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 From deBroglie wavelength, construct a differential operator for momentum: Similarly, from uncertainty principle, construct energy operator:

Energy conservation  Schrödinger eqn. E = T + V E  = T  + V  where  is the wavefunction and operators depend on x, t, and momentum: 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

Wave function and probability Probability that a measurement of the system will yield a result between x1 and x2 is

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  after a measurement.

Exercises in probability: qualitative

Uncertainty and expectation values Standard deviation  can be found from the deviation from the average: But the average deviation vanishes: So calculate the average of the square of the deviation: Last quarter we saw that we can calculate  more easily by:

Expectation values Most likely outcome of a measurement of position, for a system (or particle) in state  :

Uncertainty principle Position is well-defined for a pulse with ill-defined wavelength. Spread in position measurements =  x Momentum is well-defined for a wave with precise . By its nature, a wave is not localized in space. Spread in momentum measurements =  p We saw last quarter that

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 If an evolving wavefunction  (x,t) =  (x) f(t) can be “separated”, then the time-dependent term satisfies Everyone solve for f(t)= Separable solutions are stationary states...

Separable solutions: (1) are stationary states, because * probability density is independent of time [2.7] * therefore, expectation values do not change (2) have definite total energy, since the Hamiltonian is sharply localized: [2.13] (3)  i = eigenfunctions corresponding to each allowed energy eigenvalue E i. General solution to SE is [2.14]

Show that stationary states are separable: Guess that SE has separable solutions  (x,t) =  (x) f(t) sub into SE=Schrodinger Eqn Divide by  (x) f(t) : LHS(t) = RHS(x) = constant=E. Now solve each side: You already found solution to LHS: f(t)=_________ RHS solution depends on the form of the potential V(x).

Now solve for  (x) for various U(x) Strategy: * draw a diagram * write down boundary conditions (BC) * think about what form of  (x) will fit the potential * find the wavenumbers k n =2  * find the allowed energies E n * sub k into  (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? Inside: SE becomes * Solve this simple diffeq, using E=p 2 /2m, *  (x) =A sin kx + B cos kx: apply BC to find A and B * Draw wavefunctions, find wavenumbers: k n L=  n  * find the allowed energies: * sub k into  (x) and normalize: * Finally, the wavefunction is

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  (x) k,  (x), and E depend on V(x) (shape & BC) wavefunctions oscillate as e i  t 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 Apply to the Schrödinger eqn: E  (x,t) = T  (x,t) + V  (x,t) Find the wavefunction for a given potential V(x)

Expectation values Most likely outcome of a measurement of position, for a system (or particle) in state  x,t  : Order matters for operators like momentum – differentiate  (x,t):

H-atom: quantization of energy for V= - kZe 2 /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  210 and  200 satisfy Schrödinger’s equation.

H-atom: wavefunctions  (r,  ) for V= - kZe 2 /r R(r) ~ Laguerre Polynomials, and the angular parts of the wavefunctions for any radial potential in the spherical Schrödinger equation are

Radial probability density 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 HW: (p.233)

H-atom wavefunctions ↔ electron probability distributions: l = angular momentum wavenumber Discussion: compare Bohr model to Schrödinger model for H atom.

m l denotes possible orientations of L and L z (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  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 If there are 2s+1 possible values of m s, and only 2 orientations of m s = z-component of s (Pauli), What values can s and m s have? HW #24

Wavelengths due to energy shifts

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  =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

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 E n = E/n 2 and phenomenological Rydberg constant Quantum numbers n, l, m l (Zeeman effect) Solution to Schrodinger equation shows that E n = E/l(l+1) Pauli proposed spin (m s = ±1/2), and Dirac derived it Fine-structure splitting reveals spin quantum number