Transcript The Harmonic Oscillator in Extended Relativistic Dynamics

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.
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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)
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Transformation between accelerated
systems under ER
• Introduce a metric 𝑑𝑖𝑎𝑔(𝜇, −1, −1, −1) on
(𝑡; 𝑢) which makes the symmetry Sg self-adjoint or an
isometry.
• Conservation of interval: 𝑑𝑠 2 = 𝜇𝑑𝑡 2 − 𝑑𝑢 2
• There is a maximal acceleration 𝑎𝑚 = 𝜇, which is a
𝑔
universal constant with 𝜇 =
𝜅
• The proper velocity-time transformation (parallel axes)
• Lorentz type transformation with:
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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)
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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)
𝑎𝑚 = 1021 𝑐𝑚/𝑠 2
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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 (𝑎𝑚)
• Thermal radiation curves predicted by
ER are similar to the observed ones
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Classical Mechanics
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Classical Hamiltonian
𝑝2
𝐻 𝑝, 𝑥 =
+ 𝑉(𝑥)
2𝑚
Which can be rewritten as
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𝐻 𝑥, 𝑢 =
𝑚
𝑢 − 𝑜𝑏𝑗𝑒𝑐𝑡 ′ 𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑢
𝑥
𝑣𝑑𝑣 −
0
𝑎 𝑦 𝑑𝑦
0
𝑎 − 𝑜𝑏𝑗𝑒𝑐𝑡`𝑠 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
• The two parts of the Hamiltonian are integrals
of velocity and acceleration respectively.
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Hamiltonian System
𝑑𝑥
=𝑢
𝑑𝑡
𝑑𝑢 𝐹
= =𝑎
𝑑𝑡 𝑚
• The Hamiltonian System is symmetric in x and u as
required by Born’s Reciprocity
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Classical Harmonic Oscillator (CHO)
𝑘
𝑎 𝑥 = − 𝑥 = −𝜔2 𝑥
𝑚
• The Hamiltonian
𝑢
𝐻 𝑥, 𝑢 = 𝑚
𝑥
𝑣𝑑𝑣 − 𝑚
0
𝑢
𝑎 𝑦 𝑑𝑦 = 𝑚
0
𝜔𝑥
𝑣𝑑𝑣 − 𝑚
0
𝑦𝑑𝑦
0
• The kinetic energy and the potential energy are quadratic
expressions in the variables u and ωx.
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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?
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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?
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Special Relativity
• Rate of clock depends on the velocity
• Magnitude of velocity is bounded by c
• Proper velocity u and Proper time τ
𝑑𝑥
𝑢=
𝑑𝜏
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Special Relativity Hamiltonian
𝐻 𝑥, 𝑢 = 𝑚𝑐 2 𝛾 𝑣 𝑢
+ 𝑉 𝑥 = 𝑚𝑐 2
𝑢2
1+ 2 +𝑉 𝑥
𝑐
Special Relativity Harmonic Oscillator
(SRHO)
𝐻 𝑥, 𝑢 = 𝑚𝑐 2
𝑢2 𝑚𝜔2 𝑥 2
1+ 2 +
𝑐
2
• The kinetic energy is hyperbolic in ‘u’
The potential energy is quadratic ‘ωx’
Born’s Reciprocity is lost
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Can SRHO Explain Thermal Vibrations?
• Typical amplitude and frequencies for Thermal
Vibrations
𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 − 𝐴~10−9 𝑐𝑚
𝑣𝑚𝑎𝑥
𝜔~1015 𝑠 −1
𝑐𝑚
= 𝐴𝜔~10
≪𝑐
𝑠
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• Therefore SRHO can’t explain thermal
vibrations in the non-classical region.
