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5210_chap8 - Electron Spin and the Pauli Principle Chapter...

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Chapter 8: Slide 1 Electron Spin and the Pauli Principle
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Chapter 8: Slide 2 Outline Electron Spin and the Pauli Principle Inclusion of Spin in Helium Atom Wavefunctions Spin Angular Momentum of Ground State Helium The Wavefunctions of Excited State Helium Excited State Helium Energies: He(1s 1 2s 1 )
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Chapter 8: Slide 3 The Postulates of Quantum Mechanics Postulate 1: ψ ( x, y, z, t ) is a well-behaved, square integrable function, and | ψ ( x, y , z , t )| 2 dxdydz is the probability of finding the particle between the points ( x, y, z ) and ( x+ dx, y+ dy, z+ dz ) at time t . Postulate 2: For every observable property, there is a linear Hermitian operator that is obtained by expressing the classical form in Cartesian coordinates x and momenta p x and making the replacements: x i p p x x x x = ^ ^ and Postulate 3: The wavefunction ψ ( x, t ) is obtained by solving the equation: Eqn. dinger o Schr dependent - time The ψ ψ .. ^ H t i = Postulate 4: If ψ a is an eigenfunction of the operator  with eigenvalue a , then if we measure the property A for a system whose wave function is ψ a , we always get a as the result. Postulate 5: The average (or expectation) value of an observable A is given by: = τ τ d d A A ψ * ψ ψ * ψ ^ Where  is the operator associated with the observable. Evidently, if ψ is an eigenfunction of the operator  , then the expectation value is just the eigenvalue.
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Chapter 8: Slide 4 Electron Spin We’ve known since Freshman Chemistry or before that electrons have spins and there’s a spin quantum number (there actually are two). Yet, we never mentioned electron spin, or the Pauli Exclusion Principle (actually the Pauli Antisymmetry Principle), in our treatment of ground state Helium in Chapter 7. This is because Helium is a closed shell system. That is, its electrons fill the n=1 shell. As we shall see, in open shell systems, such as the Lithium atom (1s 2 2s 1 ) or excited state Helium (e.g. 1s 1 2s 1 ), the electron’s spin and the Pauli Principle play an important role in determining the electronic energy.
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Chapter 8: Slide 5 A Brief Review of Orbital Angular Momentum in Hydrogen ) , ( ) ( , φ θ lm nl nlm Y r R φ) (r,θ ψ = The wavefunction for the electron in a hydrogen atom is: An electron moving about the nucleus in a hydrogen atom has orbital angular momentum . nlm nlm ψ ψ L 2 2 ) 1 ( h + = ^ In addition to being eigenfunctions of the Hamiltonian (with eigenvalues E n ), the wavefunctions are eigenfunctions of the angular momentum operators, L 2 and L z : ^ ^ nlm nlm z ψ m ψ L , = ˆ nlm nlm L 2 2 ) 1 ( h + = ^ nlm m nlm L z , = ˆ Shorthand
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Chapter 8: Slide 6 Do Electrons Spin?? I don’t know. I’ve never seen an electron up close and personal. What can be said is that their magnetic properties are consistent with the hypothesis that they behave “as though” they are spinning.
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