The Harmonic Oscillator in Extended Relativistic Dynamics

Report
Geometry Days in Novosibirsk 2013
Digitization of the harmonic oscillator
in Extended Relativity
Yaakov Friedman
Jerusalem College of Technology
P.O.B. 16031 Jerusalem 91160, Israel
email: [email protected]
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:  = 1 and  = 0-Galilean
• If the time depends also directly on the
acceleration:  ≠ 0 (ER)
3
Transformation between accelerated
systems under ER
• Introduce a metric (, −1, −1, −1) on
(; ) which makes the symmetry Sg self-adjoint or an
isometry.
• Conservation of interval:  2 =  2 −  2
• There is a maximal acceleration  = , which is a

universal constant with  =

• 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)
 = 1021 / 2
6
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
2
 ,  =
+ ()
2
Which can be rewritten as
1
 ,  =

 −  ′  


 −
0
  
0
 − ` 
• The two parts of the Hamiltonian are integrals
of velocity and acceleration respectively.
9
Hamiltonian System

=

 
= =
 
• The Hamiltonian System is symmetric in x and u as
required by Born’s Reciprocity
10
Classical Harmonic Oscillator (CHO)

  = −  = −2 

• The Hamiltonian

 ,  = 

 − 
0

   = 
0

 − 
0

0
• The kinetic energy and the potential energy are quadratic
expressions in the variables u and ωx.
11
Example: Thermal Vibrations of
Atoms in Solids
• CHO models well such vibrations and predicts
the thermal radiation for small ω
• Why can’t the CHO explain the radiation for large ω?
12
CHO can not Explain the Radiation
for Large ω.
Plank introduced a postulate that can explain
the radiation curve for large ω.
Can Special Relativity Explain the
Radiation for Large ω?
13
Special Relativity
• Rate of clock depends on the velocity
• Magnitude of velocity is bounded by c
• Proper velocity u and Proper time τ

=

14
Special Relativity Hamiltonian
 ,  =  2   
+   =  2
2
1+ 2 + 

Special Relativity Harmonic Oscillator
(SRHO)
 ,  =  2
2 2  2
1+ 2 +

2
• The kinetic energy is hyperbolic in ‘u’
The potential energy is quadratic ‘ωx’
Born’s Reciprocity is lost
15
Can SRHO Explain Thermal Vibrations?
• Typical amplitude and frequencies for Thermal
Vibrations
 − ~10−9 

~1015  −1

= ~10
≪

6
• Therefore SRHO can’t explain thermal
vibrations in the non-classical region.
• But
 =
2 ~1021

2
16
Extended Relativity
17
Extended Relativistic Hamiltonian

 ,  = 
0


1+
2
 − 
2
0
()

()2
1+ 2

Extends both Classical and Relativistic Hamiltonian
• For Harmonic Oscillator
 ,  =  2
2
2

4 2
1+ 2 + 2 1+ 2



• Born’s Reciprocity is restored
• Both terms are hyperbolic
18
Effective Potential Energy
(a)
  = 5 ∗ 1014  −1
  = 7 ∗ 1014  −1
  = 9 ∗ 1014  −1
(b)
(c)
  = 1021  −1
(d)
The effective potential is linearly confined
The confinement is strong when  is significantly large
19
Harmonic Oscillator Dynamics for
Extremely Large ω
20
Harmonic Oscillator Dynamics for Extremely Large ω
  =  
• Acceleration (digitized)



  =
=−
=
−


<0
>0
21
Harmonic Oscillator Dynamics for Extremely Large ω
• Velocity
2
  =
2
∞
=0
−1 
2 2 + 1 
sin
2
2 + 1

• The spectrum of ‘u’ coincides with the spectrum of
energy of the Quantum Harmonic Oscillator
22
Harmonic Oscillator Dynamics for Extremely Large ω
• Position
 
=
=
 
 
  2
1+ 2

=
 
 
1+
2
2
23
Transition between Classical and
Extended Relativity
24
Transition between Classical and Non-classical
Regions
• Acceleration
(d)
(c)
  = 7 ∗ 1014  −1
  = 9 ∗ 1014  −1
  = 15 ∗ 1014  −1
  = 30 ∗ 1014  −1
(b)
(a)
25
Transition between Classical and Non-classical
Regions
• Velocity
  = 7 ∗ 1014  −1
  = 9 ∗ 1014  −1
  = 15 ∗ 1014  −1
(a)
(c)
  = 30 ∗ 1014  −1
(b)
(d)
26
Comparison between Classical and
Extended Relativistic Oscillations
27
Comparison between Classical and Extended
Relativistic Oscillations
 = 1015  −1
28
Comparison between Classical and Extended
Relativistic Oscillations
 = 1016  −1
29
Comparison between Classical and
Extended Relativistic Oscillations
• Comparison between the ω and the effective ω.
6E+15
effective ω
5E+15
4E+15
Clasical
3E+15
ERD
2E+15
ERD limit
1E+15
0
0
ω
5E+15
30
Acceleration for a given  at different
Amplitudes (Energies)
(c)
(b)
(a)
(d)
(a)
(b)
(c)
(d)
A=10^-10
A=10^-9
A=5*10^-9
A=10^-8
31
Comparison between Classical and
Extended Relativistic Oscillations
Classical region
Non Classical region
a(t)
Aω2cos(ωt)
square wave (slide 18)
u(t)
Aω sin(ωt)
triangle wave (slide 19)
x(t)
-A cos(ωt)
(slide 20)
T
E-E0
spectrum
2π/ω
2

16
2
+ 32  
m0A2ω2/2
m0Aam
{ω}
2π/T (2k+1) : k=0,1,2,3…
32
Testing the Acceleration of a Photon
• CL:  =


• ER:  =
• =
ER


 +

|
≈ 
CL
33
The future of ER
• More experiments
• More theory: EM, GR, QM (hydrogen),
Thermodynamics
34
Thanks
Any questions?
35

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