PH 401 Dr. Cecilia Vogel

Atoms in a Crystal  On Monday, we modeled electrons in a crystal as experiencing an array of wells, one for each atom in the crystal.  What do the atoms themselves experience?  Each atom is pretty much bound to one well –  no such thing as “conduction atoms” generally!  Is it a square well?  No, square well energy levels get further and further apart,  Crystal atom levels are nearly evenly spaced

SHO  Any well that is not flat at the bottom has some curvature  Thus can be approximated by a quadratic  A quadratic potential energy function is called a simple harmonic oscillator  If we place x=0 at the center of the well,  V(x) = ½ m  2 x 2.  where  = the classical oscillation frequency (if it were classical rather than quantum)

SHO Tricks  Let’s find the stationary states of SHO  by solving TISE  Solutions are not obvious.  1 st trick:  Let’s consider regions where |x| is very large, then  and solution is

SHO Tricks  Plugging  into  Yields a value for B  Unlike the square well, this decay constant is the same value (same B) for all energy levels, for all n.  Why then does the wavefunction for higher energy levels extend further, wider?

SHO Qualitative  We can divide space into CA and CF regions, just like we have done before  the center of the well, where E>V  is classically allowed  for large x, where E<V  it is classically forbidden  The point where it changes from CA to CF are the classical turning points  the larger E is, the farther out the classical turning points  E=V= ½ m  2 x 2 occurs for larger if E is larger

SHO Tricks  Trick #2:  We know  is true for large x  For all x, we must have  where f(x) is dominated by the exponential at large x  f(x) is thus a polynomial of degree n

SHO Qualitative  We know qualitatively that the nth excited state crosses the axis n times  as does a symmetric polynomial  We also know that (since our PE is symmetric)  our wavefunctions will be even or odd  So our polynomials contain  only even powers (even n)  or only odd powers (odd n)

Ground State Energy  n=0, polynomial of degree zero = constant, call it Ao  Plug into TISE  plug in B  Eo=(1/2)  .

2 nd Excited State Energy  n=2,  Plug into TISE  quadratic term  plug in B  E2=(5/2)  .

PAL Friday week 6 1.Without doing any integrals, find the uncertainty in x and p for the ground state of a simple harmonic oscillator in terms of parameters  and m. (HW) 2.a) What is the form of the wavefunction for the first excited state of a simple harmonic oscillator (n=1)? b) Plug this wavefunction into the TISE to determine the 1 st excited state energy in terms of parameters  and m.

Patterns for SHO 1.E n =(n + ½)    2. Energy levels are equally spaced,  E=    3.