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 (2 nd deriv opposite  )  exp in barrier (2 nd deriv same sign as  )  wavefunction looks a little like two square well gs wavefunctions side-by-side, with a little dip in the barrier region

5. Double Well  1 st excited state  odd  crosses axis once  sinusoidal on both sides (2 nd deriv opposite  )  exp in barrier (2 nd deriv same sign as  )  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 1 st 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 2 nd and 3 rd excited states  which both look a lot like two square well 1 st 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 1 st es close in energy, then a gap, the 2 nd es and 3 rd es close in energy, another gap, 4 th es and 5 th 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.