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Chapter 4
The Wavelike Properties of Particles
• The energy carried by a particle is confined to a small
region of space
• The energy carried by a wave is distributed throughout
space, but localized.
In quantum mechanics there is a clear distinction from classical
mechanics. Particles must somehow obey the rules previously
established for waves → Wave Mechanics.
4.1 The de Broglie Hypothesis
𝑝=ℎ 𝜆
Just as a wave can have momentum (a particlelike property), a
particle with momentum 𝑝 can exhibit wavelike properties
(having wavelength 𝜆).
1. Because of the smallness of ℎ, only particles of atomic or
nuclear size will exhibit the wave behavior.
2. The de Broglie wavelength 𝜆 reveals itself when a wavetype experiment is performed on it. The outcome of the
experiment depends on this wavelength.
3. The de Broglie wavelength characterizes the wave-type
behavior of particles, and is central to the quantum theory.
4.2 Experimental Evidence for de Broglie Waves
The diffraction minima are located at
𝑎𝑠𝑖𝑛𝜃 = 𝑛𝜆 n = 1, 2, 3, . . .
1926
Clinton Davisson and Lester Germer (Bell Telephone Labs)
The diffraction of electrons 𝜙 = 50°
V = 54 volts
The crystal surface acts
like a diffraction grating.
𝑑𝑠𝑖𝑛𝜙 = 𝑛𝜆
n = 1, 2, 3,
Double Slit Experiment with Particles
The electrons from a hot
filament were accelerated
through 50 kV (𝜆 =5.4 pm)
and then passed through a
double slit of separation 2.0 µm
and width 0.5 µm.
Double-slit pattern for
electrons.
𝑑Δ𝑦
𝜆 =
𝐷
Neutron Scattering and the double slit experiment
Intensity pattern observed
for double-slit interference
with neutrons.
Which slit does the particle pass?
A meter measures
which slit the electron
passes through. As a
result, no interference
fringes are observed on
the screen.
Atomic Structure
of Benzene
deduced from
neutron diffraction.
Black circles are
the carbon atoms,
while the blue
circles denote the
hydrogen atoms.
4.3 Uncertainty Relationships for Classical Waves
∆𝑥 ≈ 𝜆∆𝜆~𝜀𝜆∆𝑥∆𝜆~𝜀𝜆9
∆𝑡 ≈ 𝑇∆𝑇~𝜀𝑇∆𝑡∆𝑇~𝜀𝑇 9
Using the de Broglie Relationship:
𝑝 = <
<
Δ𝑝 = − ? Δ𝜆
then
=
=
Likewise, using Einstein's relationship:
𝐸 = ℎ𝑓
So,
∆𝐸 = ℎ∆𝑓
then
∆𝐸 = ℎ −
B
C?
∆𝑓 = −
∆𝑇
4.4 Heisenberg Uncertainty Relationships
where ℏ ≡
<
9G
∆𝑥∆𝑝~𝜀ℎ
∆𝑥∆𝑝 ≥ ℏ
∆𝐸∆𝑡~𝜀ℎ
∆𝐸∆𝑡 ≥ ℏ
and
𝜀 = B
9G
The larger the Δ𝑥, the smaller the Δ𝑝H .
B
C?
∆𝑇
∆𝒙
∆𝒑𝒙
Examples:
(1) Can electrons reside in the nucleus ?
(2) What is the lifetime of a 𝜌L meson (mass = 775𝑀𝑒𝑉/𝑐 9 ,
and Δ𝑀 = 149𝑀𝑒𝑉/𝑐 9 ) ?
4.5 Wave Packets
The purpose of this section is try and construct a wave (wave
function) that describes the position and motion of a particle.
