de broglie wave velocity

Introduction

According to the theory of Special Relativity;

m = m0/(1 - v2/c2)1/2

m = mass of particle

m0 = rest mass of particle

v = velocity of particle

c = velocity of light

p = mv

p = momentum of particle

E = mc2

E = total energy of particle

These equations have two implications:

A) Since a photon has a rest mass of zero, photons must travel at the
speed of light, otherwise the momentum and the total energy of a photon
would both be zero which can't be so! Having a velocity of the speed
of light means that a photon has an undefined mass (m), since the
formulae for mass becomes 0/0, and so their momentum and their total
energy both have non-zero values.

B) A particle with a non-zero rest mass must travel slower than the
speed of light, otherwise their: mass, momentum, and total energy would
be infinite!


According to the De Broglie theory, all particles (whether they have a
zero rest mass, i.e. are photons, or a non-zero rest mass) have dual
wave properties. The following equations come into play here;

E = hf

E = total energy

h = planck's constant

f =frequency of the wave

So mc2 = hf

So f = mc2/h

L = h/p

L = wavelength

so L = h/mv

Wave velocity = fL

So wave velocity = mc2/h. h/mv

= mc2/ mv

= c2/v


I found an excellent article on the De Broglie wave velocity, which
goes as follows:

Phase and group velocities of De Broglie waves

How fast do De Broglie waves travel? Since a De Broglie wave is
associated with a moving body, one may expect that this wave has the
same velocity as that of the body. Let us see if this is true.

We call the De Broglie wave velocity vp . As seen above, this is equal
to c2/v. The wave velocity vp is the phase velocity. The particle
velocity (v) is the group velocity also called vg. Because the
particle velocity (v) must be smaller than the speed of light (c), for
a particle with a non-zero rest mass, the De Broglie wave velocity is
always larger than c, for such particles. To understand this result,
we must look into the distinction between phase velocity and group
velocity.

We consider a harmonic wave;

.y = Acos(Wt - kx)

The De Broglie waves associated with a moving body cannot be
represented simply by a formulae like this. Instead, the wave
representation of a moving body corresponds to a wave packet or wave
group. A wave group is a superposition of individual waves of
different wavelengths. The interference of the individual waves with
one another results in the variation in amplitude that defines the
group shape. If the velocities of the individual waves are the same,
the velocity of the wave group is the common wave velocity. However,
if the phase velocity varies with wavelength, an effect called
dispersion, the different individual waves do not proceed together. As
a result, the wave group has a velocity different from the phase
velocities of the individual waves. This is the case with De Broglie
waves.
As an example, we consider the case where the wave group consists of
two waves that have the same amplitude A but differ by a small amount
DELW in angular frequency and a small amount DELk in wave number;

.y1 = Acos(Wt - kx)

.y2 = Acos((W + DELW)t - (k +
DELk)x)

The wave group is given by;

y = y1 + y2

= 2Acos(1/2(DELWt - DELkx)).cos(1/2((2W + DELW)t
- (2k + DELk)x))

Since DELW and DELk are small compared to W and k, we find

.y = 2Acos(1/2(DELWt -
DELkx)).cos(Wt - kx)

The above equation represents a wave of angular frequency W and wave
number k whose amplitude is modulated by an angular frequency DELW/2
and a wave number DELk/2.

The effect of the modulation is to produce successive wave groups. The
phase velocity is;

vp = W/k,

and the velocity of the successive wave groups is;

vg = DELW/DELk

When W and k have continuous spreads instead of the two values in the
above discussion, the group velocity is;

vg = dW/dk

Now, the angular frequency of the De Broglie waves associated with a
body of rest mass m0 moving with the velocity v is;

W = 2PIf

= 2PImc2/h

= 2PIm0c2/h(1
- v2/c2)1/2

The wave number of the De Broglie waves is;

.k = 2PI/L

= 2PImv/h

= 2PIm0v/h(1 -
v2/c2)1/2

The group velocity of the De Broglie waves is;

vg = dW/dk

=
(dW/dv)/(dk/dv)


dW/dv = 2PIm0v/h(1 -
v2/c2)3/2

dk/dv = 2PIm0/h(1 -
v2/c2)3/2

Hence, we find;

vg = v

Thus, the group velocity of De Broglie waves is the velocity of the
moving body.

The phase velocity of the De Broglie waves is;

vp = W/k

= (2PIm0c2/h(1 -
v2/c2)1/2)/ (2PIm0v/h(1 - v2/c2)1/2)

= c2/v

The fact that vp > c does not violate the special relativity theory
because vp is the motion of the phase of the wave group, not the motion
of the individual waves that make up the group, and consequently, not
the motion of the body.

END

p.s. the Greek letters: delta, lambda, omega, and pi, I had to enter
as "DEL", "L", "W", and "PI" respectively. Google
could not display them!

.

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