1. Wave mechanics in potentials Modern Ch.4, Physical Systems, 30.Jan.2003 EJZ Particle in a Box (Jason Russell), Prob.12 Overview of finite potentials Harmonic Oscillator (Don Verbeke), Prob.48 Hydrogen atom

2. Infinite Square well: V(0<x<L) = 0, V=  outside

3. Overview of finite potentials Finite well:  can spill out Tunneling through finite barriers

4. Harmonic oscillator: V(x) =1/2 kx 2

5. Hydrogen atom : Bohr model We found r n = n 2 r 1, E n = E 1 /n 2, where the “principle quantum number” n labels the allowed energy levels. Discrete orbits match observed energy spectrum

6. Hydrogen atom: Orbits are not discrete (notice different r scales)

7. Hydrogen atom: Schrödinger solutions depend on new angular momentum quantum numbers Quantization of angular momentum direction for l=2 Magnetic field splits l level in (2l+1) values of m l = 0, ±1, ± 2, … ± l

8. Hydrogen atom examples from Giancoli

12. Summary: You can calculate permitted states and energies from boundary conditions Finite wells and barriers need reflection/transmission analysis Infinite square well has E n ~n 2 E 1 Harmonic oscillator has evenly spaced  E Hydrogen atom: 3D spherical solution to Schrödinger equation yields 3 new quantum numbers: l = orbital quantum number m l = magnetic quantum number m s = spin = ±1/2