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Modern Physics Ch.7: H atom in Wave mechanics Methods of Math. Physics, 27 Jan 2011, E.J. Zita • Schrödinger Eqn. in spherical coordinates • H atom wave functions and radial probability densities • L and probability densities • Spin • Energy levels, Zeeman effect • Fine structure, Bohr magneton Recall the energy and momentum operators E hc pc p h From deBroglie wavelength, construct a differential operator for momentum: h 2 k 2 h 2 p k i x 2 h Similarly, from uncertainty principle, construct energy operator: Et E i t Energy conservation Schrödinger eqn. E=T+V Ey = Ty + Vy where y is the wavefunction and operators depend on x, t, and momentum: p̂ i x E i t y y i U y 2 t 2m x 2 2 Solve the Schroedinger eqn. to find the wavefunction, and you know *everything* possible about your QM system. Schrödinger Eqn We saw that quantum mechanical systems can be described by wave functions Ψ. A general wave equation takes the form: Ψ(x,t) = A[cos(kx-ωt) + i sin(kx-ωt)] = e i(kx-ωt) Substitute this into the Schrodinger equation to see if it satisfies energy conservation. Derivation of Schrödinger Equation i Wave function and probability Probability that a measurement of the system will yield a result between x1 and x2 is x 2 2 y ( x, t ) dx x1 Measurement collapses the wave function •This does not mean that the system was at X before the measurement - it is not meaningful to say it was localized at all before the measurement. •Immediately after the measurement, the system is still at X. •Time-dependent Schrödinger eqn describes evolution of y after a measurement. Exercises in probability: qualitative Uncertainty and expectation values Standard deviation s can be found from the deviation from the average: j j j But the average deviation vanishes: j 0 So calculate the average of the square of the deviation: s 2 j Last quarter we saw that we can calculate s more easily by: s 2 j2 j 2 2 Expectation values Most likely outcome of a measurement of position, for a system (or particle) in state y: x x y ( x, t ) 2 dx Most likely outcome of a measurement of position, for a system (or particle) in state y: d x p m i dt * y y x dx Uncertainty principle Position is well-defined for a pulse with ill-defined wavelength. Spread in position measurements = sx Momentum is well-defined for a wave with precise . By its nature, a wave is not localized in space. Spread in momentum measurements = sp We saw last quarter that s xs p Applications of Quantum mechanics Blackbody radiation: resolve ultraviolet catastrophe, measure star temperatures Photoelectric effect: particle detectors and signal amplifiers Bohr atom: predict and understand H-like spectra and energies Structure and behavior of solids, including semiconductors Scanning tunneling microscope Zeeman effect: measure magnetic fields of stars from light Electron spin: Pauli exclusion principle Lasers, NMR, nuclear and particle physics, and much more... Stationary states 2 Y ( x, t ) 2Y ( x, t ) i V ( x, t ) Y ( x , t ) 2 t 2m x If an evolving wavefunction Y(x,t) = y(x) f(t) can be “separated”, then the time-dependent term satisfies i 1 df i E f dt Everyone solve for f(t)= Separable solutions are stationary states... Separable solutions: Y ( x, t ) y ( x ) (1) are stationary states, because * probability density is independent of time [2.7] * therefore, expectation values do not change 2 2 (2) have definite total energy, since the Hamiltonian is sharply localized: [2.13] s 2 0 H (3) yi = eigenfunctions corresponding to each allowed energy eigenvalue Ei. i En t Y ( x, t ) cny n e General solution to SE is [2.14] n 1 Show that stationary states are separable: Guess that SE has separable solutions Y(x,t) = y(x) f(t) Y t 2Y 2 x sub into SE=Schrodinger Eqn Divide by y(x) f(t) : Y i t 2 Y 2 V Y 2 x LHS(t) = RHS(x) = constant=E. Now solve each side: You already found solution to LHS: f(t)=_________ 2 d 2y Vy Ey 2 2m dx RHS solution depends on the form of the potential V(x). Now solve for y(x) for various U(x) Strategy: * draw a diagram * write down boundary conditions (BC) * think about what form of y(x) will fit the potential * find the wavenumbers kn=2 / * find the allowed energies En * sub k into y(x) and normalize to find the amplitude A * Now you know everything about a QM system in this potential, and you can calculate for any expectation value Infinite square well: V(0<x<L) = 0, V= outside What is probability of finding particle outside? 