This document contains mathematical formulas and definitions related to quantum mechanics, spin, angular momentum, spherical harmonics, and other physics concepts. It includes expressions for spin operators, spherical harmonics, raising and lowering operators, eigenstates of the quantum harmonic oscillator, and Legendre polynomials. Operators, matrices, eigenstates, and other quantum mechanical notation are used throughout.

1. Az Ax − iAy 0 −i 1 0 0 1
AbrahamP rado1213521 Sj = σj σi σj = δij + i ijk σk σ = [σx , σy , σz ] σ·A= σy = σz = σ =
2 Ax + iAy −Az i 0 0 −1 x 1 0
k
1 0 1 1 1 1 a+b a−b
(σ · A)(σ · B) = (A · B)I + iσ · (A × B) χ = aχ+ + bχ− χ+ = χ = u= √ v=√ χ= √ u+ √ v [σx , σy ] = 2i ijk σk
0 − 1 2 1 2 −1 2 2
 
cos ϕ − sin ϕ 0
{σx , σy } = 2δij σ0 σi σi = 1 Sn =
ˆ ± Rz =  sin ϕ cos ϕ 0 [Si , Sj ] = i ijk Sk S 2 |s, m = 2 |s, m Sz |s, m = m|s, m S± = Sx ± iSy
2
0 0 1
√ a 0 −i ±iθ
S± |s, m = s(s + 1) − m(m ± 1)|s, m ± 1 Sx χ± = χ eiα Sy = e = cos θ ± i sin θ S = [Sx , Sy , Sz ]
2 1 − a2 eiϕ 2 i 0
cos θ e−iϕ sin θ cos θ e−iϕ sin θ
Sn = Sx sin θ cos ϕ + Sy sin θ sin ϕ + Sz cos θ
ˆ χn =
ˆ 2 n
ˆ
θ χ− =
2 S =
n
ˆ
+ iϕ
e sin 2 − cos θ2 2 eiϕ sin θ − cos θ
 2 
1− 0 −
i 2 1
f (r − δr) = f (r)[1 − δr · p] Ry ( ) =  0 1 0  Sy = (S+ − S− ) S± χ = χ± S± χ± = 0 Sx χ± = χz Sy χ± = χ S z χ = ± χ±
2 2i 2 2i 2
− 0 1− 2
 
0 bs 0 ··· 0
0 0 bs−1 ···
 0
2
3  
2 2 † † 2
(s + j)(s + 1 − j) S+ =  . . . .. .
 Sx = Re(ab∗ ) Sy = − Im(ab∗ )
. . . .
S χ± = χ± χ χ=1 Sx = χ Sx χ Sj = bj =

4 4 . . . . .
0 0 0 · · · b−s+1 
0 0 0 ··· 0
n+1

∞ ∞  2 Γ n+1 /a 2
1
2 (n > −1, a > 0)
3 1 0 1 1 2 1 · 3 · 5 · (n + 1)π 1/2 n −ax2

S2 = 2
s(s + 1) S2 = 2
↑= xn e−αx dx = , n = 2k x e (2k−1)!!
dx = 2k+1 ak π
a (n = 2k, a > 0)
4 0 1 2 2 −∞ 2n/2 α(n+1)/2 0  k!
(n = 2k + 1 , a > 0)

2ak+1
i cos β − inz sin β −(inx + ny ) sin β i cos θe−iϕ/2 eiα/2 0
ˆ ˆ ˆ
Rn (β) = exp − β n · L Rn (β) =
ˆ
2 2 2 Ta = exp − a · p a=
ˆ Rz =
−(inx − ny ) sin β
2 cos β + inz sin β
2 2
sin θ eiϕ/2
2 0 e−iα/2
 
0 0 ··· 0 0  
s 0 ··· 0
bs 0 ··· 0 0
α α   0 s−1 ··· 0
Rr = cos I + i sin n·σ S− =  0
 bs−1 ··· 0 0 Sz = 
. .

. 
2 2 . . . .

. . .. . 
. . .. . . . . . .
. . . . .
0 0 · · · −s
0 0 ··· b−s+1 0
s1 s2 s s1 s2 s s m|Sk |s l
|s1 m1 |s2 m2 = Cm1 m2 m |s m |s m = Cm1 m2 m |s1 m1 |s2 m2 (σk )m l =
s m1 +m2 =m
s
† 1 1 1 1 1
(k) (y) (y)
c+ = χ + χ χ+ =√ χ− =√ Lx = 0 Ly = 0 φ(p) = e−i(p·r)/ ψ(r)dr3
2 i 2 −i (2π )3/2
1 dn ml
P = |U U |eiω1 t + |V V |eiω2 t Pn (x) = (x2 − 1)n θ = cos−1 ω = γB0 S= [sin α cos γB0 t − sin α sin γB0 t cos α]
2n n! dxn l(l + 1) 2
iγB0 t
−iE+ t −iE− t γB0 1 0 cos α e 2 q
χ(t) = Aχ+ e + Bχ− e H = −γ S · B µ = γS H=− |χ(t) = 2
−iγB0 t X = X|σx |X τ =µ×B U = −µ · B γ=
2 0 −1 sin α e 2 2me
2
   
2 2 0 r cos α sin β
ˆ 1 ∂ ∂f 1 ∂ ∂f 1 ∂ f
HY m (θ, ϕ) = ( + 1)Y m (θ, ϕ) ∆f = r2 + sin ϕ + R(α, β, γ) 0 =  r sin α sin β  ,
2I r2 ∂r ∂r r2 sin ϕ ∂ϕ ∂ϕ r2 sin2 ϕ ∂θ2 r r cos β
...

