F2004 formulas final_v4
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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.
![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
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![( − 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](https://image.slidesharecdn.com/f2004formulasfinalv4-120514153712-phpapp01/85/F2004-formulas-final_v4-2-320.jpg)
F2004 formulas final_v4
- 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