very good book! i strongly recommend reading it! but not right now! we first have to talk…....
TRANSCRIPT
Very good book! I stronglyrecommend reading it!
But not right now!
We first have to talk….
…about troubles with some other things.
Trouble with the de Broglie wavesIn many situations particles behave like “properparticles” – i.e., like an object of very small size,
with all its mass enclosed within that size.
Examples of effects, in which particles revealtheir “particle nature”:● Electrons, for instance, can be detected by a photographic film – a single electron pro- duces a tiny dark spot on the film. Or, one can use a fluorescent screen: then a single electron produces a microscopic “flash” on it. We can determine “where it was*” as well as “when it was there”.
* Within certain accuracy limits, we will get to this shortly.
● Moving charged particles leave visible “tracks” in cloud chambers and in bubble chambers.
The list can be much longer…
Cloud chamber: first obs-ervation of the positron(electron’s antiparticle)
Particle tracks in a liquidhydrogen bubble chamber
at CERN, Switzerland
However, we have learned that in other types of experiments the same particles behave liketypical waves! (Bragg diffraction, double-slitinterference).
SO, WHAT’S GOING ON?! ARE PARTICLES REALLY PARTICLES? PERHAPS
THEY ARE WAVES?
NO – definitely one cannot say that “particlesare waves”.
Why? Because particles, as we have said, are“localized objects”. We will see shortly that a particle position cannot be determined with an infinitely high “precision” – but certainly one can determine with micrometer accuracy “where the particle is”.
And how about waves?
Recall – the simplest wave (a plane wave) propagating along certain direction (call it x) can be mathematically described as follows:
frequency"angular " thecalled - 2
;"wavenumber" thecalled - 2
:notationsimpler a useusually We
amplitude. theis and period,
noscillatio theis h, wavelengt theis where
22sin
T
π
k
A
T
tT
xA
Now, the equation has a simpler form:
)( :usecan you
numbers,complex prefer you if or,
)sin(
tkxiAe
tkxA
Now, please tell me: where is this wave?
Answer: EVERYWHERE! This function spans
from x = - to x = + . Over the entire Universe!A wave is not localized, so a particle cannot be a single wave!
PAY ATTENTION, PLEASE!I use Microsoft Equation Editor for preparing
my slide presentations (any other choice? :o) )The appearance of some Roman and Greek
characters in MSEqEd is very similar:
small , capital :Greek
small , capital :Roman
vV Velocity sym-
bol in the Text
Frequencysymbol in the Textbook
Therefore, when velocity and frequency
appear together, we will use capital V for velocity
Is this the only reason why not?No! Another reason is the velocity.
The wave propagation velocity, as you certainly remember from Ph212, is:
). velocity"phase(" velocity wave
2
2
:notation , theit to transformsLet'
wave
wave
wave
kV
kTTV
kT
V
Now, consider a non-relativistic particle:
VmV
mV
p
KmVp
mVK
2
121
and 2
22
From the de Broglie Equations we get:
khh
p
K
hKh
p
22 :So
and
Comparing the two results for K/p , we obtain for the particle: k
V
2particle
The particle velocity is twice as largeas the de Broglie wave’s velocity!
CONCLUSION
Because of its “delocalized” character,
and its velocity which is inconsistent with the velocity of the particle it
represents, a wave – at least a simple plane wave – cannot be used as a mathematicaldescription of a particle.
Epilogue: Our goal
We have to construct a mathematical description of a particle that providesa proper localization and velocity,but still accounts for the wave-likeproperties revealed by experiments.