2. ( − m)! m
AbrahamP rado1213521 P −m = (−1)m P
( + m)!
1 1 1 3 1 3 −1 3
Y00 (θ, ϕ) = Y1−1 (θ, ϕ) = sin θ e−iϕ Y10 (θ, ϕ) = cos θ Y11 (θ, ϕ) = sin θ eiϕ
2 π 2 2π 2 π 2 2π
1 15 1 15 1 5 −1 15
Y2−2 (θ, ϕ) = sin2 θ e−2iϕ Y2−1 (θ, ϕ) = sin θ cos θ e−iϕ Y20 (θ, ϕ) = (3 cos2 θ − 1) Y21 (θ, ϕ) = sin θ cos θ eiϕ
4 2π 2 2π 4 π 2 2π
√
···
   
√0 0 0 0 0 1 √0 0 ···
∞
 1 0
√ 0 0 · · · 0 0 2 √0 · · ·
1 15 1 k   
· · · a = 0 0 3 · · · H = ω a† a + 1

Y22 (θ, ϕ) = sin2 θ e2iϕ eX = X a† =  0 2 √ 0
0 0 x= a† + a

4 2π k! 2 2mω
  
k=0
 0
 0 3 0 · · ·
0 0
 0 0 · · · 
.
. .
. .
. .
. .. .
. .
. .
. .
. ..
. . . . . . . . . .
n
√ √ a† 2 d
n 2 1
p=i a† − a a|n = n|n − 1 a† |n = n + 1|n + 1 |n = √ |0 Hn (ξ) = (−1)n eξ n
e−ξ H|n = (n + ) ω|n
2mω n! dξ 2
(−1)m d +m 2 ( − m)! m
P m (x) = (1 − x2 )m/2 (x − 1) . P −m (x) = (−1)m P (x).
2 ! dx +m ( + m)!
0 0 1
P0 (cos θ) = 1 P1 (cos θ) = cos θ P1 (cos θ) = − sin θ
0 2 1
P2 (cos θ) = 1
2 (3 cos θ − 1) P2 (cos θ) = −3 cos θ sin θ P2 (cos θ) = 3 sin2 θ
2
|v3 − |e1 e1 |v3 − |e2 e2 |v3 1 a−d (a − d)2 + 4bc /2c
P (r) = [Rn (r)]2 r2 |e3 = µ± = a+d± (a − d)2 + 4bc |v± =
||v3 − |e1 e1 |v3 − |e2 e2 |v3 | 2 1
 
∞ cos θ cos ψ cos φ sin ψ + sin φ sin θ cos ψ sin φ sin ψ − cos φ sin θ cos ψ (α+γ) (α−γ)
2
−x +bx+c
√ 2
b /4+c e−i 2 cos β −e−i 2 sin β
e dx = πe A = − cos θ sin ψ cos φ cos ψ − sin φ sin θ sin ψ sin φ cos ψ + cos φ sin θ sin ψ  D(α, β, γ) = (α−γ)
2
(α+γ)
2
−∞ sin θ − sin φ cos θ cos φ cos θ ei 2 sin β
2 ei 2 cos β
2
∞
1 l+1 −ρ r 2(j + l + 1 − n)
Rnl = ρ e ν(ρ), ν(ρ) = cj ρj , ρ= , cj+1 = cj ,
r j=0
na (j + 1)(j + 2l + 2)
sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α cos β
tan α ± tan β
tan(α ± β) = cosh ix = 1 (eix + e−ix ) = cos x sinh ix = 2 (eix − e−ix ) = i sin x
2
1
1 tan α tan β
tanh ax dx = a−1 ln(cosh ax) coth ax dx = a−1 ln(sinh ax) ex = cosh x + sinh x
cos((a1 − a2 )x) cos((a1 + a2 )x) dx
e−x = cosh x − sinh x. sin a1 x cos a2 x dx = − − p =m
2(a1 − a2 ) 2(a1 + a2 ) dt
∞
kg · m2
p = dxψ ∗ (x)∂x ψ(x) = 1,054 × 10−34 J · s = 6,582 × 10−15 eV · s me = 9,10938 · 10−31 kg
i −∞ s
dk (α) (α+k)
L (x) = (−1)k Ln−k (x) H2n (x) = (−1)n 22n n! L(−1/2) (x2 )
dxk n n
ex dn 2 dn −x2 x−α ex dn
Ln (x) = e−x xn Hn (x) = (−1)n ex e Hn+1 (x) = 2xHn (x) − Hn (x) L(α) (x) =
n e−x xn+α
n! dxn dxn  n! dxn 
(ek − f h) (ch − bk) (bf − ce)
1 d −b
A−1 == A−1 = (f g − dk) (ak − cg) (cd − af )
ad − bc −c a
(dh − eg) (gb − ah) (ae − bd)
A + A† A − A† 2
n(n + 1)
U †U = 1 A = + H† = H HT = H∗ AT = ±A D = T AT −1 En = ..
2 2 ma2