Lecture9ForcesPotentialsandtheShellModel

# Lecture9ForcesPotentialsandtheShellModel - Forces,...

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Forces, Potentials, and the Shell Model Recall the Infinite Square Well (1D) Solve Shroedinger’s equation: ψ E H = E V dx d = - 2 2 Result: Consideration of boundary conditions (the behavior of the wavefunction at the walls) results in quantization. Both wavefunctions and eigenstates (energy levels) 2 2 2 8 mL h n E n = Notice the dependence of the energy levels on the size of the box, and on the principal quantum number. Harmonic oscillator (1D) Hooke’s law : ) ( 0 x x k F - - = If 0 x x = , the system is at equilibrium because there is no force. However if x is different from 0 x there is a force which acts to restore the position to the equilibrium value (Notice the negative sign.) dx dV F - = Integrating we get, 2 0 ) ( 2 1 x x k V - = Now solve Schrodinger’s equation using this potential. Solution: Wavefunctions and eigenvalues Eigenvalues: ϖ ) 2 1 ( + = n E n where m k =

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Notice the energy spacing for the harmonic oscillator. What is the minimum energy of the harmonic oscillator?
V. Nuclear Shell Model A. Quantum Properties of Nuclei 1. Discrete Energy Levels 2. Nuclear Spin - I a. Experimental Summary e-e : 0 = I ALWAYS e-o, o-e: 2 e n I = , where n is an odd integer (1/2, 3/2, . ..) o-o : e n I = , where n is an integer (0, 1, 2 . ..) WE'LL USE 1 = for our spins b. Implication e-e result implies strong pairing is energetically favorable spins must cancel c. Reason: Nuclear Force is attractive ; in contrast spins are unpaired in a atomic orbitals due to e-e repulsion ( Pauli exclusion principle ) 3. Closed Shells – Unusual Stability a. Magic Numbers 2, 8, 20, 28, 50, 82, 126 (neutrons) b. Energetics: (M LD – M), B p , B n , B α c. Lifetimes: 8 2 2 0 8 1 2 6 P b 8 2 2 0 9 1 2 7 P b 8 4 2 1 0 P o 1 2 6 8 4 2 1 2 1 2 0 P o STABLE 22y 138d 10 - 7 s Z=82 & N=126 appear to be stable

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4. Magnetic Moments Moving Charge created a magnetic field with moment μ μ = e 2 M c f ( I ) e ; μ N = nuclear magneton (M = M p ) a. Expect μ p = μ N Observe : μ p = 2.793 μ N μ n = 0 μ n = - 1.913 μ N b. Implication: nucleon has substructure, since one observes charge on periphery of particle. e.g., proton
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## This note was uploaded on 07/02/2011 for the course CHEM-C 460 taught by Professor Staff during the Spring '10 term at Indiana.

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Lecture9ForcesPotentialsandtheShellModel - Forces,...

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