Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Advanced Plasma and Laser Science プラズマ・レーザー特論E
Quick review of quantum mechanics 量子力学の復習 Takeshi Sato and Kenichi Ishikawa http://ishiken.free.fr/english/lecture.html
[email protected] [email protected] 4/19 No. 1
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Hydrogen atom 水素原子の波動関数 Atomic unit 原子単位 Rabi oscillation ラビ振動
4/19 No. 2
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Hydrogen-like atom 水素原子の波動関数
4/19 No. 3
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Schrödinger equation シュレーディンガー方程式 Particle of mass m moving in a potential V(r) ポテンシャルV(r)中の質量 m の電子 ∂ψ 2 2 i =− ∇ ψ (r,t) +V (r)ψ (r,t) ∂t 2m
ψ (r,t) ：Wave function ：波動関数
ωt ψ (r,t) = ϕ (r)e−i€ steady state 定常状態
€
2 2 − ∇ ϕ (r) +V (r)ϕ (r) = εϕ (r) Eigenvalue problem 固有値問題 2m € ε = ω : Energy eigenvalue エネルギー固有値（エネルギー準位）
ϕ (r)
€
: Eigen function 固有波動関数
€
€ 4/19 No. 4
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Hydrogen-like atom 水素様原子 Bare Coulomb potential from the nucleus Ze2 原子核のクーロンポテンシャル V (r) = V (r) = − 4πε0 r (Time-independent Schrödinger equation) シュレーディンガー方程式
2 2 Ze2 − ∇€ ϕ (r) − ϕ (r) = εϕ (r) 2m 4πε0 r cumbersome coefficients 係数が煩雑 €Introduction of atomic unit (a.u.) 原子単位の導入 1 Z − ∇ 2ϕ (r) − ϕ (r) = εϕ (r) 2 r
€ 4/19 No. 5
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Atomic unit 原子単位 Electron 電子 e2 Unit system in which = m = e = = 1 となるような単位系 4πε0 2 2 4 πε −11 0 Length a0 = = = 5.292 ×10 m 2 2 ' $ me e 長さ m & ) € % 4πε0 (
Energy エネルギー € Time 時間 €
Velocity 速度 €
e2 = 27.21 eV 4πε0 a0
3 2
$ e2 ' m& ) % 4πε0 ( a0 ÷
=
a0 = αc αc €
Bohr radius ボーア半径
2 (ionization potential of H) 1 eV = 1.602 ×10−19 J
a0 = 0.0242 fs αc fine structure constant € e2 1 微細構造 α= = 7.297 ×10−3 = 4πε0 c 137.0 定数
Atomic scale of length, energy, and time 4/19 No. 6
€
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Atomic unit is closely related to Bohr hydrogen atom Dimension length energy velocity time electric field laser intensity
Expression
a0 = 4 Eh =
0
me4 (4
v=
)2
0
e2 4
2
=
e2 4
=c
0
e 4
5.29 10-11 m
/me2
a0 = Eh v
F =
Value
2 0 a0
1 c 0F 2 2
0 a0
27.2 eV 2.19 106 m/s 24.2 attoseconds 5.14 1011 V/m 3.51 1016 W/cm2
Meaning Bohr radius 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
4/19 No. 7
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Hydrogen-like atom 水素様原子 Bare Coulomb potential from the nucleus Ze2 Z 原子核のクーロンポテンシャル V (r) = V (r) = − =− 4πε0 r r (Time-independent Schrödinger equation) シュレーディンガー方程式 2 2 Ze2 − ∇ ϕ (r) − € ϕ (r) = εϕ (r) 2m 4πε0 r
Polar coordinate 極座標系 €
Bound state 束縛状態
ε r0 短距離ポテンシャル
V (r) = 0 r > r0 €
10.48 REl (r) =
€
1 # d 2 2 d l(l +1)& − % 2+ − 2 (R(r) = εR(r) 2 $dr r dr r '
1 − ∇ 2ϕ (r) = εϕ (r) 2 2k c j jl (kr) + cy yl (kr)) ( π
Spherical Bessel function
Graphs
Figure 10.48.1: jn (x), n = 0(1)4, 0
x
263
€
12.
Figure 10.48.2: yn (x), n = 0(1)4, 0 < x
12.
