Lect12 - "Anyone who can contemplate quantum mechanics...

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“Anyone who can contemplate quantum mechanics without getting dizzy hasn’t understood it.” --Niels Bohr
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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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Angular Momentum, Atomic States, Spin, & the Pauli Principle -15 -10 -5 0 0 5 10 15 20 U(r) n= 1 n= 3 n= 2
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Overview Overview z Schrödinger’s Equation for the Hydrogen Atom z Radial wave functions z Angular wave functions z Angular Momentum z Quantization of L z and L 2 z Spin and the Pauli exclusion principle z Stern-Gerlach experiment z Nuclear spin and MRI
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Summary of s Summary of s - - states of H states of H - atom atom z To get the exact eigenstates and energies for the “s-states” of the Coulomb potential, one needs to solve the radial SEQ z Summary of wave functions for the “s-states”: z The zeros in the subscripts below are a reminder that these are states with zero angular momentum. 0 r 0 4a 0 R 10 0 / 0 , 1 ) ( a r e r R r R 20 0 10a 0 0 0 2 / 0 0 , 2 2 1 ) ( a r e a r r R R 30 r 0 15a 0 0 3 / 2 0 0 0 , 3 3 2 2 3 ) ( a r e a r a r r R + ) r ( ER ) r ( R r e r r r 1 m 2 2 2 2 2 = κ = (spherically symmetric)
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The Y lm ( θ,φ ) are known as “Spherical harmonics”. They are related to the angular momentum of the electron. Last lecture we studied the properties of the radial part. Today we will examine the angular part. Wavefunction Wavefunction of the H of the H - - atom atom z Solutions to the time-independent SEQ for this spherically symmetric potential can have simple forms in spherical coordinates: x y z r θ φ ) , ( ) ( ) , , ( φ θ lm nl nlm Y r R r = Ψ with quantum numbers: n l and m principal orbital magnetic (angular momentum)
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Quantized Angular Momentum Quantized Angular Momentum z Ordinary linear momentum depends on how fast the phase changes as you move linearly in space: z Angular momentum L Z around the z axis depends on how fast the phase changes as you rotate around the z-axis. A state with an exact value of L z is of the form: Re( ψ ) φ ρ p t r = L z = m ћ Reminder: e im φ = cos(m φ ) + i sin(m φ ) m is called the ‘orbital’ magnetic quantum number where ( ) ikx p kx e ψ =∝ = , where ( ) ( , ) im Zl m Lm rY e φ ψθ G = An integer number , m, of wavelengths must fit around the circle: m = 0, ± 1, ± 2, ± 3, . ... Otherwise the function doesn’t have a single value.
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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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Lect12 - "Anyone who can contemplate quantum mechanics...

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