This document contains mathematical expressions and definitions related to quantum mechanics, angular momentum, spherical harmonics, and other physics concepts. It includes: 1) Definitions of the Pauli spin matrices and their properties. 2) Expressions for spin operators and raising/lowering operators. 3) Spherical harmonic functions and their relationships to angular momentum quantum numbers. 4) Operators for the quantum harmonic oscillator and their eigenstates. 5) Additional equations for spin, magnetic fields, and other quantum mechanical systems.

1. Az Ax − iAy 0 −i 1 0 0 1
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
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
...