Reciprocal Symmetry and Classical Discrete Oscillator Incorporating
  Half-Integral Energy Levels
Microsoft Word - Class. oscl incorp. half-inegral.31.03.doc
Reciprocal Symmetry and Classical Discrete Oscillator Incorporating Hal-Integral 
Energy Levels 
Mushfiq Ahmad 
Department of Physics, Rajshahi University, Rajshahi, Bangladesh 
E-mail: mushfiqahmad@ru.ac.bd 
Abstract 
Classical oscillator differential equation is replaced by the corresponding (finite time) 
difference equation. The equation is, then, symmetrized so that it remains invariant under 
the change d  -d, where d is the smallest span of time. This symmetric equation has 
solutions, which come in reciprocally related pairs. One member of a pair agrees with the 
classical solution and the other is an oscillating solution and does not converge to a limit 
as d 0. This solution contributes to oscillator energy a term which is a multiple of half-
integers.  
1. Introduction 
The differential equation  
=                                                         (1.1) 
has a unique solution. The corresponding finite difference equation has more solutions1. 
When the function represents a harmonic oscillator, different solutions will contribute to 
oscillator energy in different ways. We intend to study these contributions and compare 
them to the corresponding quantum mechanical values.     
2. Oscillator Finite Difference Equation   
Classical simple harmonic oscillator function f (with angular speed w) satisfies 
differential equation (1.1) 
To exploit its symmetry properties we replace the above differential equation by the 
corresponding symmetric finite difference equation2   
± = giW
),( δ
                                                   (2.1)  
where  
),(),(
),( −−+
= ±±±
twgtwg
                                    (2.2) 
The above difference quotient has the following symmetry under the change δδ −→  
),(),( δδ tD
Dg ±± =
                                                (2.3) 
We require that at least one of the solutions, +g , of (2.1) should go over to (1.1) in the 
limit 0→δ  
⎯⎯→⎯=
0)(),( δδ
=                              (2.4) 
With  
fg ⎯⎯→⎯
→+ 0δ  and  wW ⎯⎯→⎯ →0δ                                      (2.5) 
3. Reciprocal Symmetry 
Let ±g be of the form 
δ/)( tag ±± ±= so that 
),()1(),( / twgtwg t −+ −=
δ                                            (3.1) 
Consider equation (2.1) 
),(),( −−+ ++ twgtwg ),(. twgiW +=                                (3.2) 
Using (3.1) we find that −g also satisfies the equation. This establishes reciprocal 
symmetry of (2.1), that the equation remains invariant under transformation (3.1).   
4. Reciprocal symmetric Solutions 
(2.1) has a pair of solutions 
( )iwt
±=± exp)1(
sin(1
sin(1
)2/( /
                    (4.1) 
+g and −g  satisfy (2.1) with  
δ )sin(w
W =                                                      (4.2) 
We may write 
).exp()exp(
exp tiwiwti
g ++ =⎟
                             (4.3) 
).exp()exp(
exp tiwiwti
g −− =−⎟
                         (4.4) 
where  
wywnw +=+= ++ δπ /)2(                                        (4.5) 
wywnw −=−+= −− δπ /)12(                                      (4.6) 
5. Classical and Half-Integral Energy Levels 
The energy of the oscillator is proportional to 
22222 }/)2{(2}/)2{(2)( wwnnwwyyw ++=++= +++ δπδπ               (5.1) 
22222 }/)2/1{(4}/)12{(2)( wwnnwwyyw ++++=++= −+− δπδπ          (5.2) 
For n=0 (5.1) gives the classical value. The middle term of (5.2) is a product of half-
integers and w. To this extent it corresponds to quantum mechanical value.  
4. Conclusion 
We have replaced oscillator differential equation by the corresponding symmetric 
discrete equation (2.1). This has brought to surface important parts of oscillator function, 
which were lost in the conventional solution. These parts contain discrete – integral and 
half integral -- energy levels.
                                                                                                                                                                             
1 Mushfiq Ahmad. Reciprocal Symmetry and Equivalence between Relativistic and 
Quantum Mechanical Concepts.  http://www.arxiv.org/abs/math-ph/0611024 
2  Mushfiq AhmadReciprocal Symmetric and Origin of Quantum Statistics.  
http://www.arxiv.org/abs/physics/0703194
