Table of Useful Integrals, etc.

e−ax2

dx = 12

πa

⎛

⎝⎜⎞

⎠⎟

12

0

∞

∫

xe−ax2

dx = 12a0

∞

∫

x2e−ax2

dx = 14a

πa

⎛

⎝⎜⎞

⎠⎟

12

0

∞

∫

x3e−ax2

dx = 12a2

0

∞

∫

x2ne−ax2

dx =1⋅3 ⋅5 ⋅ ⋅ ⋅ 2n −1( )

2n+1an

πa

⎛

⎝⎜⎞

⎠⎟

12

0

∞

∫

x2n+1e−ax2

dx = n!2an+1

0

∞

∫

xne−axdx = n!

an+10

∞

∫

Integration by Parts:

UdV

a

b

∫ = UV⎡⎣ ⎤⎦a

b− VdU

a

b

∫ U and V are functions of x. Integrate from x = a to x = b

sin ax( )dx = − 1

acos ax( )∫

sin2 ax( )dx = x

2−

sin 2ax( )4a∫

sin3 ax( )dx = − 1

acos ax( ) + 1

3acos3 ax( )∫

sin4 ax( )∫ dx = 3x

8−

3sin 2ax( )16a

−sin3 ax( )cos ax( )

4a

sin ax( )sin bx( )dx =

sin a − b( )x⎡⎣ ⎤⎦2 a − b( )

−sin a + b( )x⎡⎣ ⎤⎦

2 a + b( )∫ where a2 ≠ b2

cos ax( )cos bx( )dx =

sin a − b( )x⎡⎣ ⎤⎦2 a − b( )

+sin a + b( )x⎡⎣ ⎤⎦

2 a + b( )∫

sin ax( )cos bx( )dx =

−cos a − b( )x⎡⎣ ⎤⎦2 a − b( )

−cos a + b( )x⎡⎣ ⎤⎦

2 a + b( )∫

x sin2 ax( )dx = x2

4−

x sin 2ax( )4a

−cos 2ax( )

8a2∫

x2 sin2 ax( )dx = x3

6−

x2

4a−

18a3

⎛

⎝⎜⎞

⎠⎟sin 2ax( ) −

xcos 2ax( )4a2∫

x sin ax( )sin bx( )dx =cos a − b( )x⎡⎣ ⎤⎦

2 a − b( )2 −cos a + b( )x⎡⎣ ⎤⎦

2 a + b( )2 +∫x sin a − b( )x⎡⎣ ⎤⎦

2 a − b( )2 −x sin a + b( )x⎡⎣ ⎤⎦

2 a + b( )2

x sin ax( )dx =

sin ax( )a2 −

xcos ax( )a∫

xcos ax( )dx =

x sin ax( )a

+cos ax( )

a2∫

cos ax( )dx =

sin ax( )a∫

cos2 ax( )dx = x

2+

sin 2ax( )4a∫

x2 cos2 ax( )dx = x3

6+

x2

4a−

18a3

⎛

⎝⎜⎞

⎠⎟sin 2ax( ) +

xcos 2ax( )4a2∫

cos bx( )e−ax2

dx = eax

a2 + b2( )acos bx( ) + bsin bx( )⎡⎣ ⎤⎦∫

Taylor Series:

f (n) xo( )n!

x − xo( )n

n=0

n=∞∑

Geometric Series:

xn

n=0

∞

∑ =1

1− x

Euler’s Formula:

eiφ = cosφ + isinφ

Quadratic Equation and other higher order polynomials:

ax2 + bx + c = 0

x = −b ± b2 − 4ac2a

ax4 + bx2 + c = 0

x = ± −b ± b2 − 4ac2a

⎛

⎝⎜⎜

⎞

⎠⎟⎟

General Solution for a Second Order Homogeneous Differential Equation with Constant Coefficients: If: ʹ′ʹ′y + p ʹ′y + qy = 0 Assume a solution for y:

Conversions from spherical polar coordinates into Cartesian coordinates:

x = r sinθ cosφy = r sinθ sinφx = r cosθdv = r 2 sinθdrdθdφ0 < r < ∞0 < θ < π0 < φ < 2π

y = esx ʹ′y = sesx ʹ′ʹ′y = s2esx

∴ s2esx + psesx + qesx = 0and s2 + ps + q = 0Hence y = c1e

s1x + c2es2 x

Commutator Identities:

A, B⎡⎣ ⎤⎦ = − B, A⎡⎣ ⎤⎦

A, An⎡⎣ ⎤⎦ = 0 n = 1,2,3,

kA, B⎡⎣ ⎤⎦ = A,kB⎡⎣ ⎤⎦ = k A, B⎡⎣ ⎤⎦

A+ B,C⎡⎣ ⎤⎦ = A,C⎡⎣ ⎤⎦ + B,C⎡⎣ ⎤⎦

A, BC⎡⎣ ⎤⎦ = A, B⎡⎣ ⎤⎦C + B A,C⎡⎣ ⎤⎦

AB,C⎡⎣ ⎤⎦ = A,C⎡⎣ ⎤⎦ B + A B,C⎡⎣ ⎤⎦

Creation and Annihilation Operators

l± l,ml , s,ms = l l +1( ) − ml ml ±1( )( )12 l,ml±1, s,ms

s± l,ml , s,ms = s s +1( ) − ms ms ±1( )( )12 l,ml , s,ms±1

j± j,mj = j j +1( ) − mj mj ±1( )( )12 j,mj±1

Atomic Units:

Physical Constants:

Operators: