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PH 301

PH 301. Dr. Cecilia Vogel. Recall square well. The stationary states of infinite square well: sinusoidal wavefunction wavefunction crosses axis n times for nth excited state GS is even, nth excited state is even for even n, odd for odd n

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PH 301

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  1. PH 301 Dr. Cecilia Vogel

  2. Recall square well • The stationary states of infinite square well: • sinusoidal wavefunction • wavefunction crosses axis n times for nth excited state • GS is even, nth excited state is even for even n, odd for odd n • energy levels get further an further apart as you go up in energy level

  3. Double Well Now imagine a double-well For low energy levels, the central barrier is CF The wavefunction will be sinusoidal on either side, but combo of exponentials in the barrier Wavefunction will cross axis n times for nth excited state GS is even, nth excited state is even for even n, odd for odd n

  4. Double Well Ground state Even never crosses axis sinusoidal on both sides (2nd deriv opposite y) exp in barrier (2nd deriv same sign as y) wavefunction looks a little like two square well gs wavefunctions side-by-side, with a little dip in the barrier region

  5. Double Well 1st excited state odd crosses axis once sinusoidal on both sides (2nd deriv opposite y) exp in barrier (2nd deriv same sign as y) wavefunction looks a little like two square well gs wavefunctions with one of them flipped, so it crosses in the barrier region

  6. Double Well gs and 1st excited state both wavefunction looks a little like two square well gs wavefunctions on both sides they look a lot alike they have very similar curvature, very similar wavenumber, very similar energy This is also true of the 2nd and 3rd excited states which both look a lot like two square well 1st es wavefunctions on both sides and so on

  7. Double Well Energy levels The energy levels of the double-well come in pairs. gs and 1st es close in energy, then a gap, the 2nd es and 3rd es close in energy, another gap, 4th es and 5th es close… Double well has pairs of energy levels separated by gaps triple well has trios of energy levels separated by gaps and so on

  8. Band Structure Electrons in crystal see many, many wells one for each atomic nucleus in the crystal that the e is attracted to many-well has many closely-spaced energy levels separated by gaps closely-spaced levels = “bands” gaps = “band gaps” Conduction band = band of energy levels that overlaps with unbound states.

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