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Lecture9ForcesPotentialsandtheShellModel_000

# Lecture9ForcesPotentialsandtheShellModel_000 - Forces...

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Forces, Potentials, and the Shell Model Recall the Infinite Square Well (1D) Solve Shroedinger’s equation: 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: ) 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?