130_notes.dvi

HARMONIC OSCILLATOR

H= p

2

2 m+

1

2 mω

(^2) x (^2) = ̄hωA†A+ 1
2 ̄hω 10.1 En= (n+
1
2 ) ̄hω n= 0,^1 ,^2 ... 10.3
un(x) =

∑∞

k=0

akyke−y

(^2) / 2
ak+2=(k+1)(2(k−kn+2)) ak y=
√mω
̄hx
A= (
√mω
2 ̄hx+i
√p
2 m ̄hω) A
†= (√mω
2 ̄hx−i
√p
2 m ̄hω) [A,A

†] = 1

A†|n〉=

√

(n+ 1)|n+ 1〉 A|n〉=

√

(n)|n− 1 〉 u 0 (x) = (mω ̄hπ)

(^14)
e−mωx
(^2) /2 ̄h

ANGULAR MOMENTUM

[Li,Lj] =i ̄hǫijkLk [L^2 ,Li] = 0

∫

Yℓm∗Yℓ′m′dΩ =δℓℓ′δmm′

L^2 Yℓm=ℓ(ℓ+ 1) ̄h^2 Yℓm LzYℓm=m ̄hYℓm −ℓ≤m≤ℓ

L±=Lx±iLy L±Yℓm= ̄h

√

ℓ(ℓ+ 1)−m(m±1)Yℓ,m± 1

Y 00 =√^14 π Y 11 =−

√

3

8 πe

iφsinθ Y 10 =

√

3

4 π cosθ

Y 22 =

√

15

32 πe

2 iφsin (^2) θ Y 21 =−

√

15

8 πe

iφsinθcosθ Y 20 =

√

5

16 π(3 cos

(^2) θ−1)
Yℓℓ=eiℓφsinℓθ Yℓ(−m)= (−1)mYℓm∗ Yℓm(π−θ,φ+π) = (−1)ℓYℓm(θ,φ)
− ̄h^2
2 μ

[

∂^2

∂r^2 +

2

r

∂

∂r

]

Rnℓ(r) +

(

V(r) +ℓ(ℓ+1) ̄h

2

2 μr^2

)

Rnℓ(r) =ERnℓ(r)

j 0 (kr) =sin(krkr) n 0 (kr) =−cos(krkr) hℓ(1)(kr) =jℓ(kr) +inℓ(kr)

H=H 0 −~μ·B~ ~μ= 2 mce L~ ~μ= 2 gemcS~

Si= ̄h 2 σi [σi,σj] = 2iǫijkσk {σi,σj}= 0

σx=

(

0 1

1 0

)

σy=

(

0 −i

i 0

)

σz=

(

1 0

0 − 1

)

Sx= ̄h







0 √^120

√^1

2 0

√^1

2

0 √^120





 Sy= ̄h







0 √−i 2 0

√i

2 0

√−i

2

0 √i 2 0





 Sz= ̄h





1 0 0

0 0 0

0 0 − 1



 1

HYDROGEN ATOM

H=p

2

2 μ−

Ze^2

r ψnℓm=Rnℓ(r)Yℓm(θ,φ) En=−

Z^2 α^2 μc^2

2 n^2 =−

13. 6

n^2 eV

n=nr+ℓ+ 1 a 0 =αμch ̄ ℓ= 0, 1 ,...,n− 1

Rnℓ(ρ) =ρℓ

∑∞

k=0

akρke−ρ/^2 ak+1=(k+1)(k+ℓ+1k+2−ℓn+2)ak ρ=

√

− 8 μE

̄h^2 r=

2 r

na 0

R 10 = 2(aZ 0 )

(^32)
e
−aZr
(^0) R 20 = 2( 2 Za 0 )
(^32)
(1− 2 Zra 0 )e
− 2 aZr
(^0) R 21 =√^13 ( 2 Za 0 )
(^32)
(Zra 0 )e
− 2 aZr
0
Rn,n− 1 ∝ rn−^1 e−Zr/na^0 μ=mm 11 +mm^22 〈ψnℓm|e
2
r|ψnℓm〉=
Ze^2
n^2 a 0 =
Zα^2 μc^2
n^2
H 1 =− p
4
8 m^3 c^2 H^2 =
e^2
2 m^2 c^2 r^3

S~·~L ∆E 12 =−^1

2 n^3 α

(^4) mc (^2) (^1
j+^12 −
3
4 n)
H 3 = e
(^2) gp
3 mMpc^2
S~·~I 4 πδ^3 (~r) ∆E 3 =^2 gpmα
(^4) mc 2
3 Mpn^3 (f(f+ 1)−I(I+ 1)−
3
4 )
HB= 2 eBmc(Lz+ 2Sz) ∆EB=e 2 hB ̄mc(1± 2 ℓ^1 +1)mjforj=ℓ±^12