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Physics 451 Quantum mechanics I Fall 2012 Nov 16, 2012 Karine Chesnel Quantum mechanics Announcements Homework this week: • HW #20 Thursday Nov 17 by 10pm Pb 4.27, 4.28, 4.29, 4.33, 4.34, 4.35 Practice Test 3 review Mon Nov 19 – Tu Nov 20 Test 3 : Monday Nov 26 – Friday Nov 30 Quantum mechanics The Spin Types of angular momentum Lrp orbital L I spin Quantum mechanics Agular moment in the atom • Orbital moment (l) • Spin moment (s) Representation of nlm r , , Quantum mechanics Spin in elementary particles Each elementary particle is characterized by an immutable spin S • Fermions: (S half-integer) S=1/2 Leptons: electrons,… Quarks: u,b,c,s,t,b Proton, neutron • Bosons: (S integer) Photon Mesons S=1 Quantum mechanics The spin L̂ Sˆ S x , S y i S z S y , S z i S x Sz , Sx i Sy S 2 , S x S 2 , S y S 2 , S z 0 Quantum mechanics The spin S 2 sm 2 s(s 1) sm Sz sm m sm S sm s ( s 1) m(m 1) s m 1 Quantum mechanics The spin 1/2 The “spinor” 3 S 4 2 Sz 2 2 Spin up Spin down 1 0 0 1 3 S 4 2 2 Sz 2 3 S 4 2 2 1 0 0 1 1 0 Sz 2 0 1 Quantum mechanics Pauli matrices 3 S 4 2 2 1 0 0 1 0 1 Sx 2 1 0 x 0 i Sy 2i 0 y Pb 4.29 1 0 Sz 2 0 1 z Quantum mechanics State of the electron a a b b a b 1 2 Normalization Expectation values 2 Sx Sx Pb 4.27, 4.28, 4.29 etc… Quantum mechanics Quiz 29a When measuring Sx on an electron, what are the possible results? A. 0 B. C. /2 D. / 2 E. /3 Quantum mechanics Adding spins S S1 S2 1 2 S z (m1 m2 ) m : m 1 Possibilities for two spins 1/2 : m 0 : m 0 : m 1 Quantum mechanics Adding spins 1/2 Triplet possibilities : m 1 : m 0 : m 0 : m 1 11 1 10 2 1 1 Singlet 1 00 2 Quantum mechanics Adding spins 1/2 Magnitude of the spin momentum ? S 2 S12 S22 2S1S2 Triplet Singulet S 2 10 2 S 2 00 0 2 10 Quantum mechanics Adding spins S Possible values for S: S S1 S2 , S1 S2 1 , S1 S2 2 ,... S1 S2 sm s1s2 s m1m2 m C m1 m2 m s1m1 s2 m2 Clebsch- Gordan coefficients Pb 4.34, 4.35 Quantum mechanics Clebsch- Gordan coefficients Quantum mechanics Quiz 29b What is the value of the Clebsch-Gordan coefficient A. 3 / 10 B. 8 / 15 C. 6 / 15 D. 1 / 15 E. 3/ 5 C 213 10 1? Quantum mechanics The spin and magnetic field Quantum mechanics The spin and magnetic field S B S H .B Quantum mechanics Larmor precession B z S Sz y x Pb 4.32 d i H B0 Sˆz dt Sx 2 2 cos sin cos B0t S y sin sin B0t 2 Larmor frequency B0 Quantum mechanics Stern-Gerlach experiment z B S F .B (t ) ae i tB0 /2 E e i t /2 z Moves up B0 z ae 2 i tB0 /2 e i t /2 z Moves down