% ( (kr)l 1 π jl (kr) "r→0 "" → "r→∞ "" → cos'kr − (l +1)* & ) (2l +1)!! kr 2 % ( (2l −1)!! 1 π yl (kr) "r→0 "" →− " "" → sin kr − (l +1) *) r→∞ (kr)l+1 kr '& 2
Phase shift 位相シフト（位相のずれ） Figure 10.48.3: j5 (x), y5 (x),
j25 (x) + y52 (x), 0
x
12.
⇥ Figure 10.48.4: j5 (x), y5 (x), j5 2 (x) + y5 2 (x), 0 12.
x 4/19 No. 16
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Temporal evolution by an external field 外場との相互作用による時間発展 i
∂ψ 1 Z = − ∇ 2 ψ (r,t) − ψ (r,t) +VI (r,t)ψ (r,t) ∂t 2 r 相互作用 Interaction
€ i
∂ψ = (H 0 + H I )ψ (r,t) ∂t
1 Z H 0 = − ∇2 − 2 r
H I = VI (r,t)
Without the external field 相互作用項がない場合 € = εn ω n € With the external field 相互作用項がある場合 −iω n t ψ€ (r,t) = ϕ (r)e n n
H 0ϕ n (r) = ε nϕ n (r)
€
ψ (r,t) = ∑€cnϕ n (r)e−iω n t n
H0 n = ωn n
€
€ €
Eigen state 固有状態
iω n t * iω n t cn = e€ ϕ (r) ψ (r,t)dV = e nψ ∫ n
(atomic unit) 4/19 No. 17
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
i
∂ ψ = (H 0 + H I ) ψ ∂t
n ψ = cn e−iω n t
∂ n ψ = n H0 + H I ψ = n H0 ψ + n H I ψ = ωn n ψ + n H I ψ ∂t € operator 単位演算子 ic˙n = n H I ψ eiω n t ∑ m m = I Identity can be inserted anywhere m
i
€ € €
ic˙n = ∑ n H I m m ψ eiω n t = ∑ n H I m cm ei(ω n −ω m )t m
m
€ ic˙n = ∑ n H I m cm ei (ω n −ω m )t m
€ n HI m €
Image イメージ €
Transition matrix element 遷移行列要素 Transition from m to n due to the interaction HI 状態 m が相互作用HIによって状態 n に遷移する The interaction HI couples m to n. 4/19 No. 18
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Important example: Rabi oscillation 重要な例：ラビ振動 Resonance frequency 遷移振動数（共鳴振動数） ２準位系
ω0 = ε2 − ε1 C2
Two-level atom２準位系 € If the laser frequency ω is close to ω0, only the two levels are relevant. € 光の振動数がω0に近いときは、放 射過程に関与するのは選ばれた二 つの原子状態のみ。
ψ (r,t) = C1 (t)ψ1 (r,t) + C2 (t)ψ2 (r,t)
€
2
ε2 ω0
C1
2
€
ε1
€ €
€ 4/19 No. 19
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
i
∂ψ 1 Z = − ∇ 2 ψ (r,t) − ψ (r,t) +VI (r,t)ψ (r,t) ∂t 2 r
C2
2
ε2
ψ (r,t) = C1 (t)ψ1 (r,t) + C2 (t)ψ2 (r,t) €
∫
€
ω0
2 2 2 ψ (r,t) d 3r = C1 (t) + C2 (t) = 1 €
C1
€
2
ε1
€ $ ∂C ' ∂C VI (C1ψ1 + C2 ψ2 ) = i& 1 ψ1 + 2 ψ2 ) % ∂t ( ∂t € multiply with ψ1∗ from the left € and take a volume integral € ψ1∗ を左からかけて空間積分
€
i
∂C1 = C1V11 + C2V12 e−iω 0 t € ∂t
Vij = i VI j =
∗ i I
3
∫ϕ Vϕ d r j
€ ∂C Similarly i 2 = C1eiω 0 tV21 + C2V22 ∂t 同様に € €
€
4/19 No. 20
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Interaction Hamiltonian 相互作用ハミルトニアン Complete Hamiltonian for the interaction of an atom with an electromagnetic field is rather complicated. 電磁場と原子 の間の相互作用に対するハミルトニアンの完全な形は複雑 Dipole approximation is often sufficient. レーザーに関しては、多くの場合、 電気双極子近似で十分 z E cos(kx − ωt) Wave number k = 2π 0 波数 λ Wavelength 波長 r € x
