Section 2 Flashcards
mod-squared gives
a probability density
probability of density in [x,x+dx] at t =
|Ψ(x,t)|^2
the normalisation of probability density
∫ |Ψ(x,t)|^2 dx = 1
dispersion relation
E = hv = ℏω ; p = ℏk
same intensity everywhere, fully delocalised
|Ψ(x,t)| = Ψ(x,t) * Ψ(x,t) = |A|^2
well-defined k, undefined x
plane-wave space and time derivatives:
-iℏ ∂/∂x Ψ = pΨ
-iℏ ∂/∂t Ψ = EΨ
localise the wave-function using a Fourier transform
f(x) = 1/√2π (-∞ ∫ ∞) dk g(k) exp[i(kx-ω(k)t)]
wave-packet
localised sum of waves
group velocity
x/t = (dω/dk)|k=k0 = vg
wave-packet does not disperse when
ω(k) = vk, dω/dk = const
Parsaval’s theorem
(-∞ ∫ ∞) |f(x)|^2dx = (-∞ ∫ ∞) |g(k)|^2 dk
time-evolution of each definite-k / definite-ω is just
a phase exp[iω(k)t]
Domains related by FT are
conjugate
Gaussian function is special under FT
maps to another Gaussian
