applications Objectives: Solve the Schrödinger equation for: Translational motion (Particle in a box) Vibrational.
P460 - Sch. wave eqn.1 Solving Schrodinger Equation If V(x,t)=v(x) than can separate variables G is separation constant valid any x or t Gives 2 ordinary.
Lecture 2. Postulates in Quantum Mechanics Engel, Ch. 2-3 Ratner & Schatz, Ch. 2 Molecular Quantum Mechanics, Atkins & Friedman (4 th ed. 2005), Ch. 1.
Operators A function is something that turns numbers into numbers An operator is something that turns functions into functions Example: The derivative.
Ch 9 pages Lecture 22 – Harmonic oscillator.
Ch 3 Quantum Mechanics of Electrons EE 315/ECE 451 N ANOELECTRONICS I.
An Electron Trapped in A Potential Well Probability densities for an infinite well Solve Schrödinger equation outside the well.
6.852: Distributed Algorithms Spring, 2008 April 1, 2008 Class 14 – Part 2 Applications of Distributed Algorithms to Diverse Fields.
(1) Experimental evidence shows the particles of microscopic systems moves according to the laws of wave motion, and not according to the Newton laws of.
MODULE 1 In classical mechanics we define a STATE as “The specification of the position and velocity of all the particles present, at some time, and the.
The Quantum Theory of Atoms and Molecules The Schrödinger equation and how to use wavefunctions Dr Grant Ritchie.
Chapter 2 The Schrodinger Equation.  wave function of a free particle.  Time dependent Schrodinger equation.  The probability density.  Expectation.
Modern Physics (II) Chapter 9: Atomic Structure
MS310 Quantum Physical Chemistry
Quantum Chemistry: Our Agenda Birth of quantum mechanics (Ch. 1) Postulates in quantum mechanics (Ch. 3) Schrödinger equation (Ch. 2) Simple examples of.
PHY 520 Introduction Christopher Crawford
Physical Chemistry III (728342) The Schrödinger Equation
Vibrational Motion Harmonic motion occurs when a particle experiences a restoring force that is proportional to its displacement. F=-kx Where k is the.
Quantum Chemistry: Our Agenda Postulates in quantum mechanics (Ch. 3) Schrödinger equation (Ch. 2) Simple examples of V(r) Particle in a box (Ch. 4-5)
1 2. Atoms and Electrons How to describe a new physical phenomenon? New natural phenomenon Previously existing theory Not explained Explained New theoryPredicts.
Postulates Postulate 1: A physical state is represented by a wavefunction. The probablility to find the particle at within is. Postulate 2: Physical quantities.
Quantum Atom. Problem Bohr model of the atom only successfully predicted the behavior of hydrogen Good start, but needed refinement.
Chapter 3 Postulates of Quantum Mechanics. Questions QM answers 1) How is the state of a system described mathematically? (In CM – via generalized coordinates.
Principles of Quantum Mechanics P1) Energy is quantized The photoelectric effect Energy quanta E = h  where h = J-s.
1924: de Broglie suggests particles are waves Mid-1925: Werner Heisenberg introduces Matrix Mechanics In 1927 he derives uncertainty principles Late 1925:
Lectures in Physics, summer 2008/09 1 Modern physics 2. The Schrödinger equation.
PHL424: Nuclear angular momentum
The Quantum Mechanical Model of the Atom
Q. M. Particle Superposition of Momentum Eigenstates Partially localized Wave Packet Photon – Electron Photon wave packet description of light same.
Schrodinger wave equation
Quantum Mechanics.
Properties of Hermitian Operators
Lecture 13 Space quantization and spin
 Heisenberg’s Matrix Mechanics Schrödinger’s Wave Mechanics
Fundamentals of Quantum Electrodynamics
The Postulates and General Principles
Quantum One.
Central Potential Another important problem in quantum mechanics is the central potential problem This means V = V(r) only This means angular momentum.
Quantum Two.
Modern Physics Photoelectric Effect Bohr Model for the Atom
Chapter 4 Electrons as Waves
Physical Chemistry Week 5 & 6
The Stale of a System Is Completely Specified by lts Wave Function
Quantum Mechanics Postulate 4 Describes expansion
Shrödinger Equation.
Infinite Square Well.
Addition of Angular Momentum
Introductory Quantum Mechanics/Chemistry
Presentation transcript:

Nanoelectronics Chapter 3 Quantum Mechanics of Electrons

STM image of atomic “quantum corral” Atoms form a quantum corral to confine the surface state electrons.

3.1 General Postulates of Quantum Mechanics

3.1 General Postulates of Quantum Mechanics Operators

3.1.2 Eigenvalues and Eigenfunctions

3.1.3 Hermitian Operator Hermitian operators have real eigenvalues. Their eigenfunctions form an orthogonal, complete set of functions. (if normalized)

3.1.4 Operators for Quantum Mechanics Momentum operator Energy operator

3.1.4 Operators for Quantum Mechanics Position operator The eigenfunction is

3.1.4 Operators for Quantum Mechanics Commutation and the Uncertainty principle α and β operators are commute The difference operator: is commutor  So one cannot measure x and p x (along x-axis) with arbitrary precision They are not commute!

3.1.4 Operators for Quantum Mechanics Uncertainty principle So one can measure x and p y (along y-axis) with arbitrary precision

3.1.5 Measurement Probability Postulate 3: The mean value of an observable is the expectation value of the corresponding operator. Postulate 4:

3.2 Time-independent Schrodinger’s Equation Separation of variables

3.2.1 Boundary Conditions on Wavefunction Consider a one-dimensional space with electrons constrained in 0<x<L

Evidence for existence of electron wave

3.3 Analogies between Quantum Mechanics and Classical Electromagnetics Maxwell’s equations: comparison

3.4 Probabilistic current density

3.5 Multiple Particle Systems State function Joint probability of finding particle 1 in d 3 r 1 point r 1 and finding particle 2 in d 3 r 2 of point r 2 State function obeys

3.5 Multiple Particle Systems Hamiltonian: Example: two charged particles:

3.6 Spin and Angular Momentum Lorentz force If the particle has a net magnetic moment µ, passing through a magnetic field B Angular momentum: Spin is a purely quantum phenomenon that cannot be understood by appealing to everyday experience. (it is not rotating by its own axis.)

3.7 Main Points Meaning of state function Probability of finding particles at a given space Probability of measuring certain observable Operators, eigenvalues and eigenfunctions Important quantum operators Mean of an observable Time-dependent/independents Schrodinger equations Probabilistic current density Multiple particle systems

3.8 Problems 1, 3, 8, 9, 15