• But
𝑎𝑚𝑎𝑥 =
𝐴𝜔2 ~1021
𝑐𝑚
𝑠2
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Extended Relativity
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Extended Relativistic Hamiltonian
𝑢
𝐻 𝑥, 𝑢 = 𝑚
0
𝑥
𝑣
1+
𝑣2
𝑑𝑣 − 𝑚
𝑐2
0
𝑎(𝑦)
𝑑𝑦
𝑎(𝑦)2
1+ 2
𝑎𝑚
Extends both Classical and Relativistic Hamiltonian
• For Harmonic Oscillator
𝐻 𝑥, 𝑢 = 𝑚𝑐 2
2
𝑢2
𝑎𝑚
𝜔4𝑥 2
1+ 2 +𝑚 2 1+ 2
𝑐
𝜔
𝑎𝑚
• Born’s Reciprocity is restored
• Both terms are hyperbolic
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Effective Potential Energy
(a)
𝑎 𝜔 = 5 ∗ 1014 𝑠 −1
𝑏 𝜔 = 7 ∗ 1014 𝑠 −1
𝑐 𝜔 = 9 ∗ 1014 𝑠 −1
(b)
(c)
𝑑 𝜔 = 1021 𝑠 −1
(d)
The effective potential is linearly confined
The confinement is strong when 𝜔 is significantly large
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Harmonic Oscillator Dynamics for
Extremely Large ω
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Harmonic Oscillator Dynamics for Extremely Large ω
𝑉𝑞 𝑥 = 𝑎𝑚 𝑥
• Acceleration (digitized)
𝑑𝑢
𝜕𝐻
𝑎𝑚
𝑎 𝑡 =
=−
=
−𝑎𝑚
𝑑𝑡
𝜕𝑥
𝑥<0
𝑥>0
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Harmonic Oscillator Dynamics for Extremely Large ω
• Velocity
2𝑇𝑎𝑚
𝑢 𝑡 =
𝜋2
∞
𝑘=0
−1 𝑘
2𝜋 2𝑘 + 1 𝑡
sin
2
2𝑘 + 1
𝑇
• The spectrum of ‘u’ coincides with the spectrum of
energy of the Quantum Harmonic Oscillator
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Harmonic Oscillator Dynamics for Extremely Large ω
• Position
𝑑𝑥 𝜕𝐻
=
=
𝑑𝑡 𝜕𝑢
𝑢 𝑡
𝑢 𝑡 2
1+ 2
𝑐
=
𝑎𝑚 𝑡
𝑎𝑚 𝑡
1+
𝑐2
2
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Transition between Classical and
Extended Relativity
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Transition between Classical and Non-classical
Regions
• Acceleration
(d)
(c)
𝑎 𝜔 = 7 ∗ 1014 𝑠 −1
𝑏 𝜔 = 9 ∗ 1014 𝑠 −1
𝑐 𝜔 = 15 ∗ 1014 𝑠 −1
𝑑 𝜔 = 30 ∗ 1014 𝑠 −1
(b)
(a)
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Transition between Classical and Non-classical
Regions
• Velocity
𝑎 𝜔 = 7 ∗ 1014 𝑠 −1
𝑏 𝜔 = 9 ∗ 1014 𝑠 −1
𝑐 𝜔 = 15 ∗ 1014 𝑠 −1
(a)
(c)
𝑑 𝜔 = 30 ∗ 1014 𝑠 −1
(b)
(d)
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Comparison between Classical and
Extended Relativistic Oscillations
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Comparison between Classical and Extended
Relativistic Oscillations
𝜔 = 1015 𝑠 −1
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Comparison between Classical and Extended
Relativistic Oscillations
𝜔 = 1016 𝑠 −1
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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
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Acceleration for a given 𝜔 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
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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
𝐴
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𝑐2
+ 32 𝐴 𝑎𝑚
m0A2ω2/2
m0Aam
{ω}
2π/T (2k+1) : k=0,1,2,3…
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Testing the Acceleration of a Photon
• CL: 𝒕 =
𝒙
𝒄
• ER: 𝒕 =
• 𝜶=
ER
𝒄𝟐
𝒂𝒎
𝒙𝟐 +𝟐𝜶𝒙
𝒄
|
≈ 𝟏𝒄𝒎
CL
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The future of ER
• More experiments
• More theory: EM, GR, QM (hydrogen),
Thermodynamics
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Thanks
Any questions?
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