Digitization of the harmonic oscillator
in Extended Relativity
Yaakov Friedman
Jerusalem College of Technology
P.O.B. 16031 Jerusalem 91160, Israel
email:
friedman@jct.ac.il
Geometry Days in Novosibirsk 2013
Relativity principle
symmetry
•
Principle
of Special Relativity for inertial systems
•
General
Principle of relativity for accelerated
system
The transformation
will be a symmetry, provided
that the axes
are chosen
symmetrically.
2
Consequences of the symmetry
•
If the time does not depend on the
acceleration:
𝛾
=
1
and
𝜅
=
0

Galilean
•
If
the time depends also directly on the
acceleration
:
𝜅
≠
0
(
ER)
3
Transformation between accelerated
systems under ER
•
Introduce a metric
𝑔
(
𝜇
,
−
1
,
−
1
,
−
1
)
on
(
;
)
which makes
the symmetry
S
g
self

adjoint
or an
isometry
.
•
Conservation
of
interval:
2
=
𝜇
2
−
2
•
There
is a
maximal acceleration
𝑚
=
𝜇
,
which
is a
universal
constant
with
𝜇
=
𝑔
𝜅
•
The proper velocity

time transformation (parallel axes)
•
Lorentz type
transformation with:
4
The Upper Bound for Acceleration
•
If the acceleration affects the rate of the
moving clock then:
–
there is a universal maximal acceleration
(Y. Friedman, Yu. Gofman,
Physica
Scripta
,
82
(
2010
)
015004
.)
–
There is an additional Doppler shift due to
acceleration
(
Y. Friedman, Ann. Phys. (Berlin)
523
(
2011
)
408
)
5
Experimental Observations of the
Accelerated Doppler Shift
•
Kündig's experiment measured the transverse
Doppler shift
(W.
Kündig
, Phys. Rev.
129
(
1963
)
2371
)
•
Kholmetskii
et al: The Doppler shift observed
differs from the one predicted by Special
Relativity.
(A.L.
Kholmetski
, T.
Yarman
and O.V.
Missevitch
,
Physica
Scripta
77 035302
(
2008
))
•
This additional shift can be explained with
Extended Relativity
. Estimation for maximal
acceleration
(Y. Friedman
arXiv:
0910.5629
)
6
𝑚
=
10
21
/
2
Further Evidence
•
DESY (
1999
) experiment using nuclear forward
scattering with a rotating disc observed the
effect of rotation on the spectrum. Never
published. Could be explained with ER
•
ER model for a hydrogen and using the value
of ionization of hydrogen leads approximately
to the value of the maximal acceleration (
)
•
Thermal radiation curves predicted by
ER are similar to the observed ones
7
Classical Mechanics
8
Classical Hamiltonian
9
Which can be rewritten as
•
The two parts of the Hamiltonian are integrals
of velocity and acceleration respectively.
𝐻
,
=
2
2
+
𝑉
(
)
1
𝐻
,
=
𝑢
0
−
𝑥
0
−
′
−
`
Hamiltonian System
10
•
The Hamiltonian System is symmetric in
x
and
u
as
required by
Born’s
Reciprocity
=
=
𝐹
=
Classical Harmonic Oscillator (CHO)
11
•
The kinetic energy and the potential energy are quadratic
expressions in the variables u and
ω
x.
•
The Hamiltonian
=
−
=
−
2
𝐻
,
=
𝑢
0
−
𝑥
0
=
𝑢
0
−
𝜔
𝑥
0
Example: Thermal Vibrations of
Atoms in Solids
•
CHO models well such vibrations and predicts
the thermal radiation for small
ω
12
•
Why can’t the CHO explain the radiation for large
ω?
Plank introduced a postulate that can explain
the radiation curve for large
ω.
13
CHO can not Explain the Radiation
for Large ω.
Can Special Relativity Explain the
Radiation for Large ω?
•
Rate of clock depends on the velocity
•
Magnitude of velocity is
b
ounded by c
•
Proper velocity u and Proper time
τ
14
Special Relativity
=
𝜏
Special Relativity Hamiltonian
15
𝐻
,
=
2
𝛾
+
𝑉
=
2
1
+
2
2
+
𝑉
Special Relativity Harmonic Oscillator
(SRHO)
𝐻
,
=
2
1
+
2
2
+
2
2
2
•
The kinetic energy is
hyperbolic
in ‘u’
The potential energy is
quadratic
‘
ω
x’
Born’s
Reciprocity
is lost
Can SRHO Explain Thermal Vibrations?
•
Typical amplitude and frequencies for Thermal
Vibrations
•
Therefore SRHO can’t explain thermal
vibrations in the non

