PH 401 Dr. Cecilia Vogel

Review Outline  Particle in a box  expectation values  uncertainties  Bound and unbound states  Particle in a box  solve TISE  stationary state wavefunctions  eigenvalues

Infinite Square Well  Odd-n wavefunctions   since  Even-n wavefunctions   since  energy 

Infinite Square Well  For odd and even wavefunctions  =0  Expectation value of p 2   From the TISE, we know 

Infinite Square Well  So   and  p=0+  k  The wave is a combo of a wave moving right with wavelength =2  /k and a wave moving left with wavelength =2  /k  Just like a standing wave in a string - demo

Generalize  We can solve the TISE in a similar fashion for any piecewise constant potential  V(x) =V i = constant in the i th region  Wavefunction will be sinusoidal in regions where E> V i  Wavefunction will be exponential in regions where E< V i  Note: Infinite square well has only bound state  Other situations may have bound and unbound states

Dichotomy  Recall  If E> V i for the regions where x approaches infinity or negative infinity  then the state is unbound  If E< V i for the regions where x approaches infinity and negative infinity  then the state is bound  Unbound states will  have a continuous spectrum of energies  will be un-normalizable  Bound states will  have a discrete spectrum of energies  will be normalizable

PAL Monday week 5  Given the following “box”  1.For what energies will a particle be bound by this potential? 2.For what energies will a particle be in an unbound state of this potential? 3.For what energies will there be discrete energy levels for a particle subject to this potential? 4.For what energies will there be continuous energy levels for a particle subject to this potential?

PAL Monday week 5 5.Write out the general form of a wavefunction corresponding to a bound state with E>2eV. 6.Sketch the wavefunction for a bound state with E>2eV with ___ nodes. 7.Sketch the wavefunction for an unbound state.