Table of Useful Integrals and Other Quantities of Interest

Table of Useful Integrals and Other Quantities of Interest

Table of Useful Integrals, etc.
∞
∫e
− ax 2
0
∞
1 ⎛ π ⎞
dx = ⎜ ⎟
2 ⎝ a ⎠
2 − ax 2
∫x e
0
∞
0
n − ax
∫ xe
dx =
0
− ax 2
dx =
0
1
∞
2
3 − ax 2
∫x e
dx =
0
(
)
1⋅ 3⋅ 5 ⋅ ⋅ ⋅ 2n − 1 ⎛ π ⎞
dx =
⎜⎝ a ⎟⎠
2 n+1 a n
∞
∫x e
∞
2
1 ⎛ π ⎞
dx =
4a ⎜⎝ a ⎟⎠
2n − ax 2
∫x e
1
1
∞
2
2
1
2a
1
2a 2
2n+1 − ax
∫ x e dx =
0
n!
2a n+1
n!
a n+1
Integration by Parts:
b
b
b
∫ UdV = ⎡⎣UV ⎤⎦ − ∫ VdU
a
a
U and V are functions of x. Integrate from x = a to x = b
a
1
∫ sin ( ax ) dx = − a cos ( ax )
( )
2
∫ sin ax dx =
( )
x sin 2ax
−
2
4a
1
1
∫ sin ( ax ) dx = − a cos ( ax ) + 3a cos ( ax )
3
3
( )
( ) ( )
3
3x 3sin 2ax sin ax cos ax
∫ sin ax dx = 8 − 16a −
4a
4
( )
( ) ( )
∫ sin ax sin bx dx =
( ) ( )
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
sin ⎡⎣ a − b x ⎤⎦ sin ⎡⎣ a + b x ⎤⎦
−
2 a−b
2 a+b
∫ cos ax cos bx dx =
sin ⎡⎣ a − b x ⎤⎦ sin ⎡⎣ a + b x ⎤⎦
+
2 a−b
2 a+b
where a 2 ≠ b2
(
(
( )
(
(
)
)
(
(
)
)
( ) − x cos ( ax )
sin ax
a2
a
( ) + cos ( ax )
x sin ax
a2
a
( )
sin ax
a
x
∫ cos ( ax ) dx = 2 +
( )
sin 2ax
4a
( )
x cos 2ax
x 3 ⎛ x 2
1 ⎞
x
cos
ax
dx
=
+
−
sin
2ax
+
∫
6 ⎜⎝ 4a 8a 3 ⎟⎠
4a 2
2
2
( )
∫ cos ( bx ) e
− ax 2
( )
dx =
Taylor Series:
( n)
xo
n= ∞ f
∑
( )
n=0
n!
eax
(
Geometric Series:
∞
1
∑ xn = 1 − x
n=0
( )
( )
⎡ a cos bx + bsin bx ⎤
⎦
a 2 + b2 ⎣
)
(x − x )
o
(
)
(
)
cos ⎡⎣ a − b x ⎤⎦ cos ⎡⎣ a + b x ⎤⎦ x sin ⎡⎣ a − b x ⎤⎦ x sin ⎡⎣ a + b x ⎤⎦
−
+
−
2
2
2
2
2 a−b
2 a+b
2 a−b
2 a+b
( ) ( )
2
( )
( )
∫ x sin ax sin bx dx =
( )
)
)
x cos 2ax
x 3 ⎛ x 2
1 ⎞
− ⎜
− 3 ⎟ sin 2ax −
6 ⎝ 4a 8a ⎠
4a 2
( )
∫ cos ax dx =
)
( )
2
2
∫ x sin ax dx =
∫ x cos ( ax ) dx =
(
(
x 2 x sin 2ax cos 2ax
−
−
4
4a
8a 2
( )
2
∫ x sin ax dx =
∫ x sin ( ax ) dx =
)
− cos ⎡⎣ a − b x ⎤⎦ cos ⎡⎣ a + b x ⎤⎦
−
2 a−b
2 a+b
( ) ( )
∫ sin ax cos bx dx =
n
(
)
(
)
Euler’s Formula:
eiφ = cos φ + isin φ
Quadratic Equation and other higher order polynomials:
ax 2 + bx + c = 0
−b ± b2 − 4ac
x=
2a
ax 4 + bx 2 + c = 0
⎛ −b ± b2 − 4ac ⎞
x = ± ⎜
⎟
⎜⎝
⎟⎠
2a
General Solution for a Second Order Homogeneous Differential Equation with
Constant Coefficients:
If: y ʹ′ʹ′ + py ʹ′ + qy = 0
Assume a solution for y:
y = esx
y ʹ′ = sesx
y ʹ′ʹ′ = s 2 esx
∴ s 2 esx + psesx + qesx = 0
and
s 2 + ps + q = 0
Hence
sx
s x
y = c1e 1 + c2 e 2
Conversions from spherical polar coordinates into Cartesian coordinates:
x = r sin θ cos φ
y = r sin θ sin φ
x = r cosθ
dv = r 2 sin θ drdθ dφ
0<r <∞
0 <θ <π
0 < φ < 2π
Commutator Identities:
⎡ Â, B̂ ⎤ = − ⎡ B̂, Â⎤
⎣
⎦
⎣
⎦
n
⎡ Â, Â ⎤ = 0 n = 1,2,3,
⎣
⎦
⎡ kÂ, B̂ ⎤ = ⎡ Â, kB̂ ⎤ = k ⎡ Â, B̂ ⎤
⎣
⎦ ⎣
⎦
⎣
⎦
⎡ Â + B̂, Ĉ ⎤ = ⎡ Â, Ĉ ⎤ + ⎡ B̂, Ĉ ⎤
⎣
⎦ ⎣
⎦ ⎣
⎦
⎡ Â, B̂Ĉ ⎤ = ⎡ Â, B̂ ⎤ Ĉ + B̂ ⎡ Â, Ĉ ⎤
⎣
⎦ ⎣
⎦
⎣
⎦
⎡ ÂB̂, Ĉ ⎤ = ⎡ Â, Ĉ ⎤ B̂ + Â ⎡ B̂, Ĉ ⎤
⎣
⎦ ⎣
⎦
⎣
⎦
Creation and Annihilation Operators
1
l± l, ml , s, ms = ( l ( l + 1) − ml ( ml ± 1)) 2  l, ml ±1 , s, ms
1
s± l, ml , s, ms = ( s ( s + 1) − ms ( ms ± 1)) 2  l, ml , s, ms ±1
(
(
))
j± j, m j = j ( j + 1) − m j m j ± 1
Atomic Units:
1
2
 j, m j ±1
Physical Constants:
Operators: