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The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring.
If a particle is confined to the motion of an entire ring ranging from 0 to , the particle is subject only to a periodic boundary condition (see particle in a ring). If a particle is confined to the motion of − π 2 {\textstyle -{\frac {\pi }{2}}} to π 2 {\textstyle {\frac {\pi }{2}}} , the issue of even and odd parity becomes important.
In the general time-independent case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form: ^ = ^ + Here, is the mass of the particle, ^ is the momentum operator, and the potential () depends only on the vector magnitude of the position vector, that is, the radial distance from the ...
Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the Lyman (n f =1), Balmer (n f =2), and Paschen (n f =3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted. [30]: 34
Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wave function.
For positive energies, the particle is free to move in either half-space: x < 0 or x > 0. It may be scattered at the delta-function potential. The quantum case can be studied in the following situation: a particle incident on the barrier from the left side (A r). It may be reflected (A l) or transmitted (B r).
Single ions can be investigated in a Penning trap held at a temperature of 4 K. For this the ring electrode is segmented and opposite electrodes are connected to a superconducting coil and the source and the gate of a field-effect transistor. The coil and the parasitic capacitances of the circuit form a LC circuit with a Q of about 50 000. The ...
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field.A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center.