why can’t a plane wave psi = Ae^i(kx-wt)
be normalised?
a plane wave cannot be normalised because its probability density is constant over all space, so the integral of |psi|^2 diverges.
what are two fixes to the normalisation problem?
- box normalisation - confine length to 2L, giving |A| = 1/sqrt2L
- wave packet - superpose many waves so the particle is naturally localised
what is a wave packet mathematically?
integral : learn
a continuous superposition of plane waves with wavenumbers grouped around a central value k0, weighted by the amplitude function a(k).
what is a(k) and what does its shape represent
a(k) is the amplitude distribution in wavenumber space.
describes how much each plan wave contributes to the wave packet
it is peaked around k0 with width sigma
what is sigma and how is it defined precisely?
sigma is the standard deviation ( spread ) of |a(k)|^2 in k-space,
measuring how widely the wavenumbers are distributed
what taylor expansion is used to simplify the wave packet integral?
expand w(k) about k0:
w(k)≈w0 + (k-k0)dw/dk
after the taylor expansion, what form does psi take?
e^i(k0x-w0t) f(x-vgt)
e^i(k0x-w0t) is carrier wave
f(x-vgt) is envelope
vg = dw/dk, packet moves at group velocity
what is the group velocitt and what does it describe physically?
Vg = dw/dk
it is the speed at which the wave packet envelope - and hence the probability distribution - moves
derive the group velocity for a free quantum particle
from dispersion relation
ħw = (ħk)^2 / 2m
w = ħk^2 / 2m
dw/dk = 2ħk/2m = ħk/m = p/m
matches classical velocity p/m
why does the group velocity matching p/m matter?
it confirms the correspondence principle
quantum mechanics must reproduce classical physics in the appropriate limit
A WAVE PACKET MOVES EXACTLY AS A CLASSICAL PARTICLE WOULD
what is the phase velocity and how does it differ from group velocity?
Vp = w/k = speed of individual wave crests
Vg = dw/dk = speed of the wave packet
for a quantum particle Vg = p/m gives the physical particle velocity
what is the key reciprocal relation between k-space and real space?
width in k-space is sigma; width in real space ≈ 1 / sigma
a narrow distribution in one space corresponds to a broad distribution in the other
why is the Gaussian wave packet a special and useful case?
A Gaussian in k-space gives a Gaussian in real space, making the mathematics tractable and preserving the functional form
what does the shape of the wave packet in real space depend on?
the shape of the wave packet is determined by the amplitude distribution a(k)
why is |psi|^2 for a wave packet normalisable when a plane wave isnt?
A wave packet is localised, so |psi|^2 integrates to a finite value.
A plane wave has constant probability density, so the integral diverges
what does it mean physically that the wave packet moves at Vg = p/m
the centre of the probability distribution moves at the classical velocity p/m, so a wave packet behaves like a classical particle on average
how does the two-wave superposition demonstrate group velocity?
superposing two nearby waves produces an interference pattern that moves with group velocity Vg = dw/dk
list four reasons wave packets are physically important
- Normalisable - finite total probability
- Localised - particle confined in space
- Correct velocity - Vg = p/m
- Leads to uncertainty principle
