Forces, Potentials, and the Shell Model Recall the Infinite Square Well (1D) Solve Shroedinger’s equation: EHEVdxd22Result: Consideration of boundary conditions (the behavior of the wavefunction at the walls) results in quantization. Both wavefunctions and eigenstates (energy levels) 2228mLhnEnNotice the dependence of the energy levels on the size of the box, and on the principal quantum number. Harmonic oscillator (1D) Hooke’s law : )(0xxkFIf 0xx, the system is at equilibrium because there is no force. However ifxis different from 0xthere is a force which acts to restore the position to the equilibrium value (Notice the negative sign.) dxdVFIntegrating we get, 20)(21xxkVNow solve Schrodinger’s equation using this potential. Solution: Wavefunctions and eigenvalues Eigenvalues: )21(nEnwhere mk
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