Identities, relations, and methods Flashcards
Trig addition formulae
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB - cosAsinB
cos(A+B) = cosAcosB - sinAsinB
cos(A-B) = cosAcosB + sinAsinB
Double angle formulae
sin(2x) = 2sinxcosx
cos(2x) = cos²x - sin²x
= 2cos²x - 1
= 1 - 2sin²x
Fourier series (sin/cos)
For a function with period 2L in the range -L to +L
f(x) = a₀/2 + Σ(1to∞) (aₙcos(nπx/L) + bₙsin(nπx/L))
where
a₀ = 1/L ∫±L f(x) dx
aₙ = 1/L ∫±L f(x)cos(nπx/L) dx
bₙ = 1/L ∫±L f(x)sin(nπx/L) dx
Orthogonality relations
∫±L sin(nπx/L) cos(mπx/L) dx = 0
∫±L sin(nπx/L) sin(mπx/L) dx = Lδₙₘ
where
δₙₘ =
1 if m = n
0 if m ̸= n
∫±L cos(nπx/L) cos(mπx/L) dx =
0 if m ̸= n
L if m = n
2L if m = n = 0
Fourier series (complex)
f(x) = Σ±∞ cₙexp(inπx/L)
cₙ = 1/2L ∫±L f(x)exp(−inπx/L) dx
Fourier transform
F(k) = ∫±∞ f(x) exp(−ikx) dx
f(x) = 1/2π ∫±∞ F(k) exp(ikx) dk
Laplace equation
∇²φ = 0
Wave equation
∇²φ = 1/c² ∂²φ/∂t²
Schrödinger equation
−ħ²/2m ∇²ψ + Vψ = iħ ∂ψ/∂t
Solution to
a d²f/dx² + b df/dx + cf = 0
f = Aexp(λ₁x) + Bexp(λ₂x)
where aλᵢ² + bλᵢ + c = 0
Solution to
df/dx = −αf
f = Aexp(−αx)
Solution to
d²f/dx² = −α²f
f = A cos(αx) + B sin(αx)
Solution to
d²f/dx² = α²f
f = Aexp(αx) + Bexp(-αx)
Trig to exponential
exp(ix) = cosx + isinx
exp(-ix) = cos-x - isinx
cosx = (exp(ix) + exp(-ix)) / 2
sinx = (exp(ix) - exp(-ix)) / 2i
Hyperbolic definitions
sinhx = (exp(x) - exp(-x)) / 2
coshx = (exp(x) + exp(-x)) / 2
General solution to wave equation
φ = f(x − ct) + g(x + ct)
Method for solving wave equations
- Investigate solutions in the separable form φ = X(x)T(t).
- Build the general solution as a superposition of these.
- Fix the undetermined constants by applying the initial conditions.
sinA sinB as a sum
1/2 (cos(A−B) − cos(A+B))
sinA cosB as a sum
1/2 (sin(A−B) + sin(A+B))
cosA cosB as a sum
1/2 (cos(A−B) + cos(A+B))
When is the Fourier series expansion valid
Valid for any function satisfying Dirichlet’s conditions:
* Single valued
* Finite number of discontinuities
* ∫±L |f(x)| dx must be finite
Note that sin(nπx/L) and cos(nπx/L) are orthogonal basis vectors.
Define orthogonality
Two functions u(x) and v(x) are orthogonal on an interval a ≤ x ≤ b if
∫ₐᵇ u(x) v(x) dx = 0
Mean of a function
f̄ ≡ 1/2L ∫±L f(x) dx = a₀/2
RMS of a function
∫±L [f(x)]² dx
= 1/2 a₀ ∫±L f(x) dx
+ ∫±L Σ(1to∞) f(x) aₙ cos(nπx/L) dx
+ ∫±L Σ(1to∞) f(x) bₙ sin(nπx/L) dx
Evaluate each term
f̄² ≡ 1/2L ∫±L [f(x)]² dx = (a₀/2)² + 1/2 Σ(1to∞) aₙ² + bₙ²
rms is given by √f̄²
Diffusion equation
∇²T = 1/α² ∂T/∂t
T can refer to the temperature in a region or the density of particles as they diffuse.
