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 to π where it cannot escape, because the potential from π to 2 π is infinite. Instead there is total reflection, meaning the particle bounces back and forth between 0 to π . The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle S 1 ) is
− ℏ 2 2 m ∇ 2 ψ = E ψ ( 1 ) Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so
∇ 2 = 1 s 2 ∂ 2 ∂ ϕ 2 ( 2 ) Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as
− ℏ 2 2 m s 2 d 2 ψ d ϕ 2 = E ψ ( 3 ) 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 . Solving the integral, one finds that the moment of inertia of a semicircle is I = m s 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 , which is easily solvable.
Since the particle cannot escape the region from 0 to π , the general solution to this differential equation is
ψ ( ϕ ) = A cos ( m ϕ ) + B sin ( m ϕ ) ( 4 ) Defining m = 2 I E ℏ 2 , we can calculate the energy as E = m 2 ℏ 2 2 I . We then apply the boundary conditions, where ψ and d ψ d ϕ are continuous and the wave function is normalizable:
∫ 0 π | ψ ( ϕ ) | 2 d ϕ = 1 ( 5 ) .
Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both ϕ = 0 and ϕ = π . Basically
ψ ( 0 ) = ψ ( π ) = 0 ( 6 ) .
Since the wave function ψ ( 0 ) = 0 , the coefficient A must equal 0 because cos ( 0 ) = 1 . The wave function also equals 0 at ϕ = π so we must apply this boundary condition. Discarding the trivial solution where B=0, the wave function ψ ( π ) = 0 = B sin ( m π ) only when m is an integer since sin ( n π ) = 0 . This boundary condition quantizes the energy where the energy equals E = m 2 ℏ 2 2 I where m is any integer. The condition m=0 is ruled out because ψ = 0 everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out.
We then normalize the wave function, yielding a result where B = 2 π . The normalized wave function is
ψ ( ϕ ) = 2 π sin ( m ϕ ) ( 7 ) .
The ground state energy of the system is E = ℏ 2 2 I . Like the particle in a box, there exists nodes in the excited states of the system where both ψ ( ϕ ) and ψ ( ϕ ) 2 are both 0, which means that the probability of finding the particle at these nodes are 0.
Since the wave function is only dependent on the azimuthal angle ϕ , the measurable quantities of the system are the angular position and angular momentum, expressed with the operators ϕ and L z respectively.
Using cylindrical coordinates, the operators ϕ and L z are expressed as ϕ and − i ℏ d d ϕ respectively, where these observables play a role similar to position and momentum for the particle in a box. The commutation and uncertainty relations for angular position and angular momentum are given as follows:
[ ϕ , L z ] = i ℏ ψ ( ϕ ) ( 8 ) ( Δ ϕ ) ( Δ L z ) ≥ ℏ 2 where
Δ ψ ϕ = ⟨ ϕ 2 ⟩ ψ − ⟨ ϕ ⟩ ψ 2 and
Δ ψ L z = ⟨ L z 2 ⟩ ψ − ⟨ L z ⟩ ψ 2 ( 9 ) As with all quantum mechanics problems, if the boundary conditions are changed so does the wave function. If a particle is confined to the motion of an entire ring ranging from 0 to 2 π , 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 to π 2 , the issue of even and odd parity becomes important.
The wave equation for such a potential is given as:
ψ o ( ϕ ) = 2 π cos ( m ϕ ) ( 10 ) ψ e ( ϕ ) = 2 π sin ( m ϕ ) ( 11 ) where ψ o ( ϕ ) and ψ e ( ϕ ) are for odd and even m respectively.
Similarly, if the semicircular potential well is a finite well, the solution will resemble that of the finite potential well where the angular operators ϕ and L z replace the linear operators x and p.
