https://doi.org/10.1140/epjp/i2019-12834-4
Regular Article
From classical periodic orbits in integrable 
-rational billiards to quantum energy spectrum
1
Department of Physics, N.I.T. Silchar, 788010, Cachar, India
2
Department of Physics, IIT Kharagpur, 721302, Kharagpur, India
3
Saha Institute of Nuclear Physics, 1/AF Bidhannagar, 700064, Kolkata, India
4
Department of Physics, City College, 102/1, Raja Rammohan Sarani, 70009, Kolkata, India
5
Centre for Theoretical Studies, IIT Kharagpur, 721302, Kharagpur, India
* e-mail: subhasis@phy.nits.ac.in
Received:
20
March
2019
Accepted:
16
April
2019
Published online:
28
June
2019
In the present paper, we establish a remarkable connection between the length of the periodic orbit of a classical particle enclosed in a class of 2-dimensional planar billiards and the energy of a quantum particle confined to move in an identical region with infinitely high potential wall on the boundary. We observe that the quantum energy spectrum of the particle is in exact one-to-one correspondence with the spectrum of the amplitude squares of the periodic orbits of a classical particle for the class of integrable billiards considered. We have established the results by geometric constructions and exploiting the method of reflective tiling and folding of classical trajectories. We have further extended the method to 3-dimensional billiards, for which exact analytical results are scarcely available --exploiting the geometric construction, we determine the exact energy spectra of two new tetrahedral domains which we believe are integrable. We test the veracity of our results by comparing them with numerical results.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2019
