Outlines of Quantum Physics

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1 Duality S. Eq Hydrogen Theorems Perturbation Spin Atoms Radiation Outlines of 1 Wave-Particle Duality 2 The Schrödinger Equation 3 The Hydrogen Atom 4 Theorems of Quantum Mechanics 5 The Variation Method and Perturbation Theory 6 Electron Spin and the Pauli Principle 7 Many-Electron Atoms 8 Time-dependent Perturbation Theory, Radiation Invariant Time-Dependent Perturbation Theory Optical Transitions, Selection Rules Line Profiles, Laser Cooling

2 þf žmüz þfåæ üa K NXŒUG ÿþåæþš! K NX žmüz K =üz Kµ i t ψ(t) = Hψ(t) ehøw¹t {φ n} 8 Hφ n = E n φ n ψ(0) = n a n φ n ψ(t) = n a n e ient/ φ n

3 Åðþ åæþa NXHamiltonian H A3? ψ e²þ ŠA = (ψ, Aψ) k d dt d dt A(t) = ( ψ t A(t) = (Hψ i A, Aψ) + (ψ, A ψ ) + (ψ, t t ψ) ψ t = Hψ i, H+ = H A, Aψ) + (ψ, AHψ ) + (ψ, i t ψ) = 1 1 A (ψ, HAψ) + (ψ, AHψ) + i i t = 1 A [A, H] + i t eaøw¹t kehrenfest'x d dt Ā = 1 [A, H] i? Ú/ ek[a, H] = 0 K d dt Ā = 0 åæþa Å ðþ"

4 Åðþ Åðþ åæþžîaøw¹t [A, H] = 0 A Åðþ" 3? e ÅðåÆþ²þŠ!VÇ ÙØ žmuc" 1 5 ½ Vg«O 2 ØC VÇ Ù šnxtåæþ ½ ½~ê ²;åÆÅðþ«O" 3 NXŒUGØ ½ Åðþ Œ± Ü " 4 ŒUÓž3õ Åðþ Ø ½3 kåðþó U Ñ~fíº

5 þf[ 1 H 0 ψ 0 k = E 0 k ψ0 k ψ 0 k ψ0 m = δ mk 2 H = H 0 + H (t) 3 Ψ(x, t) = k c k(t)e ie 0 k t/ ψk 0 4 i Ψ t = (H 0 + H )Ψ 5 i k ċke ie 0 k t/ ψ 0 k + k c ke 0 k e ie 0 k t/ ψ 0 k = k c k(h 0 + H )e ie 0 k t/ ψ 0 k = i k ċke ie 0 k t/ ψ 0 k = k c kh e ie 0 k t/ ψ 0 k 6 i ċ m = k c ke i(e 0 m E 0 k )t/ ψ 0 m H ψ 0 k dτ 7 i ċ m = k c kh mk exp(iω mkt) 5 dvkú\?ûcq 1 'uh 0 ) 2 H 0 Ø¹ž H žmcz 3 tž Å¼ê 4 ¹žSchrödinger 5 \Ψ(x, t) 6 ü> e ie 0 m t/ ψm 0 dτ 7 - H mk ψ0 m H ψk 0 ω mk (Em 0 Ek 0)/

6 þf[ 6?n ƒéu¹žüz '% ƒé{ü ½m[" b½t = 0ž NX?uH 0 ½ψnþ c 0 m (t = 0) = δ mn 6H lt = 0Š^tž ^c m (t) δ mn Úµ i ċ m = k c kh mk exp(iω mkt) éum ne kµ c m (t) 1 t dτh mn exp(iω mn τ) i 0 =tž NX?u [ ψ 0 m þaçò P m n = c m (t) 2 = 1 t 2 dτh mn exp(iω mn τ) 0 2 B+[µXJH mn = 0 KP m n = 0 m[açp m n = P n m U?mQº

7 ¹ž 6 {?n1áâ 1 Œ²; {µf! f þf 1 À ²;² >^Å 2 1Å fº Ñ>^ m Ù À þ!>^ 3 Ä>^Å> Š^ Ñ^ å Ä>ó4Š ^ 4 NX>ó4ÝX = i q ix i HamiltonianþH = X E cos ωt H mn = X mn E cos ωt = 1 2 X mne(e iωt + e iωt ) X mn = ψm 0 i q ix i ψn 0 [>ó4ý Transition Electric Dipole Moment 3x > þ" 5 \c ¹ž 6Š^(J c m (t) = E 2i X mn t 0 [e i(ωmn+ω)t + e i(ωmn ω)t ]dt = E 2i X mn[ ei(ωmn+ω)t 1 + ei(ωmn ω)t 1 i(ω mn + ω) i(ω mn ω) ] 6 Äü m[aç [ªÇω m n Äω ω m n [ Çµw m n = d dt P m n = d dt c m(t) 2

