Lect17 - Le cture17: Angular Mom ntum Atom S s, S & e ,...

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Lecture 17, p 1 Lecture 17: Angular Momentum, Atomic States, Spin, & Selection Rules
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Lecture 17, p 2 Today Schrödinger’s Equation for the Hydrogen Atom Radial wave functions Angular wave functions Angular Momentum Quantization of L z and L 2 Spin and the Pauli exclusion principle Stern-Gerlach experiment
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Lecture 17, p 3 Summary of S-states of H-atom The “s-states” ( l =0, m =0) of the Coulomb potential have no angular dependence. In general: because Y 00 ( θ , φ ) is a constant. Some s-state wave functions (radial part): r R 20 5 0 10a 0 R 30 r 0 15a 0 0 5 1 r 0 4a 0 R 10 ( 29 ( 29 ( 29 ( 29 ( 29 00 0 , , , : , t , bu nlm nl lm n n r R r Y r R r ψ θ φ = S-state wave functions are spherically symmetric. | ψ 20 (r, θ , φ )| 2 : http://www.falstad.com/qmatom/
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Lecture 17, p 4
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Lecture 17, p 5 The Y lm ( θ , φ ) are known as “spherical harmonics”. They are related to the angular momentum of the electron. Total Wave Function of the H-atom We will now consider non-zero values of the other two quantum numbers: l and m . n “principal” ( n 1) l “orbital” (0 l < n ) m “magnetic” (- l m + l ) ( 29 ( 29 ( 29 , , , nlm nl lm r R r Y ψ θ φ = x y z r θ φ * The constraints on l and m come from the boundary conditions one must impose on the solutions to the Schrodinger equation. We’ll discuss them briefly. *
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Lecture 17, p 6 Quantized Angular Momentum Linear momentum depends on the wavelength (k=2 π / λ ): Angular momentum depends on the tangential component of the momentum. Therefore L z depends on the wavelength as one moves around a circle in the x-y plane. Therefore, a state with L z has a similar form: An important boundary condition: An integer number of wavelengths must fit around the circle. Otherwise, the wave function is not single-valued. This implies that m = 0, ±1, ±2, ±3, … and L z = 0, ±ħ, ±2ħ, ±3ħ, … Angular momentum is quantized!! Reminder: e im φ = cos(m φ ) + i sin(m φ ) where ( ) ikx p k x e ψ = h where ( ) ( , ) im Z lm L m r Y e φ θ φ = r h http://www.falstad.com/qmatom/ We’re ignoring R(r) for now. Re( ψ ) φ L z
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Lecture 17, p 7 Summary of quantum numbers for the H-atom orbitals: The l Quantum Number In the angular wave function ψ lm (θ,φ29 the quantum number l tells us the total angular momentum L. L 2 = L x 2 + L y 2 + L z 2 is also quantized. The possible values of L 2 are: The quantum number m reflects the component of angular momentum about a given axis. 2 2 ( 1) where 0,1, 2,. .. L = + = h l l l Principal quantum number: n = 1, 2, 3, …. Orbital quantum number: l = 0, 1, 2, …, n-1 Orbital magnetic’ quantum number: m = - l , -( l -1), … 0, … ( l -1), l Wave functions can be eigenstates of both L 2 and L Z . For spherically symmetric potentials, like H-atom, they can also be eigenstates of E. Such states are called “orbitals”.
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This note was uploaded on 02/21/2011 for the course PHYS 214 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lect17 - Le cture17: Angular Mom ntum Atom S s, S &amp; e ,...

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