Lect11 - "It was almost as incredible as if you fired a...

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“It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper, and it came back to hit you!” --E. Rutherford (on the ‘discovery’ of the nucleus)
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Final Exam – March 3! Overview of the rest of the course Up to now – general properties and equations of quantum mechanics Time-dependent and time-independent Schr. Eqs. Eigenstates, superposition of eignestates and time- dependence, tunneling, Schrodinger’s cat, . . . This week – quantum states in 3 dimensions, electron spin, H atom, exclusion principle, periodic table of atoms Next week – molecules and solids, consequences of Q. M. Metals, insulators, semiconductors, superconductors, lasers, . . HW 6 due next Thursday, 8am
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Particles in 3D Potentials and the Hydrogen Atom P(r) 0 4a 0 0 1 r r = a 0 z x L L L ( ) 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 ψ = o a / r 3 o e a 1 ) r ( = π 2 6 13 n eV . E n =
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Overview Overview z 3-Dimensional Potential Well z Product Wavefunctions z Concept of degeneracy z Early (Incorrect!) Models of the Hydrogen Atom z Planetary Model z Schrödinger’s Equation for the Hydrogen Atom z Semi-quantitative picture from uncertainty principle z Ground state solution* z Spherically-symmetric excited states (“s-states”)* z *contain details beyond what we expect you to learn here
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Quantum Particles in 3D Potentials Quantum Particles in 3D Potentials z One consequence of confining a quantum particle in two or three dimensions is “degeneracy” -- the occurrence of several quantum states at the same energy level. z 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 many “real” 3D quantum systems , such as atoms and artificial quantum structures: (www.kfa-juelich.de/isi/) A real (3D) “quantum dot” z To illustrate this important point in a simple system, we extend our favorite potential -- the infinite square well -- to three dimensions.
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Particle in a 3D Box (1) Particle in a 3D Box (1) outside box, x or y or z < 0 outside box, x or y or z > L 0 inside box U(x,y,z) = z Let’s solve this SEQ for the particle in a 3D box: y z The extension of the Schrödinger Equation (SEQ) to 3D is straightforward in cartesian (x,y,z) coordinates: ψψ ψ E ) z , y , x ( U dz d dy d dx d m = + + + 2 2 2 2 2 2 2 2 = ) , , ( z y x ψ≡ where This simple U(x,y,z) can be “separated” A special feature U(x,y,z) = U(x) + U(y) + U(z) z x L L L Kinetic energy term in the Schrödinger Equation () 2 2 2 2 1 z y x p p p m like + + Easy to see in this case since U = 0 or Very useful in general as illustrated in the solution on the following slide
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Particle in a 3D Box (2) Particle in a 3D Box (2) z So, the whole problem simplifies into three one-dimensional equations that we’ve already solved in Lecture 7.
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This note was uploaded on 09/12/2009 for the course PHYS 214 taught by Professor Debevec during the Spring '07 term at University of Illinois at Urbana–Champaign.

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Lect11 - "It was almost as incredible as if you fired a...

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