8 ÀJ½K Selection Rules ü>fó4[µd f i = ψ f e r ψ i NXoÄþÅð 1fÄþ 1 Cz éufµψ = φ nlml φ nlml ( r) = ( 1) l φ nlml ( r) Š m üc D f i AØCµ( 1) li +l f +1 = 1 =[c 7LUC A5 e> z z = r cos θ m l = 0 e> x, y x, y = r sin θe iφ m l = ±1 g^øc [Ý g^ã' Ïd[c ^;ÍÜ f>ó4[àj½k UC j = ±1, 0 (0 0) l = ±1, s = 0, m l = 0, ±1 Ag^ØC

9 Fine structures of Helium S S P 8.1GHz 0.66GHz P 0 (1s3p) P P P S 0 20ms 8000s , P , GHz 2.29GHz P 0 (1s2p) P P S 1 (1s2s) 1 1 S 0 (1s 2 )

10 E(cm -1 ) Lowest energy levels of neutral Helium (He I) S 1/2 (He II) P 1 (1s4p) P 1 (1s3p) D 2 (1s3d) S 0 (1s3s) , , , , , D 1 (1s3d) D D P 0 (1s3p) P P S 1 (1s3s) P 1 (1s2p) S 0 (1s2s) 58.43nm, , ms (M1) , s , P 0 (1s2p) P P S 1 (1s2s) S 0 (1s 2 )

11 Ì ÝÚ /

12 Ì \ guëµdn f i = A f i N i dt N i = N i0 e t/τ τ = 1/A Γ = /τ Briet-WignerµP(E) = P(E) = 1 FWHM = Γ g, Lorentzian /" Γ/2π (E E 0) 2 +(Γ/2) 2 Äµτ = -u ÄŒUkõ guëe µa i = Σ k A ik τ i = 1/A i -E\ Γ c þ! Homogeneous \ Doppler Shift: ν = ν 0 (1 + v c ) Doppler\ µγ D = E 0 8 ln 2kT Mc T A E 0 Gaussian /µf ( E) = 2 ln 2 Γ D π exp[ 4 ln 2( E Γ D ) 2 ], šþ!\ Voigt /µgaussian /ÚLorentzian /òè"

13 Ú\ B 12 B 21 A 21 Absorption coefficient E 2 ; g 2 ; N 2 ρ(ω) E 1 ; g 1 ; N 1 κ(ω, I ) = Nσ(ω) 1 + I /I s N 1 N 2 = N 1+I /I s (ω) Saturation intensity I s (ω) = ωa21 2σ(ω), its minimum value hc λ 3 τ I sat I s (ω 0 ) = π 3 (take σ(ω) = 3π2 c 2 ω0 2 Γ 2 /4 = Nσ 0 (ω ω 0 ) 2 + Γ 2 / I = Nσ 0 Γ 2 /4 (ω ω 0 ) Γ2 (1 + I /I sat ) ω FWHM = Γ 1 + I /I sat A 21 g H (ω) ) Γ 2 /4 I sat (ω ω 0) 2 +Γ 2 /4

14 Duality S. Eq Hydrogen Theorems Perturbation Spin Atoms Radiation Invariant TD-Perturbation SelectionRules LaserCooling Saturation Spectroscopy

15 Laser Cooling Scattering Force MOT Magneto-Optical Trap F scatt = k Γ 2 a max = Fmax M I /I sat 1+I /I sat+4 2 /Γ 2 = k Gamma M 2 = v recoil 2τ For Rb atom, λ = 780nm laser, v recoil 0.006m/s, a max m/s 2. F = K r α r

16 The Nobel Prize in Physics 1997 The Royal Swedish Academy of Sciences has awarded the 1997 Nobel Prize in Physics jointly to Steven Chu, Claude Cohen-Tannoudji and William D. Phillips for their developments of methods to cool and trap atoms with laser light.

17 The Nobel Prize in Physics 2001 for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates

18 The Nobel Prize in Physics 2005 for his contribution to the quantum theory of optical coherence for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique Roy J. Glauber Harvard University, Cambridge, MA, USA John L. Hall University of Colorado, JILA; NIST, Boulder, CO, USA Theodor W. Hänsch MPI für Quantenoptik, Garching, Germany

École Normale Supérlieure, Paris, France

19 The Nobel Prize in Physics 2012 for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems Serge Haroche École Normale Supérlieure, Paris, France David J. Wineland NIST, Univ Colorado, Boulder, CO, USA

20 ësnjj SK SN ëö SK Åðþ =Q> 5.1 ¹ž 6Ø =Lv> 9.9 =Q> [ ÀJ½K =Lv> 9.10

The Nobel Prize in Physics 2012

The Nobel Prize in Physics 2012 Serge Haroche Collège de France and École Normale Supérieure, Paris, France David J. Wineland National Institute of Standards and Technology (NIST) and University of Colorado