particle in a ring について

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particle in a ring ：ウィキペディア英語版

particle in a ring

In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle

S^1

) is
:

-\frac\nabla^2 \psi = E\psi

== Wave function ==

Using polar coordinates on the 1-dimensional ring of radius R, the wave function depends only on the angular coordinate, and so
:

\nabla^2 = \frac \frac

Requiring that the wave function be periodic in

\ \theta

with a period

2 \pi

(from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions
:

\int_^ \left| \psi ( \theta ) \right|^2 \, R d\theta = 1\

,
and
:

\ \psi (\theta) = \ \psi ( \theta + 2\pi)

Under these conditions, the solution to the Schrödinger equation is given by
:

\psi_(\theta) = \frac \sqrt \theta }

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