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The Quantum Mechanics of a Particle in a Central Field

Consider a particle moving in a central field whose force function is F(r), where r is the distance of the particle from the center of the field. The force is directed along the radial vector from the particle to the center of the field. Let p be the linear momentum of the particle.

The potential energy function V(r) for the field is given by

V(r) = ∫rF(s)ds

The Hamiltonian function H for the particle is its total energy; i.e., the sum of its kinetic energy K and its potential energy V. For a single particle in a potential field V it is then given by

H = p²/(2m) + V(r)

where m is the mass of the particle.

The trajectory for a point particle can be determined using classical mechanics methods. The wave function for a particle can be determined using quantum mechanical methods.

The Relevant Cases

Although the problem may be analyzed using a general central force function F(r) it should be noted that there are only about two relevant cases. The field is carried by some particles such as photons in the case of the electrostatic force and pi mesons in the case of the nuclear strong force. This means there will be a (1/r²) dependence. If the force-carrying particles do not decay, as is the case for photons, then F(r) is of the form K/r² where K is a constant. If they do decay, as is the case with the pi mesons, then the surviving proportions is a negative exponential function of time which translates into a negative exponential function of distance from the source; i.e.,

F(r) = H*exp(−r/r0)/r²

where H and r0 are constants. Generally there should be constant factors Qq in the equations, where Q is the charge of the force field and q is the charge of the particle with respect to the field.

The Hamiltonian Operator

The Hamiltonian operator is constructed from the Hamiltonian function for a system by substituting −hi∇ for the momentum p, where ∇ is the gradient operator for the coordinate system, i is the imaginary unit and h is Planck's constant divided by 2π.

Thus for a particle in a central field with a potential energy function V(r)

H = −(h/(2m))²∇² + V(r)

The expresssion ∇² is called the Laplacian operator of the system.

Schrödinger's Equation for
the Wave Function for the System

The wave function ψ(r,θ,φ,t) is such that the product of it times its complex conjugate ψc is the the probability density of finding the particle at (r,θ,φ) at time t. This means that the integral of ψ*ψc over all space is always equal to unity.

The wave function satisfies the equation

Hψ = ih∂ψ/∂t

Suppose ψ(r,θ,φ,t) is of the form Ψ(r,θ,φ)T(t). Then

T(t)HΨ(r,θ,φ) = ihΨ(r,θ,φ)dT/dt
which may be put into the form
HΨ/Ψ = ih(dT/dt)/T

The left-hand side (LHS) of the above equation is a function of the space variables r, φ and θ only. The right-hand side (RHS) is a function of t alone. Obviously the common value of the two sides is a constant, which will be denoted as E.

Thus, from the right side of the equation,

ih(dT/dt)/T = E
and thus
dT/dt = (E/ih)T = −i(E/h)T

The solution to the above equation is

T(t) = T(0)exp(−(E/h)t)

This is a periodic solution with a frequency equal to (E/h).

From the LHS of the previous equation


Thus Ψ is an eigenfunction of the Hamiltonian operator and E is an eigenvalue. The above equation is called the time-independent Schrödinger equation for the system.

The Laplacian Operator for Spherical Coordinates

∇²Ψ = [(1/r²)∂/∂r(r²∂/∂r) + (1/(r²sin(θ))∂/∂θ(sin(θ∂/∂θ) + (1/(r²sin²(θ))∂²/∂φ²]Ψ

Solution by Separation of Variables

It is assumed that Ψ(r,θ,φ)=R(r)Θ(θ)Φ(φ). When this form is substituted into the the previous equations and the differentiation carried out, the result is

(−h²/(2m))[(ΘΦ/r²)d(r²(dR/dr))/dr + (RΦ/(r²sin(θ))d(sin(θ)(dΘ/dθ))/dθ
+ (RΘ/(r²sin²(θ))(d²Φ/dφ²)] + V(r)RΘΦ = ERΘΦ

If this equation is first divided by RΘΦ and then multiplied by −(2mr²sin²(θ)/h² the result can be expressed as

−(sin²(θ)/R)d(r²(dR/dr)) − (sin(θ)/Θ)d(sin(θ)(dΘ/dθ)) −(2m/h²)r²sin²(θ)(E−V(r)) = (1/Φ)(d²Φ/dφ²)

This is an equation in which the LHS depends only upon φ and the RHS does not depend upon φ. Therefore the common value is a constant. Let this constant be denoted as M. Thus

(1/Φ)(d²Φ/dφ²) = M
and hence
d²Φ/dφ² = MΦ

This equation has the solution

Φ(φ) = exp(M½φ)

Because φ is an angle it must be that

Φ(2π) = Φ(0)
which requires that
exp(M½2π) = 1

This cannot occur for M½ being real. Therefore M½ must be imaginary, say im. This means that

exp(im2π) = 1

which requires that m be an integer, positive, negative or zero. Therefor M=−m².

An equation that results from the RHS of the previous equation being equal to M=−m² is

(1/R)(d(r²(dR/dr)) + (2mr²/h²)(E−V(r)) = −(M/sin²(θ)) − (1/(Θsin(θ))d(sin(θ)(dΘ/dθ))/dθ

The LHS of the above equation depends only upon r and the RHS only upon θ. Let the common value be denoted as Λ. There are then the two equations

d(r²(dR/dr))/dr + (2m/h²)r²(E−V(r)) = ΛR
(1/sin(θ))d(sin(θ)(dΘ/dθ))/dθ) + (MΘ/sin²(θ)) = −ΛΘ

It turns out that for the second equation to have a physically acceptable solution it must be that Λ is of the for form l(l+1) where l is of the form |m|+k, for k being a nonnegative integer. For the derivation of this result see Quantum Eigenvalues.

(To be continued.)

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