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4/15 No. 1

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4/15 No. 3

Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem ε = ω ϕ(r) : Energy eigenvalue : Eigen function 4/15 No. 4

Bare Coulomb potential from the nucleus V(r) = V(r) = Ze 4πε 0 r (Time-independent Schrödinger equation) m ϕ(r) Ze ϕ(r) = εϕ(r) 4πε 0 r cumbersome coefficients Introduction of atomic unit (a.u.) 1 ϕ(r) Z r ϕ(r) = εϕ(r) 4/15 No. 5

Electron Unit system in which = m = e = e =1 4πε 0 Length Energy a 0 = $ ' m& e ) % 4πε 0 ( = 4πε 0 me e 4πε 0 a 0 = 7.1 ev = 5.9 10 11 Bohr radius m (ionization potential of H) 1 ev =1.60 10 19 J Time Velocity 3 $ ' = a 0 αc m& e ) % 4πε 0 ( a 0 a 0 αc = αc = 0.04 fs α = fine structure constant e 4πε 0 c = 7.97 10 3 = 1 137.0 Atomic scale of length, energy, and time 4/15 No. 6

Atomic unit is closely related to Bohr hydrogen atom Dimension Expression Value Meaning length a 5.9 10-11 0 =4 0 /me m Bohr radius energy velocity time electric field laser intensity E h = me4 (4 0 ) = e 4 0a 0 v = e 4 0 = a 0 E h v F = e = c 4 0a 0 1 c 0F 7. ev.19 10 6 m/s 4. attoseconds 5.14 10 11 V/m Coulomb potential energy at the Bohr radius electron orbital velocity time during which the electron proceeds 1 radian field at the Bohr radius laser field = electric 3.51 10 16 W/cm field at the Bohr radius 4/15 No. 7

Bare Coulomb potential from the nucleus V(r) = V(r) = Ze 4πε 0 r = Z r (Time-independent Schrödinger equation) m ϕ(r) Ze ϕ(r) = εϕ(r) 4πε 0 r Polar coordinate 1 ϕ(r) Z r r = (r,θ,φ) ϕ(r) = εϕ(r) Bound state ε < 0 Energy eigenvalue ε n = Z me 4 4πε 0 Eigen function ϕ(r) = R nl (r)y lm (θ,φ) ( ) 1 n = Z n n =1,,3 0 l n 1 l n l Radial wave function Spherical harmonics 4/15 No. 8

Energy eigenvalue ε = Z me 4 n 4πε 0 ( ) 1 n = Z n =1,,3 n Energy (ev) Coulomb potential r in a 0 (Bohr radius ) a 0 = 4πε 0 me = 5.3 10 11 m = 0.053 nm ϕ(r) = R nl (r)y lm (θ,φ) 0 l n 1 l n l Ground state me 4 ε 1 = 4πε 0 ( ) = 13.6 ev 4/15 No. 9

Energy eigenvalue ε n = Z n =1,,3 n Balmer series Lyman series 4/15 No. 10

Z = 1 3/ " R 1s = 1 % $ ' e r / a 0 # a 0 & R s = 1 3/ " % $ ' # a 0 & 1 e r /a 0 " 1 r % $ ' # a 0 & R p = 1 3/ " % 1 $ ' # a 0 & 6 e r /a 0 r a 0 R 3s = 1 3/ " % ) $ ' # a 0 & 3 3 e r / 3a 0 1 r + " r % + $ ' * + 3 a 0 7 # a 0 & Orthonormality (r)r n # l (r) r dr = δ n n # R 0 nl ϕ nlm ϕ n % l % %,. -. Y 1,0 = Y 00 = 3 4π cosθ Y,0 = 1 4π Y 1,±1 = 5 16π 3cos θ 1 ( ) Y,±1 = 15 sinθ cosθ e±iφ 8π 15 Y,± = 3π sin θ e ±iφ Orthonormality Y lm (θ,φ) m r sinθdrdθdφ = δ n % Y l & & n δ l l % δ m m % 3 sinθ e±iφ 8π m (θ,φ)sinθdθdφ = δ l & l δ m m & 4/15 No. 11

Radial wave function Probability density r (atomic unit) 4/15 No. 1

ε > 0 Necessary when ionization is considered ε > 0 Arbitrary positive number ϕ(r) = R εl (r)y lm (θ,φ) l 0 l n l Radial wave function Coulomb wave function R εl (r) = l Z s + % 1 e π n % s=1 wave number k = me / = E n " = Z k ε ε $ R εl (r)r ε $ l (r) r dr = 0 n 0 (kr) l (l +1)! e ikr F(i n % +l +1,l +,ikr) confluent hypergeometric function R εl (r) r dr > 0 Density of states 0 4/15 No. 13

Bound states Continuum states r a 0 ( ) 4/15 No. 14

V(r) = 0 In a free space R El (r) = 1 ϕ(r) = εϕ(r) 1 # d % dr + $ r k π j (kr) l % % r πk d dr l(l +1) & r ( R(r) = εr(r) ' 1 r cos ' kr π * (l +1) () +, Spherical Bessel function 1 Coulomb wave function R El (r) $ $ r πk r cos ( kr + Z k logkr π (l +1) σ + l )*,- rr El (r) l =1 (p-wave) E = 13.6 ev 0.5 0.5 Phase shift σ l = argγ(l +1+iZ /k) 10 0 30 40 50 r Couomb V(r)=0 4/15 No. 15

