In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from 0 {\displaystyle 0} to π {\displaystyle \pi } where it cannot escape, because the potential from π {\displaystyle \pi } to 2 π {\displaystyle 2\pi } is infinite. Instead there is total reflection, meaning the particle bounces back and forth between 0 {\displaystyle 0} to π {\displaystyle \pi } . The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle S 1 {\displaystyle S^{1}} ) is
Wave function Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so
Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as
The moment of inertia for a semicircle, best expressed in cylindrical coordinates, is I = d e f ∭ V r 2 ρ ( r , ϕ , z ) r d r d ϕ d z {\textstyle I\ {\stackrel {\mathrm {def} }{=}}\ \iiint _{V}r^{2}\,\rho (r,\phi ,z)\,rdr\,d\phi \,dz\!} . Solving the integral, one finds that the moment of inertia of a semicircle is I = m s 2 {\displaystyle I=ms^{2}} , exactly the same for a hoop of the same radius. The wave function can now be expressed as − ℏ 2 2 I d 2 ψ d ϕ 2 = E ψ {\displaystyle -{\frac {\hbar ^{2}}{2I}}{\frac {d^{2}\psi }{d\phi ^{2}}}=E\psi } , which is easily solvable. Since the particle cannot escape the region from 0 {\displaystyle 0} to π {\displaystyle \pi } , the general solution to this differential equation is
Defining m = 2 I E ℏ 2 {\textstyle m={\sqrt {\frac {2IE}{\hbar ^{2}}}}} , we can calculate the energy as E = m 2 ℏ 2 2 I {\textstyle E={\frac {m^{2}\hbar ^{2}}{2I}}} . We then apply the boundary conditions, where ψ {\displaystyle \psi } and d ψ d ϕ {\displaystyle {\frac {d\psi }{d\phi }}} are continuous and the wave function is normalizable:
Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both ϕ = 0 {\displaystyle \phi =0} and ϕ = π {\displaystyle \phi =\pi } . Basically
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