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