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Nanoelectronics Chapter 3 Quantum Mechanics of Electrons [email protected]@2015.3 1 STM image of atomic βquantum corralβ Atoms form a quantum corral to confine the surface state electrons. [email protected]@2015.3 2 3.1 General Postulates of Quantum Mechanics β’ P1: To every quantum system there is a state function, Ξ¨(π,π‘) , that contains everything that can be known about the system β’ P2: (a) Every physical observable O (position, momentum, energy, etc.) is associated with a linear Hermitian operator. π (b) Eigenvalue problem: πππ = ππ ππ (c) If a system is in the initial state Ξ¨, measurement of O will yield one of the eigenvalues Ξ»n of π with probability π ππ = | Ξ¨(π,π‘) ππ β π3 π₯ |2 [email protected]@2015.3 3 3.1 General Postulates of Quantum Mechanics β’ 3.1.1 Operators [email protected]@2015.3 4 3.1.2 Eigenvalues and Eigenfunctions [email protected]@2015.3 5 3.1.3 Hermitian Operator β’ Hermitian operators have real eigenvalues. Their eigenfunctions form an orthogonal, complete set of functions. (if normalized) [email protected]@2015.3 6 3.1.4 Operators for Quantum Mechanics β’ Momentum operator β’ Energy operator [email protected]@2015.3 7 3.1.4 Operators for Quantum Mechanics β’ Position operator The eigenfunction is [email protected]@2015.3 8 3.1.4 Operators for Quantum Mechanics β’ Commutation and the Uncertainty principle Ξ± and Ξ² operators are commute The difference operator: is commutor ο¨ They are not commute! So one cannot measure x and px (along x-axis) with arbitrary precision [email protected]@2015.3 9 3.1.4 Operators for Quantum Mechanics So one can measure x and py (along y-axis) with arbitrary precision β’ Uncertainty principle [email protected]@2015.3 10 3.1.5 Measurement Probability β’ Postulate 3: The mean value of an observable is the expectation value of the corresponding operator. β’ Postulate 4: [email protected]@2015.3 11 3.2 Time-independent Schrodingerβs Equation Separation of variables [email protected]@2015.3 12 3.2.1 Boundary Conditions on Wavefunction Consider a one-dimensional space with electrons constrained in 0<x<L [email protected]@2015.3 13 Evidence for existence of electron wave [email protected]@2015.3 14 3.3 Analogies between Quantum Mechanics and Classical Electromagnetics β’ Maxwellβs equations: comparison [email protected]@2015.3 15 3.4 Probabilistic current density [email protected]@2015.3 16 3.5 Multiple Particle Systems β’ State function β’ Joint probability of finding particle 1 in d3r1 point r1 and finding particle 2 in d3r2 of point r2 β’ State function obeys [email protected]@2015.3 17 3.5 Multiple Particle Systems β’ Hamiltonian: β’ Example: two charged particles: [email protected]@2015.3 18 3.6 Spin and Angular Momentum β’ Lorentz force β’ If the particle has a net magnetic moment µ, passing through a magnetic field B β’ Angular momentum: β’ Spin is a purely quantum phenomenon that cannot be understood by appealing to everyday experience. (it is not rotating by its own axis.) [email protected]@2015.3 19 3.7 Main Points β’ β’ β’ β’ β’ β’ β’ 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 β’ Probabilistic current density β’ Multiple particle systems [email protected]@2015.3 20 3.8 Problems β’ 1, 3, 8, 9, 15 [email protected]@2015.3 21
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