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Intermediate Quantum Mechanics PHYS307 Professor Scott Heinekamp 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 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…) 2 1 n 1 1 nm .01097 2 nm 1 364 .6 2 n 4 4 n • it is a miracle that we can only SEE the Balmer series • the other series are given by 1 1 .01097 2 2 nm 1 n n f i 1 • Lyman: nf = 1 (all UV) • Paschen: nf = 3 (all IR) 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): Ze e mv 2 mv 2 Ze 2 which yields KE 2 40 r r 2 80 r Ze 2 1 we also get v(r ) [I] m 40 r 1 Ze 2 PE • more classical theory: 40 r • assuming a circular orbit of radius r, both PE and KE are constants 1 1 Ze 2 1 Ze 2 1 Ze 2 E PE KE 40 r 80 r 80 r 1 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/2 = 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: 1 speed v 1 f orb Tor circumference 2r 2 Ze 2 m 40 • [Kepler’s third law: (period)2 ~ (radius)3] • so, we equate |E| to ½ nhforb: 11 r r Ze 2 m16 3 0 r 3 2 Ze 2 nhforb 1 Ze 2 nh v Ze 2 E [II] vn 80 r 2 80 r 2 2r 2 0 nh 1 1 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: Ze 2 1 Ze 2 Ze 2 1 Z 2e 4 m 40 r 2 0 nh m 40 r 4 02 n 2 h 2 0 h 2 2 a0 n 2 0h2 rn n : where a0 .053 nm 2 2 Z mZe me • so the orbital radii are quantized… as are the orbital speeds… as are the energies of the orbits! Ze 2 1 Ze 2 Ze 2 Z 2e 4 1 1 2 En Z E 0 2 80 rn 80 0 h 2 n 2 802 h 2 n 2 n where E0 e4 802 h 2 13.6 eV • one can show that angular momentum is quantized: L = nħ • this is equivalent to n de Broglie wavelengths around the orbit circumference The ‘old’ theory of the hydrogen-like atom à la Niels Bohr • electron energies En = – Z2 E0 n– 2 and 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 4 1 1 m e Z RH 2 2 where the Rydberg is RH 2 3 1.097 x 10 2 nm 1 n 8 0 h c f ni 1 2 • 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