2 Inside: SE becomes d 2y Ey 2m dx 2 * Solve this simple diffeq, using E=p2/2m, * y(x) =A sin kx + B cos kx: apply BC to find A and B * Draw wavefunctions, find wavenumbers: kn L= n * find the allowed energies: (n ) 2 2 2 En ,A 2 * sub k into y(x) and normalize: 2mL L * Finally, the wavefunction is 2 n y n ( x) sin x L L Square well: homework Repeat the process above, but center the infinite square well of width L about the point x=0. Preview: discuss similarities and differences Infinite square well application: Ex.6-2 Electron in a wire (p.256) Summary: • Time-independent Schrodinger equation has stationary states y(x) • k, y(x), and E depend on V(x) (shape & BC) • wavefunctions oscillate as eiwt • wavefunctions can spill out of potential wells and tunnel through barriers That was mostly review from last quarter. Moving on to the H atom in terms of Schrödinger’s wave equation… Review energy and momentum operators p̂ i x E i t Apply to the Schrödinger eqn: EY(x,t) = T Y(x,t) + V Y(x,t) y y i Vy 2 t 2m x 2 Find the wavefunction for a given potential V(x) 2 Y ( x, t ) cny n e n 1 i En t Expectation values f y * f y dx Most likely outcome of a measurement of position, for a system (or particle) in state yx,t: x x y ( x, t ) dx 2 where y y *y 2 Order matters for operators like momentum – differentiate y(x,t): d x p m i dt * y y x dx H-atom: quantization of energy for V= - kZe2/r Solve the radial part of the spherical Schrödinger equation (next quarter): Do these energy values look familiar? Continuing Modern Physics Ch.7: H atom in Wave mechanics Methods of Math. Physics, 10 Feb. 2011, E.J. Zita • Schrödinger Eqn. in spherical coordinates • H atom wave functions and radial probability densities • L and probability densities • Spin • Energy levels, Zeeman effect • Fine structure, Bohr magneton Spherical harmonics solve spherical Schrödinger equation for any V(r) You showed that Y210 and Y200 satisfy Schrödinger’s equation. H-atom: wavefunctions Y(r,q,f) for V= - kZe2/r R(r) ~ Laguerre Polynomials, and the angular parts of the wavefunctions for any radial potential in the spherical Schrödinger equation are Y (r ,q , ) R(r )Ylm (q , ) where Ylm (q , ) spherical harmonics Radial probability density P(r ) r Rn, (r ) 2 2 Look at Fig.7.4. Predict the probability (without calculating) that the electron in the (n,l) = (2,0) state is found inside the Bohr radius. Then calculate it – Ex. 7.3. HW: 11-14 (p.233) H-atom wavefunctions ↔ electron probability distributions: l = angular momentum wavenumber Discussion: compare Bohr model to Schrödinger model for H atom. ml denotes possible orientations of L and Lz (l=2) Wave-mechanics L ≠ Bohr’s n HW: Draw this for l=1, l=3 QM H-atom energy levels: degeneracy for states with different Y and same energy Selections rules for allowed transitions: n = anything (changes in energy level) l must change by one, since energy hops are mediated by a photon of spin-one: l = ±1 m = ±1 or 0 (orientation) DO #21, HW #23 Stern-Gerlach showed line splitting, even when l=0. Why? l = 1, m=0,±1 ✓ l = 0, m=0 !? Normal Zeeman effect “Anomalous” A fourth quantum number: intrinsic spin Since L l (l 1), let S s(s 1) If there are 2s+1 possible values of ms, and only 2 orientations of ms = z-component of s (Pauli), What values can s and ms have? HW #24 Wavelengths due to energy shifts E hc dE _________ d ________ E Spinning particles shift energies in B fields Cyclotron frequency: An electron moving with speed v perpendicular to an external magnetic field feels a Lorentz force: F=ma (solve for w=v/r) Solve for Bohr magneton… Magnetic moments shift energies in B fields Spin S and orbit L couple to total angular momentum J = L+ S Spin-orbit coupling: spin of e- in magnetic field of p Fine-structure splitting (e.g. 21-cm line) (Interaction of nuclear spin with electron spin (in an atom) → Hyper-fine splitting) Total J + external magnetic field → Zeeman effect Total J + external magnetic field → Zeeman effect Total J + external magnetic field → Zeeman effect History of atomic models: • Thomson discovered electron, invented plum-pudding model • Rutherford observed nuclear scattering, invented orbital atom • Bohr quantized angular momentum, improved H atom model. • Bohr model explained observed H spectra, derived En = E/n2 and phenomenological Rydberg constant • Quantum numbers n, l, ml (Zeeman effect) • Solution to Schrodinger equation shows that En = E/l(l+1) • Pauli proposed spin (ms= ±1/2), and Dirac derived it • Fine-structure splitting reveals spin quantum number