The Harmonic Oscillator in Extended Relativistic Dynamics

baconossifiedMechanics

Oct 29, 2013 (3 years and 9 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
οƒ 

symmetry

β€’
Principle
of Special Relativity for inertial systems

β€’
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





β€’
If the time does not depend on the
acceleration:
𝛾

=
1

and
πœ…
=
0
-
Galilean

β€’
If
the time depends also directly on the
acceleration
:
πœ…
β‰ 
0

(
ER)

3

Transformation between accelerated
systems under ER

β€’
Introduce a metric
 𝑔
(
πœ‡
,
βˆ’
1
,
βˆ’
1
,
βˆ’
1
)

on
(

;

)

which makes
the symmetry
S
g

self
-
adjoint

or an
isometry
.

β€’
Conservation
of
interval:


2
=
πœ‡

2
βˆ’

2

β€’
There
is a
maximal acceleration

π‘š
=
πœ‡
,
which
is a
universal
constant
with
πœ‡
=
𝑔
πœ…

β€’
The proper velocity
-
time transformation (parallel axes)



β€’
Lorentz type
transformation with:

4

The Upper Bound for Acceleration

β€’
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

β€’
KΓΌndig's experiment measured the transverse
Doppler shift
(W.
KΓΌndig
, Phys. Rev.
129
(
1963
)
2371
)


β€’
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
))


β€’
This additional shift can be explained with
Extended Relativity
. Estimation for maximal
acceleration
(Y. Friedman
arXiv:
0910.5629
)

6


π‘š
=
10
21

/

2


Further Evidence

β€’
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

β€’
ER model for a hydrogen and using the value
of ionization of hydrogen leads approximately
to the value of the maximal acceleration (


)

β€’
Thermal radiation curves predicted by



ER are similar to the observed ones

7

Classical Mechanics

8

Classical Hamiltonian

9







Which can be rewritten as

β€’
The two parts of the Hamiltonian are integrals
of velocity and acceleration respectively.

𝐻

,

=

2
2

+
𝑉
(

)

1

𝐻

,

=


𝑒
0
βˆ’




π‘₯
0


βˆ’


β€²


 











βˆ’

`


  

Hamiltonian System

10

β€’
The Hamiltonian System is symmetric in
x
and
u

as
required by
Born’s

Reciprocity




=



=
𝐹

=


Classical Harmonic Oscillator (CHO)

11







β€’
The kinetic energy and the potential energy are quadratic
expressions in the variables u and
Ο‰
x.

β€’
The Hamiltonian



=
βˆ’



=
βˆ’

2


𝐻

,

=



𝑒
0
βˆ’





π‘₯
0
=



𝑒
0
βˆ’




πœ”
π‘₯
0

Example: Thermal Vibrations of
Atoms in Solids

β€’
CHO models well such vibrations and predicts
the thermal radiation for small
Ο‰

12

β€’
Why can’t the CHO explain the radiation for large
Ο‰?

Plank introduced a postulate that can explain
the radiation curve for large
Ο‰.

13

CHO can not Explain the Radiation
for Large Ο‰.

Can Special Relativity Explain the
Radiation for Large Ο‰?

β€’
Rate of clock depends on the velocity

β€’
Magnitude of velocity is
b
ounded by c

β€’
Proper velocity u and Proper time
Ο„

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

β€’
The kinetic energy is
hyperbolic

in β€˜u’

The potential energy is
quadratic

β€˜
Ο‰
x’

Born’s

Reciprocity

is lost

Can SRHO Explain Thermal Vibrations?

β€’
Typical amplitude and frequencies for Thermal
Vibrations



β€’
Therefore SRHO can’t explain thermal
vibrations in the non
-
classical region.

