02-量子力学の復習

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1 4/17 No. 1

2 4/17 No. 2

3 4/17 No. 3

4 Particle of mass m moving in a potential V(r) V(r) m i ψ t = 2 2m 2 ψ(r,t)+v(r)ψ(r,t) ψ(r,t) Wave function ψ(r,t) = ϕ(r)e iωt steady state 2 2m 2 ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem ε = ω ϕ(r) : Energy eigenvalue : Eigen function 4/17 No. 4

5 Bare Coulomb potential from the nucleus V(r) = V(r) = Ze2 4πε 0 r (Time-independent Schrödinger equation) 2 2m 2 ϕ(r) Ze2 ϕ(r) = εϕ(r) 4πε 0 r cumbersome coefficients Introduction of atomic unit (a.u.) ϕ(r) Z r ϕ(r) = εϕ(r) 4/17 No. 5

6 Unit system in which = m = e = Electron e2 4πε 0 =1 Length Energy a 0 = 2 $ ' m& e2 ) % 4πε 0 ( = 4πε 0 2 me 2 e 2 4πε 0 a 0 = ev = Bohr radius m 2 (ionization potential of H) 1 ev = J Time Velocity 3 $ ' = a 0 = fs 2 αc fine structure constant m& e2 ) % 4πε 0 ( e 2 α = 4πε 0 c = = a 0 a 0 αc = αc Atomic scale of length, energy, and time 4/17 No. 6

7 Atomic unit is closely related to Bohr hydrogen atom Dimension Expression Value Meaning length a =4 0 2 /me 2 m Bohr radius energy velocity time electric field laser intensity E h = me4 (4 0 ) 2 = e2 4 0a 0 v = e2 4 0 = a 0 E h v F = e = c 4 0a c 0F ev m/s 24.2 attoseconds V/m W/cm 2 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 field at the Bohr radius 4/17 No. 7

8 Bare Coulomb potential from the nucleus V(r) = V(r) = Ze2 4πε 0 r = Z r (Time-independent Schrödinger equation) 2 2m 2 ϕ(r) Ze2 ϕ(r) = εϕ(r) 4πε 0 r Polar coordinate ϕ(r) Z r r = (r,θ,φ) ϕ(r) = εϕ(r) Bound state ε < 0 Energy eigenvalue ε n = Z 2 me 4 4πε 0 n = Z 2 2 2n 2 n =1,2,3 Eigen function ϕ(r) = R (r)y (θ,φ) nl lm 0 l n 1 l n l Radial wave function Spherical harmonics 4/17 No. 8 ( )

9 Energy eigenvalue ε = Z 2 me 4 n 4πε 0 ( ) n = Z 2 n =1,2,3 2 2n 2 r in a 0 (Bohr radius ) a 0 = 4πε 0 2 me 2 = m = nm Energy (ev) Coulomb potential 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 4/17 No. 9

10 Energy eigenvalue ε n = Z n =1,2,3 2n 2 Balmer series Lyman series 4/17 No. 10

11 Z = 1 3/2 " R 1s = 1 % $ ' 2e r / a 0 # a 0 & R 2s = 1 3/2 " % $ ' # a 0 & 12 e r /2a 0 " 1 r % $ ' # 2a 0 & R 2 p = 1 3/2 " % 1 $ ' # a 0 & 2 6 e r /2a 0 r a 0 R 3s = 1 3/2 " % 2 ) $ ' # a 0 & 3 3 e r / 3a r " r %, + $ '. * + 3 a 0 27 # a 0 & -. Orthonormality (r)r n # l (r) r 2 dr = δ n n # R 0 nl ϕ nlm ϕ n % l % % Y 1,0 = Y 00 = 3 4π cosθ Y 2,0 = 1 4π Y 1,±1 = 5 16π 3cos2 θ 1 ( ) Y 2,±1 = 15 sinθ cosθ e±iφ 8π 15 Y 2,±2 = 32π sin2 θ e ±2iφ Orthonormality Y lm (θ,φ) m r 2 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 & 4/17 No. 11

12 Radial wave function Probability density r (atomic unit) 4/17 No. 12

13 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 2 Z s 2 + % 1 e 2π n % s=1 wave number k = 2mE / = 2E n " = Z k ε ε $ R εl (r)r ε $ l (r) r 2 dr = 0 n 2 0 (2kr) l (2l +1)! e ikr F(i n % +l +1,2l + 2,2ikr) confluent hypergeometric function R εl (r) 2 r 2 dr > 0 Density of states 0 4/17 No. 13

14 V(r) = ϕ(r) = εϕ(r) 1 # d 2 % 2 dr + 2 $ 2 r In a free space R El (r) = 2k π j (kr) l % % r 2 πk d dr l(l +1) & r 2 ( R(r) = εr(r) ' 1 r cos ' kr π * (l +1) () 2 +, Spherical Bessel function 2 1 Coulomb wave function R El (r) $ $ r πk r cos ( kr + Z k log2kr π 2 (l +1) σ + l )*,- rr El (r) l =1 (p-wave) E = 13.6 ev Phase shift σ l = argγ(l +1+iZ /k) r Couomb V(r)=0 4/17 No. 14

