Physics 451

Physics 451

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 L r p 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 L S S x , S y iS z S y , S z iS x S z , S x iS y 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 S z sm m sm S sm s ( s 1) m(m 1) s m 1 Quantum mechanics The spin 1/2 The spinor Spin up Spin down 1 0

0 1 3 2 S 4 3 2 S 4 3 2 1 0 S 4 0 1 Sz 2

S z 2 1 0 Sz 2 0 1 2 2 2 Quantum mechanics Pauli matrices

3 2 1 0 S 4 0 1 2 0 1 Sx 2 1 0 x 0 i Sy 2 i 0 y Pb 4.29

1 0 Sz 2 0 1 z Quantum mechanics State of the electron a a b b 2 Normalization Expectation values

2 a b 1 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 S 2 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 S S 2S1S 2 2 Triplet Singulet

2 1 2 2 2 2 S 10 2 10 S 2 00 0 Quantum mechanics Adding spins S Possible values for S:

S S1 S 2 , S1 S 2 1 , S1 S 2 2 ,... S1 S 2 sm C m1 m2 m s1s2 s m1m2 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 S Quantum mechanics

The spin and magnetic field S B S H .B Quantum mechanics Larmor precession B z S

y x Pb 4.32 d i H B0 Sz dt S z cos 2 S x sin cos B0t 2 S y sin sin B0t

2 Larmor frequency B0 Quantum mechanics Stern-Gerlach experiment z B S F .B

E B0 z 2 (t ) aeitB0 /2 ei t /2 z ae itB0 /2 e i t /2 z Moves up Moves down

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