Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)

Advanced Laser and Photon Science レーザー・光量子科学特論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]

4/17 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

4/17 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

4/17 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

ψ (r,t) = ϕ (r)e−i€ωt 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

€ € 4/17 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

€ 4/17 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 4/17 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

4/17 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