Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Homework Phys 452 Wednesday Feb 22: assignment # , 8.2, 8.7, 8.14 extended to Thursday Feb 23 Friday Feb 24: assignment # , 8.4, 8.16

Techniques to find approximate solutions to the Schrodinger equation Phys The perturbation theory 2. The variational principle 3. The WKB approximation

The WKB approximation Wentzel- Kramers - Brillouin Phys 452 Hendrik Kramers Dutch Leon Brillouin French Gregor Wentzel German

Phys 452 The WKB approximation The WKB approximation is based on the idea that for any given potential, the particle can be locally seen as a free particle with a sinusoidal wave function, but whose wavelength varies very slowly in space.

Phys 452 The free particle Infinite space Finite box

Phys 452 Flat potential Scattering state Bound state E V

Phys 452 Varying potential The WKB approximation V(x) E Classical region (E>V) Turning points

Phys 452 The WKB approximation V(x) E Classical region (E>V) Locally constant or varying very slowly In respect to wavelength

Phys 452 The WKB approximation Classical region with

Phys 452 The WKB approximation Classical region solution real part imaginary part

Phys 452 The WKB approximation Classical region solution assumption and

Phys 452 The WKB approximation Classical region solution where Incidentally

Quiz 17a Phys 452 In the WKB approximation, what can we say about the solution for the wave function ? A. The amplitude and the wavelength are fixed B. The amplitude is fixed but the wavelength varies C. The wavelength varies but the amplitude is fixed D. Both the wavelength and the amplitude vary E. There are multiple wavelengths for a given position

Phys 452 The WKB approximation Classical region solution Phase is a function of x

Phys 452 Pb 8.2 The WKB approximation Classical region Another way to write the solution: where f(x) is a complex function Develop the function as power of

Phys 452 The WKB approximation Classical region How to use this Formula? When the phase is known at specific points: Gives information on the allowed energies

Quiz 17b Phys 452 A. For any type of potential and any energy value B. Only when C. Only when D. Only when the potential exhibits 1 turning point E. Only when the potential exhibits 2 turning points In which situation can we apply the formula ?

Phys 452 Example Infinite Square well The WKB approximation Classical region

Phys 452 Pb 8.1 The WKB approximation Classical region

Phys 452 The WKB approximation at turning points V(x) E Classical region (E>V) Turning points

Phys 452 The WKB approximation at turning points V(x) E Classical region (E>V) Connection formula (eq 8.51)

Phys 452 The WKB approximation at turning points Pb 8.7 Harmonic Oscillator Pb 8.14 Hydrogen atom