Intermediate Quantum Mechanics PHYS307 Professor Scott Heinekamp

1. Goals of the course • by speculating on possible analogies between waves moving in a uniform medium and the so-called free particle, to develop some calculational tools for describing matter waves, including the de Broglie wavelength for a moving particle, and the Born interpretion of the wave function • to ‘derive’ the Schrödinger equation(s) for said wave function for a particle in (or not in) a potential V(x) • to discuss (review?) several important potential energy cases • to explore the alternative methodology of Heisenberg’s operator algebra for the case of the harmonic oscillator potential • to work in three dimensions, and solve problems of practical importance, including the hydrogen atom • to introduce the quantum mechanical treatment of spin and orbital angular momentum • to briefly apply these ideas to many-body systems Intermediate Quantum Mechanics PHYS307 Professor Scott Heinekamp

2. The Spectrum of Hydrogen • bright-line (emission) spectrum: hot glowing sample of H emits light • dark-line (absorption) spectrum: cool sample of H removes light • in the visible, one sees only the Balmer series, with wavelengths given by the famous Rydberg formula (n = 3,4,5…) • it is a miracle that we can only SEE the Balmer series • the other series are given by • Lyman: nf = 1 (all UV) • Paschen: nf = 3 (all IR)

3. Explaining this result by quantizing something I • we assume that the orbits of the electrons are quantized, in the sense that if an orbiting electron in ‘orbit level’ n absorbs a photon of the correct energy, it may be ‘kicked’ all the way off to ∞ • classical orbit theory: equate Coulomb force to centripetal force for an atom of atomic number Z with only one electron left on it, to get KE (m is reduced mass, which is almost the electron mass but slightly less): • more classical theory: • assuming a circular orbit of radius r, both PE and KE are constants

4. Explaining this result by quantizing something II • Einstein explained the photoelectric effect by arguing that light’s energy is proportional to its frequency, and that light can only be emitted or absorbed in ‘packets’ (quanta) now called photons: E = hf • h is Planck’s constant: h = 6.626 x10–34 J∙s = 4.136 x10–15 eV∙s • incidentally, we often use ‘hbar’: ħ:=h/2p = 1.046 x10–34 J∙s • we assume that the energy to ionize requires a photon whose frequency f is half of the orbital frequency of the ‘starting’ state n, times n: • orbital frequency is forb: • [Kepler’s third law: (period)2 ~ (radius)3] • so, we equate |E| to ½ nhforb:

5. Explaining this result by quantizing something III • now connect all of this together by relating the radius of the orbit to n: take the expression from [I] for v(r) and the expression from [II] for vn and equate the two: • so the orbital radii are quantized… as are the orbital speeds… as are the energies of the orbits! • one can show that angular momentum is quantized: L = nħ • this is equivalent to n de Broglie wavelengths around the orbit circumference

6. The ‘old’ theory of the hydrogen-like atom à la Niels Bohr • electron energies En= – Z2E0n– 2and that is very good! • they crowd closer and closer together and there are an infinite number of them  ionization at zero energy • the speeds get smaller as n goes up ~ n– 1… that’s sort of OK • the radii get larger as n goes up ~ n2… that’s sort of not so OK • in a transition from ni to nf, a photon is emitted or absorbed whose energy is precisely the difference in the electron’s energy • it misses completely the angular dependence of ‘where’ the electron is, and it oversimplifies greatly the radial position • the electrons DO NOT ‘orbit’… they are ‘everywhere’ at once • still, the theory was a smashing success and earned a Nobel Prize