Quantum mechanics of the Tonks gas

A model one-dimensional system of N unpenetrable particles of mass M and diameter d, with elastic collisions only has been considered classically by L. Tonks, Phys. Rev. 50,955, (1936). Here the same system is treated quantum-mechanically. A separation in center of-mass- and relative motion is made. The eigenfunctions of the Hamilton operator and the energy eigenvalues of the relative motion are discussed. Then a transition to periodic boundary conditions is made. From the energy eigenvalues of the relative motion the partition function and then the thermodynamic quantities internal energy, pressure, entropy are derived. This procedure is tested in the particular cases N = 3, 5, 10. (qui)