Two waves of slightly different wavelengths:
𝑦 𝑥 = 𝐴B cos 𝑘B 𝑥 + 𝐴9 cos 𝑘9 𝑥
𝑦 𝑥 = 𝐴B cos 2𝜋𝑥/𝜆B + 𝐴9 cos(2𝜋𝑥/𝜆9 )
𝜋𝑥 𝜋𝑥
𝜋𝑥 𝜋𝑥
𝑦 𝑥 = 2𝐴𝑐𝑜𝑠
−
𝑐𝑜𝑠
+
𝜆B 𝜆9
𝜆B 𝜆9
If 𝜆B and 𝜆9 are close together, then:
𝑦 𝑥 = 2𝐴𝑐𝑜𝑠
𝛥𝜆𝜋𝑥
2𝜋𝑥
𝑐𝑜𝑠
𝜆bc
𝜆9bc
1. The first cosine term is the shaping envelope.
2. The second cosine term represents a wavelength 𝜆bc .
1.
2.
3.
This results in a repeating “beat” pattern.
See example.
This does not localize the wave packet.
How can we localize the “continuous” wave into a single
wave packet?
The book describes three different ways of doing this:
1.
modulating the wave using the
𝑦 𝑥 =
2.
9d
H
𝑠𝑖𝑛
e=GH
𝑐𝑜𝑠
=?f
B
H
technique:
9GH
=f
modulating the wave using
the Gaussian technique:
𝑦 𝑥 =
e=GH
g9
=?f
𝐴𝑒
?
cos
2𝜋𝑥
𝜆L
3. Adding together waves of differing amplitude and
wavelength, but the wavelengths form a continuous rather
than a discrete set of waves.
𝑦 𝑥 =
𝐴h cos 𝑘h 𝑥 → 𝐴 𝑘 𝑐𝑜𝑠 𝑘𝑥 𝑑𝑘
This is called a Fourier cosine transformation.
See the example on my website:
A better approximation of the shape of the wave packet can be
found by letting 𝐴 𝑘 vary according to a Gaussian distribution:
𝐴 𝑘 = 𝐴L 𝑒 g
igif ? /9 ei
?
See the comparison of 𝛥𝑝 to 𝛥𝑥 shown in the previous plot
earlier in this lecture:
Using the Fourier cosine transform with the Gaussian 𝐴 𝑘 :
𝑦 𝑥 = 𝐴L 𝛥𝑘 2𝜋𝑒
eiH ? /9
cos 𝑘L 𝑥
4.6
The Motion of a Wave Packet
A traveling wave:
𝐴cos(𝑘𝑥 − 𝜔𝑡)
1. Has a unique momentum 𝑝 = ℏ𝑘
2. Has an infinite extent
𝛥𝑥 = ∞
Let’s look at the structure of a traveling wave packet made up of
two waves close in wave-number and close in frequency.
𝛥𝑘
𝛥𝜔
𝑘B + 𝑘9
𝜔B + 𝜔9
𝑦 𝑥, 𝑡 = 2𝐴
𝑥 −
𝑡 𝑐𝑜𝑠
𝑥 −
𝑡
2
2
2
2
1. The first term describes the overall shape of the waveform.
2. The second term represents the rapid variation of the wave
within the envelopes described by the first term.
The speed of the overall waveform is:
𝛥𝜔
𝑣mnLop = 2 = 𝛥𝜔
𝛥𝑘
𝛥𝑘
2
The speed of the underlying wave is:
𝜔B + 𝜔9
𝜔bc
𝜔
2
𝑣p<bqr = = = 𝑘B + 𝑘9
𝑘bc
𝑘
2
The Group Speed of a deBroglie Wave
𝑣mnLop = 𝑑𝜔
𝑑𝐸 ℏ
𝑑𝐸
= = 𝑑𝑘
𝑑𝑝 ℏ
𝑑𝑝
For a classical particle having only kinetic energy
𝐸 = 𝐾 =
So,
𝑣mnLop = 𝑝9
2𝑚
𝑑𝐸
𝑝
= = 𝑣
𝑑𝑝
𝑚
Thus,
𝑣mnLop = 𝑣pbnuhvwr