The Harmonic Oscillator in Extended Relativistic Dynamics

Mechanics

Oct 29, 2013 (4 years and 8 months ago)

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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
-

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 (


)

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
ω

12

Why can’t the CHO explain the radiation for large
ω?

Plank introduced a postulate that can explain
ω.

13

CHO can not Explain the Radiation
for Large ω.

Can Special Relativity Explain the

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

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
)

-

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