Today Monday, March 21, 2005 Event: The Elliott W. Montroll Lecture Speaker: Prof. David Gross, University of California, Santa Barbara Title: The future of physics (25 questions that might guide physics in the next 25 years) Talk:3:45 pm, Hoyt Hall Tea:3:15 pm, B&L Foyer  Recipient of the 2004 Nobel Prize In Physics 1/18/2019 Lecture XVII

Hydrogen Atom PHY123 1/18/2019 Lecture XVII

Concepts Electron distributions Quantum numbers, quantum state Zeeman effect 1/18/2019 Lecture XVII

Schrödinger equation for Hydrogen atom Potential energy – electron is in Coulomb’s potential of the nucleus: Spherically symmetric potential 3-D Schrödinger's equation 1/18/2019 Lecture XVII

Hydrogen atom Energy levels in H n- principle quantum number – determines the energy level What about electron distribution in atom? 1/18/2019 Lecture XVII

Electron distributions Same n, different shapes. Other quantum numbers play a role Quantum state of electron in atom is defined by a set of 4 numbers Principle quantum number n Orbital quantum number l Magnetic quantum number ml (same as lz) Spin projection ms Wave functions depend on 3 quantum numbers 1/18/2019 Lecture XVII

Electron quantum state Principle quantum number n=1,2,3,4,… determines energy level, higher E for higher n Orbital quantum number l For each n l can be 0,1,2,3, …(n-1) l states are leveled by letters s: l=0; p: l=1; d: l=2; f: l=3; g:l=4 E.g. n=5, then l can be 0, 1, 2, 3, 4 Possible l states are s,p,d,f,g n=1, only l=0 s-state is possible 1/18/2019 Lecture XVII

Electron quantum state Orbital quantum number is a vector length l Orbital angular momentum: Its projection on z axis is Lz =mlh another q.n. – magnetic quantum number ml ml can be only integer z 1/18/2019 Lecture XVII

Zeeman effect Normally energy does not depend on ml, but under magnetic field energy levels split- fine structure Magnetic moment (dipole) associated with orbital angular momentum Potential energy of the magnetic dipole in magnetic field 1/18/2019 Lecture XVII

Electron quantum state z All electrons have spin=1/2 It is a vector Its projection on z axis is another q.n. – spin ms ms can be only 1/18/2019 Lecture XVII

Ground state wave function n=1l=0ml=0y100 spin could be up or down No “preferred direction”  system is spherically symmetric  expect wave function to depend only on r Not true if l≠0 n=10 knots Wave function Bohr’s radius: 1/18/2019 Lecture XVII

Probability to find e at r dV=4pr2dr dP=|y|2dV= |y|24pr2dr=Prdr Most probable radius – where Pr has maximum 1/18/2019 Lecture XVII

n=2 wave function n=2l=0,1 First consider y200 Still no “preferred direction”  system is spherically symmetric  expect wave function to depend only on r n=21 knots Wave function 1/18/2019 Lecture XVII

Probability to find e at r for n=2 Most probable radius – where Pr has global maximum 1/18/2019 Lecture XVII

n=2 wave function n=2l=0,1 Now consider y21m There is a “preferred direction” – direction of vector L system is NOT spherically symmetric, but axial symmetry is there  expect wave function to depend not only on r 1/18/2019 Lecture XVII