Ch. 6. Quantum Theory of the H Atom Ch. 6. Quantum Theory of the H Atom
… Schrödinger gave a talk on de Broglie’s notion that a moving particle has a wave character. A colleague remarked to him afterward that to deal properly with a wave, one
needs a wave equation. Schrödinger took this to heart, and a few weeks later he was “struggling with a new atomic
theory. If only I knew more mathematics! I am very
optimistic about this thing and expect that if I can only … solve it, it will be very beautiful.”
The struggle was successful, and in January 1926 the first of four papers on “Quantization as an Eigenvalue Problem”
was completed. In this epochal paper Schrödinger
introduced the equation that bears his name and solved it for
Spherical Coordinates
Spherical Coordinates
2
2
( ) ( ) ( ) ( ) 2
( ) ( )
( ) ( , , ) ( ) ( ) ( )
U E
m
U U r
r R r
ψ ψ ψ
ψ ψ θ φ θ φ
− ∇ + =
=
= = Θ Φ
r r r r
r
r
=
일 경우
변수분리
Time-independent 3D Schrödinger Equation
When Potential Depends on r Only
When Potential Depends on r Only
[ ]
2 2
2 2 2 2
2
2 2 2
2
1
2 1 2
sin cot ( )
l
d d
r d R dR d d mr
E U r
R dr r dr d d
m
φ
θ θ
θ θ
Φ Φ
⎧ ⎛ ⎞ ⎛ Θ Θ ⎞ ⎫
⎪ ⎪
= − ⎨ ⎪ ⎩ ⎜ ⎝ + ⎟ ⎠ + Θ ⎜ ⎝ + ⎟ ⎠ + − ⎬ ⎪ ⎭
= −
=
Magnetic quantum number
m l
When Potential Depends on r Only
When Potential Depends on r Only
2
2
2
( )
( ) 1
2
, 2, 1, 0,1, 2,
l
l
im
l
d m
d
e m
φ
φ φ
φ π
Φ = − Φ
Φ =
= ⋅⋅⋅ − − ⋅⋅⋅
When Potential Depends on r Only
When Potential Depends on r Only
[ ]
22 2 2 2
2 2 2 2
2 2 1
( ) cot
sin ( 1)
m
lr d R dR mr d d
E U r
R dr r dr d d
l l
θ θ θ θ
⎛ ⎞ ⎛ Θ Θ ⎞
+ + − = − + +
⎜ ⎟ Θ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= +
=
Orbital quantum number
l
When Potential Depends on r Only
When Potential Depends on r Only
2
2 2
2
cot
lcsc ( ) ( 1) ( )
d d
m l l
d θ d θ θ θ
θ θ
Θ Θ
+ − Θ = − + Θ
0,1, 2,
l
l
l m
= ⋅⋅⋅
≥
Associated Legendre polynomials
When Potential Depends on r Only
When Potential Depends on r Only
( ) ( ) θ φ Y
lml( , ) θ φ
Θ Φ =
Spherical harmonics
2 2 2
2 2
2 ( 1)
( ) ( ) ( ) ( )
2 2
d R dR l l
R r U r R r ER r
m dr r dr mr
⎛ ⎞ +
− ⎜ + ⎟ + + =
⎝ ⎠
= =
For
2
0
( ) ,
4 U r Ze
πε r
= −
Radial wave equation
2 2 4 2 2
2 2 2 2 2 2
0 0
2 2 0
13.6eV , 1, 2, 3,
8 32
Bohr radius 4
n
o
o
e Z me Z Z
E n
a n n n
a me
πε π ε
πε
= − = − = − ⋅ = ⋅⋅⋅
⎛ ⎞
⎜ = ⎟
⎝ ⎠
=
= Principal quantum number
Hydrogen-like Atom
Hydrogen-like Atom
4 2 2
2 2 2 2 2
0
13.6eV , 32
( , , , ) ( ) ( , ) 1, 2, 3,
0,1, 2, , ( 1)
, ( 1), , 2, 1, 0,1, 2, , ( 1), 1 1 ,
2 2
n l
l
n
i E t m
nlm nl l
l
s
me Z Z
E n n
r t R r Y e
n
l n
m l l l l
m
π ε
θ φ θ φ −
= − = − ⋅
Ψ =
= ⋅⋅⋅
= ⋅⋅⋅ −
= − − + ⋅⋅⋅ − − ⋅⋅⋅ −
= −
=
=
Principal quantum number Orbital quantum number
Magnetic quantum number Spin magnetic quantum number
Hydrogen-like Atom
Hydrogen-like Atom
표 6.1 수소원자의 규격화된 파동함수; n = 1, 2, 3
Table 8-5, p.280
표 6.2 원자 전자 상태
= × L r p
Angular Momentum
Angular Momentum
2 2 2
2 2
2 ( 1)
( ) ( ) ( ) ( )
2 2
d R dR l l
R r U r R r ER r
m dr r dr mr
⎛ ⎞ +
− ⎜ + ⎟ + + =
⎝ ⎠
= =
2 2
2
1
2 2
orb
K mv L
= = mr
Radial wave equation
When Potential Depends on r Only
When Potential Depends on r Only
ˆ ˆ ˆ ˆ
/ / /
ˆ sin cot cos
ˆ cos cot sin
ˆ
x
y
z
x y z
i x i y i z
L i
L i
L i
φ θ φ
θ φ
φ θ φ
θ φ
φ
= ×
=
− ∂ ∂ − ∂ ∂ − ∂ ∂
⎛ ∂ ∂ ⎞
= ⎜ ⎝ ∂ + ∂ ⎟ ⎠
⎛ ∂ ∂ ⎞
= − ⎜ ⎝ ∂ − ∂ ⎟ ⎠
= − ∂
∂
x y z
L r p
a a a
L
= = =
=
=
=
Angular Momentum
Angular Momentum
2 2
ˆ ( , ) ( 1) ( , )
ˆ ( , ) ( , )
( 1)
l l
l l
m m
l l
m m
z l l l
z l
L Y l l Y
L Y m Y
L l l
L m
θ φ θ φ
θ φ θ φ
= +
=
= +
=
=
=
=
=
Angular Momentum
Angular Momentum
그림 6.4 궤도 각운동량의 공간 양자화
그림 6.6 각운동량 벡터 L은 z축 주위로 계속하여 세차 운동을 한다.
Space Quantization
Space Quantization
그림 6.5 불확정성 원리에 의해 각운동량 벡터 L은 공간에서 확정된 방향을 가질 수 없다.
그림 6.8 다양한 양자 상태에 서의 수소원자 지름 파동함수 의 핵으로부터의 거리에 따른 변화.
그림 6.9 구면 극좌표계에서의 부피 요소 dV
핵으로부터의 거리가 r과 r+dr 사이인 공 껍질에서 수소원자 의 전자를 발견할 확률은
P(r)dr이다.
2 2
2 2
( ) 4
( ) ( )
P r dr r dr
P r r R r
ψ π
=
=
그림 6.11 그림 6.8에서 나타낸 상태들에 대한, 핵으로부터 의 거리 r과 r+dr 사이에서 수소원자의 전자를 발견할 확률.
