Lect16 - Lecture 16 p 1 Lecture 16 3D Potentials and the...

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Unformatted text preview: Lecture 16, p 1 Lecture 16: 3D Potentials and the Hydrogen Atom P(r) 4a 1 r r = a z x L L L ( 29 2 2 2 2 2 8 z y x n n n n n n mL h E z y x + + ⋅ = ) ( ) ( ) ( ) , , ( z y x z y x ϕ ϕ ϕ ψ = o a / r 3 o e a 1 ) r (- = π ψ 2 6 13 n eV . E n- = Lecture 16, p 2 Final Exam: Monday, March Homework 6: Due last Saturday before Final, 8 am Up to now: • General properties and equations of quantum mechanics • Time-independent Schrodinger’s Equation (SEQ) and eigenstates. • Time-dependent SEQ, superposition of eigenstates, time dependence. • Collapse of the wave function, Schrodinger’s cat • Tunneling This week: • 3 dimensions, angular momentum, electron spin, H atom • Exclusion principle, periodic table of atoms Next week: • Molecules and solids, consequences of Q. M. • Metals, insulators, semiconductors, superconductors, lasers, . . Overview of the Course Lecture 16, p 3 Today 3-Dimensional Potential Well: • Product Wave Functions • Degeneracy Schrödinger’s Equation for the Hydrogen Atom: • Semi-quantitative picture from uncertainty principle • Ground state solution * • Spherically-symmetric excited states (“s-states”) * * contains details beyond what we expect you to know on exams. Lecture 16, p 4 Lecture 16, p 5 Quantum Particles in 3D Potentials So far, we have considered quantum particles bound in one-dimensional potentials. This situation can be applicable to certain physical systems but it lacks some of the features of most real 3D quantum systems, such as atoms and artificial structures. One consequence of confining a quantum particle in two or three dimensions is “degeneracy” -- the existence of several quantum states at the same energy. To illustrate this important point in a simple system, let’s extend our favorite potential - the infinite square well - to three dimensions . A real (2D) “quantum dot” http://pages.unibas.ch/phys-meso/Pictures/pictures.html Lecture 16, p 6 Particle in a 3D Box (1) ∞ outside box, x or y or z < 0 ∞ outside box, x or y or z > L inside box U(x,y,z) = Let’s solve this SEQ for the particle in a 3D cubical box: The extension of the Schrödinger Equation (SEQ) to 3D is straightforward in Cartesian (x,y,z) coordinates: This U(x,y,z) can be “separated”: U(x,y,z) = U(x) + U(y) + U(z) y z x L L L ( , , ) x y z ψ ψ ≡ where Kinetic energy term: ( 29 2 2 2 1 2 x y z p p p m + + U = ∞ if any of the three terms = ∞ . 2 2 2 2 2 2 2 ( , , ) 2 U x y z E m x y z ψ ψ ψ ψ ψ ∂ ∂ ∂- + + + = ∂ ∂ ∂ ℏ Lecture 16, p 7 http://www.falstad.com/qm2dbox/ Particle in a 3D Box (2) 2 2 ( ) sin 2 2 x x x n nx n n h f x N x E L m L π = = ⋅ Whenever U(x,y,z) can be written as the sum of functions of the individual coordinates, we can write some wave functions as products of functions of the individual coordinates: (see the supplementary slides) Similarly for y and z....
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Lect16 - Lecture 16 p 1 Lecture 16 3D Potentials and the...

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