130_notes.dvi

37 Formulas

̄h= 1. 05 × 10 −^27 erg sec c= 3. 00 × 1010 cm/sec e= 1. 602 × 10 −^19 coulomb

1 eV= 1. 602 × 10 −^12 erg α=e

2

̄hc= 1/137 =

e^2

4 πǫ 0 hc ̄ (SI) ̄hc= 1973 eV ̊A= 197.3 MeV F

1 ̊A= 1. 0 × 10 −^8 cm 1 Fermi = 1. 0 × 10 −^13 cm a 0 =αm ̄hec= 0. 529 × 10 −^8 cm

mp= 938.3 MeV/c^2 mn= 939.6 MeV/c^2 me= 9. 11 × 10 −^28 g = 0.511 MeV/c^2

kB= 1. 38 × 10 −^16 erg/◦K ge= 2 +απ gp= 5. 6

μBohr= 2 meh ̄ec= 0. 579 × 10 −^8 eV/gauss

∫∞

−∞

dx f(x)δ(g(x)) =

[

1

|dgdx|f(x)

]

g(x)=0

∞∫

−∞

dx f(x)δ(x−a) =f(a)

∫∞

−∞

dx e−ax

2

=

√π

a use

∂

∂afor other forms

eA=

∑∞

n=0

An

n! sinθ=

∑∞

n=1, 3 , 5 ...

θn

n!(−1)

n− 1

(^2) cosθ=

∑∞

n=0, 2 , 4 ...

θn

n!(−1)

n 2

P(x) =√ 21 πσ 2 e−x

(^2) / 2 σ 2 ∞∫
0
dr rne−ar=ann+1! E=

√

m^2 c^4 +p^2 c^2

GENERAL WAVE MECHANICS

E=hν= ̄hω λ=h/p p= ̄hk

∆p∆x≥ ̄h 2 ∆A∆B≥〈i 2 [A,B]〉 ∆A=

√

〈A^2 〉−〈A〉^2

ψ(x) =√ 21 π ̄h

∞∫

−∞

dp φ(p)eipx/ ̄h φ(p) =√ 21 π ̄h

∞∫

−∞

dx ψ(x)e−ipx/ ̄h

pop= ̄hi∂x∂ Eop=i ̄h∂t∂ xop=i ̄h∂p∂

Huj(x) =Ejuj(x) ψj(x,t) =uj(x)e−iEjt/ ̄h − ̄h

2

2 m

∂^2 ψ

∂x^2 +V(x)ψ=i ̄h

∂ψ

∂t

ψ(x) continuous dψdxcontinous ifVfinite

∆dψdx=^2 mλ ̄h 2 ψ(a) forV(x) =λδ(x−a)

〈φ|ψ〉=

∞∫

−∞

dxφ∗(x)ψ(x) 〈ui|uj〉=δij

∑

i

|ui〉〈ui|= 1

φ=

∑

i

aiui ai=〈ui|φ〉 ψ(x) =〈x|ψ〉

〈φ|A|ψ〉=〈φ|Aψ〉=〈A†φ|ψ〉=〈ψ|A†|φ〉∗ φ(p) =〈p|ψ〉

[ 21 m(~p+ceA~)^2 +V(~r)]ψ(~r) =Eψ(~r) Hψ=Eψ

[px,x] = ̄hi [Lx,Ly] =i ̄hLz [L^2 ,Lz] = 0

ψi=〈ui|ψ〉 Aij=〈ui|A|uj〉 d〈dtA〉=〈∂A∂t〉+ ̄hi〈[H,A]〉