classical region.
•
But
16
𝐴
−
𝐴
~
10
−
9
~
10
15
−
1
𝑚𝑎𝑥
=
𝐴
~
10
6
≪
𝑚𝑎𝑥
=
𝐴
2
~
10
21
2
Extended Relativity
17
Extended Relativistic Hamiltonian
18
•
For Harmonic Oscillator
•
Born’s
Reciprocity is restored
•
Both terms are hyperbolic
Extends both Classical and Relativistic Hamiltonian
𝐻
,
=
1
+
2
2
𝑢
0
−
(
)
1
+
(
)
2
𝑚
2
𝑥
0
𝐻
,
=
2
1
+
2
2
+
𝑚
2
2
1
+
4
2
𝑚
2
Effective Potential Energy
19
(a)
(b)
(c)
(d)
=
5
∗
10
14
−
1
=
7
∗
10
14
−
1
=
9
∗
10
14
−
1
=
10
21
−
1
The effective potential is linearly confined
The confinement is strong when
is significantly large
20
Harmonic Oscillator Dynamics for
Extremely Large
ω
Harmonic Oscillator Dynamics for Extremely Large
ω
•
Acceleration (digitized)
21
𝑉
𝑞
=
𝑚
=
=
−
𝐻
=
𝑚
<
0
−
𝑚
>
0
•
Velocity
22
Harmonic Oscillator Dynamics for Extremely Large
ω
•
The spectrum of ‘u’ coincides with the spectrum of
energy of the
Quantum
Harmonic Oscillator
=
2
𝑇
𝑚
𝜋
2
−
1
𝑘
2
+
1
2
sin
2
𝜋
2
+
1
𝑇
∞
𝑘
=
0
•
Position
23
Harmonic Oscillator Dynamics for Extremely Large
ω
=
𝐻
=
1
+
2
2
=
𝑚
1
+
𝑚
2
2
Transition
b
etween Classical and
Extended Relativity
24
•
Acceleration
25
Transition between Classical and Non

classical
Regions
(a)
=
30
∗
10
1
4
−
1
=
7
∗
10
14
−
1
=
9
∗
10
14
−
1
=
15
∗
10
14
−
1
(b)
(c)
(d)
•
Velocity
26
Transition between Classical and Non

classical
Regions
=
30
∗
10
1
4
−
1
=
7
∗
10
14
−
1
=
9
∗
10
14
−
1
=
15
∗
10
14
−
1
(a)
(b)
(c)
(d)
Comparison between Classical and
Extended Relativistic Oscillations
27
28
Comparison between Classical and Extended
Relativistic Oscillations
=
10
15
−
1
29
Comparison between Classical and Extended
Relativistic Oscillations
=
10
16
−
1
Comparison between Classical and
Extended Relativistic Oscillations
•
Comparison between the
ω
and the effective
ω
.
30
0
1E+15
2E+15
3E+15
4E+15
5E+15
6E+15
0
5E+15
effective ω
ω
Clasical
ERD
ERD limit
Acceleration for a given
at different
Amplitudes (Energies)
31
(a)
A=
10
^

10
(b)
A=
10
^

9
(c)
A=
5
*
10
^

9
(d)
A=
10
^

8
(a)
(d)
(c)
(b)
Comparison between Classical and
Extended Relativistic Oscillations
32
Non Classical region
Classical
region
(slide
18
)
square wave
?
A
ω
2
cos(
ω
t)
a(t)
triangle wave (slide
19
)
A
ω
sin(
ω
t)
u(t)
(slide
20
)

䄠
cos
(
ω
t)
x(t)
16
𝐴
2
2
+
32
𝐴
𝑚
2
π
/
ω
T
m
0
Aa
m
m
0
A
2
ω
2
/
2
E

E
0
2
π
/T (
2
k+
1
)
: k=
0
,
1
,
2
,
3
…
{
?˜
}
spectrum
Testing the Acceleration
of a Photon
33
•
CL
:
𝒕
=
𝒙
𝒄
•
ER
:
𝒕
=
𝒙
+
𝜶𝒙
𝒄

•
𝜶
=
𝒄
𝒂
𝒎
≈
𝒄𝒎
ER
CL
The future of ER
•
More experiments
•
More theory: EM, GR, QM (hydrogen),
Thermodynamics
34
Thanks
Any questions?
35
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