V(r) = 0 1 ϕ(r) = εϕ(r) 1 # d % dr + r > r $ 0 r R El (r) = k π ( c j (kr)+ c y (kr) j l y l ) d dr Spherical Bessel function l(l +1) & r ( R(r) = εr(r) ' Figure 10.48.1: j n (x),n= 0(1)4, 0 x 1. Figure 10.48.: y n (x),n= 0(1)4, 0 <x 1. j l (kr) y l (kr) """ (kr)l """ 1 r 0 r (l +1)!! kr cos % kr π & ' (l 1)!! """ """ 1 r 0 (kr) l+1 r kr sin % kr π & ' ( (l +1) ) * ( (l +1) ) * Phase shift 4/15 No. 16

i ψ t = 1 ψ(r,t) Z r ψ(r,t)+v I (r,t)ψ(r,t) Interaction i ψ t = (H 0 + H I )ψ(r,t) H 0 = 1 Z r H I = V I (r,t) Without the external field H 0 ϕ n (r) = ε n ϕ n (r) ψ n (r,t) = ϕ n (r)e iω n t With the external field ω n = ε n Eigen state ψ(r,t) = c n ϕ n (r)e iω n t c n = e iω n t ϕ * n (r)ψ(r,t)dv = e iω n t n ψ n H 0 n = ω n n (atomic unit) 4/15 No. 17

i t ψ = (H 0 + H I ) ψ n ψ = c n e iω nt i t n ψ = n H 0 + H I ψ = n H 0 ψ + n H I ψ = ω n n ψ + n H I ψ i c n = n H I ψ e iω nt i c n = m m = I Identity operator can be inserted anywhere n H I m m ψ e iω n t = n H I m c m e i(ω n ω m )t m m m i c n = n H I m c m e i(ω n ω m )t m n H I m Image Transition matrix element Transition from m to n due to the interaction H I m H I n The interaction H I couples m to n. 4/15 No. 18

Resonance frequency ω 0 = ε ε 1 Two-level atom C ε If the laser frequency ω is close to ω 0, only the two levels are relevant. ω 0 C 1 ω 0 ε 1 ψ(r,t) = C 1 (t)ψ 1 (r,t)+c (t)ψ (r,t) 4/15 No. 19

i ψ t = 1 ψ(r,t) Z r ψ(r,t)+v I (r,t)ψ(r,t) C ε ψ(r,t) = C 1 (t)ψ 1 (r,t)+c (t)ψ (r,t) ω 0 ψ(r,t) d 3 r = C 1 (t) + C (t) =1 $ V I ( C 1 ψ 1 +C ψ ) = i C 1 t ψ 1 + C ' & ψ % ) t ( multiply with ψ 1 from the left and take a volume integral ψ 1 C 1 ε 1 i C 1 t = C 1 V 11 +C V 1 e iω 0t V ij = i V I j = ϕ i V I ϕ j d 3 r Similarly i C t = C 1 e iω 0 t V 1 +C V 4/15 No. 0

Dipole approximation is often sufficient. Wave number z k = π E 0 cos(kx ωt) λ Wavelength r x << λ kx <<1 E 0 cos(kx ωt) E 0 cosωt V I = ze 0 cosωt Dipole approximation " x Ze H 0 cos(kx ωt) k y 4/15 No. 1

i C 1 t i C t = C 1 V 11 +C V 1 e iω 0 t = C 1 e iω 0 t V 1 +C V V I = ze 0 cosωt V ij = i V I j = ϕ i V I ϕ j d 3 r = cosωt ze 0 ϕ i ϕ j d 3 r = X ij cosωt X 11 = X = 0 How V I couples the two levels. V I j i V ij = i V I j = ϕ i V I ϕ j d 3 r X 1 = X 1 = γ Real i C 1 t = γc e iω 0 t cosωt i C t = γc 1 e iω 0 t cosωt i C 1 t ( )t [ ] i C = γc e i ( ω ω 0 )t + e i ω+ω 0 t ( )t [ ] = γc 1 e i ( ω+ω 0 )t + e i ω ω 0 4/15 No.

i C 1 t ( )t [ ] i C = γc e i ( ω ω 0 )t + e i ω+ω 0 Rotating wave approximation t ( )t [ ] = γc 1 e i ( ω+ω 0 )t + e i ω ω 0 i C 1 t = γe i ( ω ω 0 )t C i C t Initial condition C 1 =1, C = 0 = γe i ( ω ω 0 )t C 1 ( ) % C 1 (t) = cosωt i ω ω 0 ' & Ω sinωt ( + * exp i ( ) ω ω. 0)t,- / 0 C ε C (t) = iγ Ω sinωt exp & i ( ω ω ) 0)t '( Ω = γ *+ + (ω ω 0) 4 C 1 ω 0 ε 1 4/15 No. 3

ω ω Population 0 = 3.5γ C 1 (t) =1 C (t) C (t) = γ Ω sin Ωt C 1 (t) Ω = γ + (ω ω 0 ) 4 Absorption-emission cycle γ t ω = ω 0 C (t) γ t γ t ω ω 0 = 0.9γ 4/15 No. 4

Dipole interaction can be expressed in either the length or velocity gauge Length gauge velocity gauge i L t = p + V (r)+r E(t) L i V t = (p + A(t)) + V (r) V gauge transformation L = e ir A(t) V vector potential All physical observables are gauge invariant. probability density L = V projection on eigenstate i (or population of eigenstate i) depends on gauge! i Ldr 3 = i V dr 3 Level population (such as C 1 and C ) is meaningful only if or 4/15 No. 5