β€’
But

16

𝐴 
βˆ’
𝐴
~
10
βˆ’
9



~
10
15

βˆ’
1


π‘šπ‘Žπ‘₯
=
𝐴
~
10
6


β‰ͺ



π‘šπ‘Žπ‘₯
=
𝐴

2
~
10
21


2

Extended Relativity

17

Extended Relativistic Hamiltonian




18

β€’

For Harmonic Oscillator

β€’
Born’s

Reciprocity is restored

β€’
Both terms are hyperbolic

Extends both Classical and Relativistic Hamiltonian

𝐻

,

=



1
+

2

2

𝑒
0
βˆ’



(

)
1
+

(

)
2

π‘š
2

π‘₯
0

𝐻

,

=


2
1
+

2

2
+


π‘š
2

2
1
+

4

2

π‘š
2

Effective Potential Energy

19

(a)

(b)

(c)

(d)



=
5
βˆ—
10
14

βˆ’
1



=
7
βˆ—
10
14

βˆ’
1



=
9
βˆ—
10
14

βˆ’
1



=
10
21

βˆ’
1

The effective potential is linearly confined

The confinement is strong when


is significantly large

20

Harmonic Oscillator Dynamics for
Extremely Large
Ο‰

Harmonic Oscillator Dynamics for Extremely Large
Ο‰

β€’
Acceleration (digitized)

21

𝑉
π‘ž

=

π‘š




=


=
βˆ’
𝐻

=


π‘š















<
0
βˆ’

π‘š











>
0

β€’
Velocity


22

Harmonic Oscillator Dynamics for Extremely Large
Ο‰

β€’
The spectrum of β€˜u’ coincides with the spectrum of
energy of the
Quantum

Harmonic Oscillator



=
2
𝑇

π‘š
πœ‹
2

βˆ’
1
π‘˜
2

+
1
2
sin
2
πœ‹
2

+
1

𝑇
∞
π‘˜
=
0

β€’
Position

23

Harmonic Oscillator Dynamics for Extremely Large
Ο‰



=
𝐻

=


1
+


2

2
=

π‘š

1
+

π‘š

2

2

Transition
b
etween Classical and
Extended Relativity

24

β€’
Acceleration


25

Transition between Classical and Non
-
classical
Regions

(a)



=
30
βˆ—
10
1
4

βˆ’
1



=
7
βˆ—
10
14

βˆ’
1




=
9
βˆ—
10
14

βˆ’
1



=
15
βˆ—
10
14

βˆ’
1

(b)

(c)

(d)

β€’
Velocity


26

Transition between Classical and Non
-
classical
Regions



=
30
βˆ—
10
1
4

βˆ’
1



=
7
βˆ—
10
14

βˆ’
1




=
9
βˆ—
10
14

βˆ’
1



=
15
βˆ—
10
14

βˆ’
1

(a)

(b)

(c)

(d)

Comparison between Classical and
Extended Relativistic Oscillations

27

28

Comparison between Classical and Extended
Relativistic Oscillations


=
10
15

βˆ’
1

29

Comparison between Classical and Extended
Relativistic Oscillations


=
10
16

βˆ’
1

Comparison between Classical and
Extended Relativistic Oscillations

β€’
Comparison between the
Ο‰

and the effective
Ο‰
.

30

0
1E+15
2E+15
3E+15
4E+15
5E+15
6E+15
0
5E+15
effective Ο‰

Ο‰

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
Ο‰
2
cos(
Ο‰
t)

a(t)

triangle wave (slide
19
)

A
Ο‰

sin(
Ο‰
t)

u(t)

(slide
20
)

-
δ„ 
cos
(
Ο‰
t)

x(t)

16
𝐴
2

2

+
32
𝐴

π‘š


2
Ο€
/
Ο‰

T

m
0
Aa
m

m
0
A
2
Ο‰
2
/
2

E
-
E
0

2
Ο€
/T (
2
k+
1
)

: k=
0
,
1
,
2
,
3
…

{
?˜
}

spectrum

Testing the Acceleration
of a Photon



33

β€’
CL
:
𝒕
=
𝒙
𝒄

β€’
ER
:
𝒕
=
𝒙

+
πœΆπ’™
𝒄



|

β€’


𝜢
=
𝒄

𝒂
π’Ž
β‰ˆ
π’„π’Ž

ER

CL

The future of ER

β€’
More experiments

β€’
More theory: EM, GR, QM (hydrogen),
Thermodynamics



34

Thanks


Any questions?

35