Functional forms of Dirac delta
Rectangular function
δ(x) = lim(k→∞) kΠ(kx)
Triangle function
Λ(x) = (1−|x| for −1 ≤ x ≤ 1, 0 otherwise)
δ(x) = lim(k→∞) kΛ(kx)
Gaussian
δ(x) = lim(k→∞) √(k/π)’ exp(−kx²)
Sinc
δ(x) = lim(k→∞) k/π sin(kx)/kx
Exponential (derived from sinc)
δ(x) = 1/2π ∫±∞ exp(−ikx) dk = 1/2π ∫±∞ exp(ikx) dk = δ(−x)
Properties of Dirac delta (7)
lim(x→0) δ(x) = ∞
∫±∞ δ(x) dx = 1
δ(x) = δ(−x)
xδ(x) = 0
x d/dx δ(x) = −δ(x)
∫±∞ δ(x−a) f(x) dx = f(a)
δ(ax) = 1/|a| δ(x)
Bandwidth product relation
∆k∆x ≥ 1/2
Heisenberg uncertainty relation from bandwidth product relation
Bandwidth product relation
∆k∆x ≥ 1/2
k is essentially the wavenumber
de Broglie’s theorem states
λp = h
⇒ k = 2π/λ = p/ħ
Substitute in
∆x∆p ≥ ħ/2
Practical check for Fourier transform analysis
F(0) = ∫±∞ f(x) dx
The total area under f(x) is given by the value of the Fourier transform at the origin.
Parseval’s theorem
∫±∞|f(x)|² dx = a ∫±∞|F(k)|² dk
where a = 1 for the mathematical (symmetric) transform
and a = 1/2π for the physics transform
Define convolution
h(x) = f(x) ∗ g(x)
= ∫±∞ f(x−x′) g(x′) dx′ = ∫±∞ f(x′) g(x−x′) dx′
Fourier transform link to convolution
H(k) = F(k)G(k)
Fowards and backwards travelling waves
Forwards: ϕ(x, t) = exp(−i(kx−ωt))
Backwards: ϕ(x, t) = exp(i(kx+ωt))
where ω = ck
(can be found by substituting into wave equation)
Solution to the 3-d wave equation
ϕ(x, t) = exp(i(k.x−ωt))
ω/|k| = c, plane wave traveling along k
Note that k.x = α is the equation of a plane perpendicular to k and whose distance from the origin is α/|k|.
Phase velocity
vₚ(k) = ω(k)/k
Group velocity
v₉ = dω/dk|ₖ₀ = ω’₀
Wave equation including dispersion
ϕ(x, t) = 1/√2π’ ∫±∞ G(k) exp(i(kx−ω(k)t)) dk
where G(k) is the Fourier transform of ϕ(x, 0)
and ω(k) is determined after solving the wave equation for ϕ
Shifting property of the Fourier transform
∫±∞ g(x−x₀) exp(−ikx) dx = exp(−ikx₀) ∫±∞ g(x) exp(−ikx) dx
= exp(−ikx₀) G(k)
Transform of f(ax)
1/|a| F(k/a)
Transform of f(x + a)
exp(ika) F(k)
Transform of f(x) exp(−ixa)
F(k + a)
Transform of d/dx f(x)
ikF(k)
Transform of ∫(-∞tox) f(x′) dx′
F(k)/ik + πF(0)δ(k)
Transform of F(x)
2π f(−k)
Transform of −ixf(x)
d/dk F(k)
Transform of δ(x − a)
exp(−ika)
Transform of exp(iax)
2πδ(k − a)
Transform of H(x)
πδ(k) + 1/ik
Transform of 1/iπx
sgn(k)
Transform of sgn(x)
2/ik
Transform of a/2π sinc(ka/2)
Π(k/a)
Transform of Π(ax)
1/a sinc(k/2a) = 1/a sin(k/2a)/(k/2a)
Transform of Λ(x)
sinc²(k/2)
Transform of cos(ax)
π[δ(k−a) + δ(k+a)]
Transform of sin(ax)
iπ[δ(k+a) − δ(k−a)]
Transform of H(x)cos(ax)
π/2 [δ(k−a) + δ(k+a)] + ik/a²−k²
Transform of H(x)sin(ax)
π/2i [δ(k−a) − δ(k+a)] + a/a²−k²
Transform of Π(x/a)cos(πx/a)
2π/a cos(ka/2)/(π/a)²−k²
Transform of exp(−x²/2σ²)
√2π’ σ exp(−1/2 k²σ²)
Transform of 1/2a exp(−a|x|)
(a² + k²)⁻¹
Transform of F(−x)
2πf(k)
Rectangular, triangular, and sgn functions in terms of Heaviside
H(x) = 1 for x≥0, 0 for x<0
Π(x) = H(x+1/2) − H(x−1/2)
= 1 for |x|<1/2, 0 otherwise
Λ(x) = Π(x) ∗ Π(x)
= 1 − |x|
sgn(x) = H(x) − H(−x)
= 1 for x > 0, −1 for x < 0
Taylor expansion
f(x) = f(x₀) + f’(x₀) (x − x₀) + f’‘(x₀) (x − x₀)²/2! + …
= Σ(0to∞) 1/n! f⁽ⁿ⁾ (x₀) (x − x₀)ⁿ
Hermite’s equation
d²y/dx² − 2x dy/dx + 2ny = 0
for n >0
