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.