Quantum-Mechanical Particle in a Cylinder

This is one of the three classic particle-in-a-box problems in elementary quantum mechanics, along with the cuboid and the sphere. For a particle of mass in a right circular cylinder of radius and altitude , the Schrödinger equation in cylindrical coordinates , , can be written as

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The equation is separable, as . The and factors are elementary functions: , with and , with . The equation for can be reduced to with the boundary condition . The solutions are Bessel functions such that is the zero of the Bessel function . The total energy is then given by .

This Demonstration shows contour plots of the wavefunction through horizontal cross sections of the cylinder, representing constant values of between 0 and . The wavefunction is positive in the light blue regions.