Chapter 4 The Wavelike Properties of Particles

Chapter 4
The Wavelike Properties of Particles
Read Chapter 4 of the hand-written notes.
4.1 De Broglie’s Hypothesis
De Broglie’s Wavelength
The wavelength of any particle, with or without mass, is related to the magnitude of its
momentum. p, by the relation:
λ = h p . (4.1)
Energy of a Particle in Terms of its Frequency
E = hν
= ω (4.2)
Momentum of a Particle in Terms of its Wavenumber
c⃗p = hνˆn
= ⃗ k
= cˆn/λ (4.3)
where ˆn is a unit vector in the direction of the particle.
1

2 CHAPTER 4. THE WAVELIKE PROPERTIES OF PARTICLES
4.2 Uncertainty Relationships for Classical Waves
4.3 Heisenberg Uncertainty Relationships
Heisenberg’s Uncertainty Relationships
∆p x ∆x ≥ /2
∆p y ∆y ≥ /2
∆p z ∆z ≥ /2
∆E∆t ≥ /2 (4.4)
4.4 Wave Packets
The general form for a wavepacket moving along the x-axis is given by:
f(x, t) = 1 √
2π
∫ ∞
−∞
dk A(k) e i[kx−ω(k)t] , (4.5)
where A(k) represents the “strength” of the contribution of wave number k to the formation
of the packet. In complete generality, the frequency ω can be a function of the wavenumber,
so we write its dependence on k explicitly as ω(k).
The expression (4.5) is a “Fourier” transform. If we know the wavenumber spectrum, A(k),
we can form its wavepacket, f(x, t). Conversely, if we know the shape of the wavepacket at
t = 0, we can determine the wavenumber spectrum by using the inverse property of Fourier
transforms. and computing:
A(k) = 1 √
2π
∫ ∞
−∞
dxf(x, 0) e −ikx . (4.6)
If we set t = 0 in (4.5) and substitute for A(k) from (4.6), we get, after some rearranging:
f(x, 0) =
∫ ∞
−∞
dx ′ f(x ′ , 0)
( ) ∫ 1 ∞
dk e ik(x−x′) . (4.7)
2π −∞
We must get the same function back, so this identifies one form of the Dirac “δ-function”
commonly seen when doing Fourier transforms, namely:
δ(x − x ′ ) = 1
2π
∫ ∞
−∞
dk e ik(x−x′) . (4.8)

4.4. WAVE PACKETS 3
There are many forms of Dirac’s δ-function. It has the peculiar property that:
f(x) =
∫ x2
as long as x 1 < x < x 2 , and zero otherwise.
x 1
dx ′ δ(x ′ − x)f(x ′ ) , (4.9)
Another form of the δ-function commonly seen in Fourier transform analysis is
δ(k − k ′ ) = 1 ∫ ∞
dxe i(k−k′ )x . (4.10)
2π −∞
Uncertainty Relationship
It can be shown from the general properties of Fourier transforms that:
(Note to self: Proof to follow at some later date.)
Multiplying by gives Heisenberg’s uncertainty relationship:
∆x∆k ≥ 1 2 . (4.11)
∆x∆p ≥ 2 . (4.12)
Group Velocity
Consider now, (4.5), and imagine that the frequencies are strongly centered on some principle
frequency k 0 , and that it is weakly dispersive, so that we may write in a Taylor expansion:
ω(k) = ω(k 0 ) +
and substitute this back into (4.5) and obtain:
f(x, t) ≈ 1 √
2π
∫ ∞
−∞
( ) dω
(k − k 0 ) · · · , (4.13)
dk
k=k 0
dk A(k) e i[kx−ω 0t−v g(k−k 0 )t] , (4.14)
where ω 0 = ω(k 0 ) and v g = ( )
dω
dk k=k 0
. The above can be rearranged to give:
f(x, t) ≈ ei(k 0v g−ω 0 )t
√
2π
∫ ∞
−∞
dk A(k) e ik(x−vgt) , (4.15)

4 CHAPTER 4. THE WAVELIKE PROPERTIES OF PARTICLES
or,
f(x, t) ≈ f(x − v g t, 0)e i(k 0v g−ω 0 )t , (4.16)
which, apart from the overall phase term in the exponential, looks like the original shape of
f(x, t) at t = 0 moving with velocity v g . This is the “group” velocity for a general, weakly
dispersive wavepacket.
Does it make sense? If we identify the energy of a wave as E = ω and the magnitude of its
momentum as p = k, we can identify the group velocity as
v g = dω
dk = dω
dk = dE
dp . (4.17)
The group velocity of a photon (E = cp) is c, for a non-relativistic particle with mass m
[E = p 2 /(2m)] it is p/m (or what we have called v), and for a relativistic particle with mass
m [E 2 − (cp) 2 = (mc 2 ) 2 ] it is pc 2 /E, also what we have been calling v. So, it does make
sense!
4.5 Probability and Randomness
4.6 The Probability Amplitude