15 i ψ t = 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 = Z r H I = V I (r,t) Without the external field ψ n (r,t) = ϕ n (r)e iω n t ω n = ε n H 0 ϕ n (r) = ε n ϕ n (r) With the external field 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) 4/17 No. 15

16 i t ψ = (H 0 + H I ) ψ n ψ = c n e iω n t i t n ψ = n H 0 + H I ψ = n H 0 ψ + n H I ψ = ω n i c n = n H I ψ e iω n t i c n = m m m = I 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 i c n = n H I m c m e i(ω n ω m )t m n ψ + n H I ψ Identity operator can be inserted anywhere 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. 4/17 No. 16

17 Resonance frequency ω 0 = ε 2 ε 1 Two-level atom If the laser frequency w is close to w 0, only the two levels are relevant. w 0 ψ(r,t) = C 1 (t)ψ 1 (r,t)+c 2 (t)ψ 2 (r,t) C 2 2 C 1 2 ω 0 ε 2 ε 1 4/17 No. 17

18 i ψ t = ψ(r,t) Z r ψ(r,t)+v I (r,t)ψ(r,t) C 2 2 ε 2 ψ(r,t) = C 1 (t)ψ 1 (r,t)+c 2 (t)ψ 2 (r,t) ω 0 ψ(r,t) 2 d 3 r = C 1 (t) 2 + C 2 (t) 2 =1 $ V I ( C 1 ψ 1 +C 2 ψ 2 ) = i C 1 t ψ + C ' 2 & 1 ψ % 2 ) t ( multiply with ψ 1 from the left and take a volume integral ψ 1 C 1 2 ε 1 i C 1 t = C 1 V 11 +C 2 V 12 e iω 0 t V ij = i V I j = ϕ i V I ϕ j d 3 r Similarly i C 2 t = C 1 e iω 0 t V 21 +C 2 V 22 4/17 No. 18

19 Dipole approximation is often sufficient. Wave number x k = 2π λ Wavelength E x (z,t) =E 0 cos(!t E 0 cos(!t x << λ kx <<1 kz) E 0 cosωt V I = r E = xe 0 cos(!t) Dipole approximation y Ze r k z B y (z,t) = E 0 c cos(!t kz) kz) 4/17 No. 19

20 i C 1 t i C 2 t = C 1 V 11 +C 2 V 12 e iω 0 t = C 1 e iω 0 t V 21 +C 2 V 22 V ij = i V I j = ϕ i V I ϕ j d 3 r = cosωt xze 0 ϕ i ϕ j d 3 r = X ij cosωt X 11 = X 22 = 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 12 = X 21 = 2γ Real V I = xe 0 cos(!t) i C 1 t = 2γC 2 e iω 0 t cosωt i C 2 t = 2γC 1 e iω 0 t cosωt i C 1 t ( )t [ ] i C 2 = γc 2 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 4/17 No. 20

21 i C 1 t ( )t [ ] i C 2 = γc 2 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 2 i C 2 t Initial condition C 1 =1, C 2 = 0 = γe i ( ω ω 0 )t C 1 ( ) % C 1 (t) = cosωt i ω ω 0 ' & 2Ω sinωt ( + * exp i ( ) 2 ω ω. 0)t,- / 0 C 2 2 ε 2 C 2 (t) = iγ Ω sinωt exp & i ( 2 ω ω ) 0)t '( Ω = γ *+ 2 + (ω ω 0 )2 4 C 1 2 ω 0 ε 1 4/17 No. 21

22 Population ω ω 0 = 3.5γ C 1 (t) 2 =1 C 2 (t) 2 C 2 (t) 2 = γ 2 Ω 2 sin2 Ωt C 1 (t) 2 Ω = γ 2 + (ω ω 0 )2 4 Absorption-emission cycle γ t ω = ω 0 C 2 (t) 2 γ t γ t ω ω 0 = 0.92γ 4/17 No. 22

23 Dipole interaction can be expressed in either the length or velocity gauge Length gauge velocity gauge i L t = p2 2 + V (r)+r E(t) L i V t = (p + A(t))2 2 + V (r) V gauge transformation L = e ir A(t) V vector potential All physical observables are gauge invariant. probability density L 2 = V 2 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 2 ) is meaningful only if or after the pulse 4/17 No. 23

24 4/17 No. 24

25 Bound states Continuum states r a 0 ( ) 4/17 No. 25

26 V(r) = 0 r > r ϕ(r) = εϕ(r) 1 # d 2 % 2 dr + 2 $ 2 r d dr l(l +1) & r 2 ( R(r) = εr(r) ' R El (r) = 2k π ( c j (kr)+ c y (kr) j l y l ) Spherical Bessel function Figure : j n (x),n= 0(1)4, 0 x 12. Figure : y n (x),n= 0(1)4, 0 <x 12. j l (kr) y l (kr) """ (kr)l """ 1 r 0 r (2l +1)!! kr cos % kr π & ' 2 (2l 1)!! """ """ 1 r 0 (kr) l+1 r kr sin % kr π & ' 2 ( (l +1) ) * ( (l +1) ) * Phase shift 4/17 No. 26

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