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From classical periodic orbits in integrable \( \pi\)-rational billiards to quantum energy spectrum

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Abstract.

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.

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Authors and Affiliations

  1. Department of Physics, N.I.T. Silchar, 788010, Cachar, India

    Subhasis Panda

  2. Department of Physics, IIT Kharagpur, 721302, Kharagpur, India

    Sabyasachi Maulik & S. Pratik Khastgir

  3. Saha Institute of Nuclear Physics, 1/AF Bidhannagar, 700064, Kolkata, India

    Sabyasachi Maulik

  4. Department of Physics, City College, 102/1, Raja Rammohan Sarani, 70009, Kolkata, India

    Somdeb Chakraborty

  5. Centre for Theoretical Studies, IIT Kharagpur, 721302, Kharagpur, India

    S. Pratik Khastgir

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  1. Subhasis Panda
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  4. S. Pratik Khastgir
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Correspondence to Subhasis Panda.

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Panda, S., Maulik, S., Chakraborty, S. et al. From classical periodic orbits in integrable \( \pi\)-rational billiards to quantum energy spectrum. Eur. Phys. J. Plus 134, 308 (2019). https://doi.org/10.1140/epjp/i2019-12834-4

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