RELATIVISTIC HARMONIC OSCILLATOR

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RELATIVISTIC HARMONIC OSCILLATOR

• Li Zhifeng Wolfgang Lucha Franz F.
  Schoeberl
• University of Vienna

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Introduction and Motivation

• Generalization of Schroedinger operator towards
  relativistic Hamiltonian H,
  Spinless Salpeter Equation
• Its difficult to obtain the analytical
  solution of this equation.
• The relativistic harmonic oscillator problem
  with Hamiltonian
  .
• A semianalytical approach was presented to this
  problem by using a recurrence scheme and get a
  compact expression of eigenfunction.

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Contents

• 1. The Relativistic Harmonic-Oscillator(RHO)
• 2. Analytical approach to the solution of RHO
  bound-state eigenfunction
• 3. Energy eigenvalues of the bound states
• 4. Summary and Conclusion

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1.The Relativistic Harmonic-Oscillator(RHO)

• Hamiltonian of RHO
• 
  
• the eigenstates equation
  
• We do some transformations

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Transformation

• 
  
• We get
• 
  
• In momentum-space
• with

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2. Analytical approach to the solution of RHO
bound-state eigenfunction

• Introduce the radial wave function
• 
  
• which satisfies a radial equation
• 
  
• We have and

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• Reconstruction of
• We construct in form of
  Taylor-series expansion
• 
  
• with the expansion coefficients
• 
  
  
• The first three terms
• 
  

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• Analysis of
• Using the Leibnizs theorem we get the recurrence
  relation
• 
  
• 
  
  
• 
  
  
• 
  

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here By inspection of the function
we find
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• We obtain and for
• 
  
  
• where
• 
  
  
• So the decomposes into two parts
• 
  

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• The expression of
• 
  

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• The analytical expressions for
• 
  
• 
  

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3. Energy eigenvalues of the bound states.

• In order to fulfill the normalization condition
• 
  
• the solution must vanish in the limit
• 
  
• for the practical purposes, the infinite
  series has to be cut to a large number N
• 
  

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• According to minimum-maximum principle,
• 
  
  
• where
• 
  
  
• 
  has to be computed numerically

• According to minimum-maximum principle,
• 
  
  
• where
• 
  
  
• 
  has to be computed numerically

0 1.37608 1 3.18131 2 4.99255 3 6.80514 4 8.61823
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• Comparison of the values
  
• with

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4. Summary and Conclusions

• We use power series to express the reduced
  eigenfunctions of the Hamiltonian
  
• with expression coefficients
  which determined by the recurrence
  relation.
• And we hope to find a way to construct the
  complete analytic solution.
• hep-ph/0501268
• Journal of Mathematical Physics 46, 1 (2005)

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• Thank you very much!