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A Quantum Mechanical Analysis of Positronium

Positronium is the system consisting of an electron and positron revolving about their center of mass. Positronium is an interesting subject in its own right but it also is of interest as a simpler subject for analysis than the deuteron which involves the nuclear force as well as the electrostatic force. Positronium units do exist physically although they do not last long.

The procedure for quantum mechical analysis of a system is to first formulate its total energy function. Then the kinetic energy is expressed in terms of the momentum. Since in nonrelativistic physics the kinetic energy of a particle of mass m is ½mv² and its momentum p is mv the kinetic energy is then p²/(2m). The total energy of the system with kinetic energy expressed in terms of momnentum is called the Hamiltonian function of the system.

The next step in the quantum mechanical analysis is to replace p in the Hamiltonian with ih∇, where i is the imaginary unit (-1)½, h is Planck's constant divided by 2π and ∇ is the gradient of the system. The exponent of momentum p becomes the order of the differentiation. The Hamiltonian function so transformed is known as the Hamiltonian operator for the system. The Hamiltonian operator is used to generate the Schroedinger equation for the system. The Schroedinger equation involves the wave function for the system. The Schroedinger equation is then subjected to analysis to find quantization rules.

When this procedure was applied to the electron in a hydrogen atom and the wave function was assumed to be separable (ψ(r,θ,φ)=R(r)Θ(θΦ(φ)) it yielded three separate equation each one of which required an integral-valued for its solution. These are the quantum numbers (n,l, m), called the principal, the orbital and the magnetic quantum numbers. There is a fourth quantum number which can have only the values ±½ indicating the spin orientation of the electron.

Although in the hydrogen atom there are two charges, the angular momentum of the proton is so small compared to that of the electron that it can be ignored. That is not the case with positronium since the positron and electron carry equal amounts of the angular momentum. Also the probability density function for positronium must take into account the conditional probability distributions. The probability of the electron being in any location other than opposite the position of the positron is zero. Thus the wave function for the pair effectively reduces to the wave function for either one of them.

The Hamiltonian Function for Positronium

Let v be the common velocity for the positron and the electron and m their common mass. The potential energy of the positron and electron separated by a distance s is

V(s) = −K/s

where K is the electrostatic constant.

Thus the total energy of positronium is

H = 2(½mv²) − K/s

Since the common momentum p is mv the above equation can be rewritten as

H = 2p²/(2m) − K/(2r)
or, equivalently
H = p²/(2m/2) − (K/2)/r)

,

This is essentially the same as for the hydrogen atom but with one half the mass and one half the coefficient of the potential function. For the case of two particles of equal mass m the reduced mass is equal to m/2. The above formula with m/2 written as the reduced mass μ is:

H = p²/(2μ) − (K/2)/r)

The appropriate substitution of ih∇ for p in the Hamiltonian function yields the Hamiltonian operator

H = −(h²/2μ)∇² − (K/2)/r

The time-independent Schroedinger equation for positronium is then

−(h²/2μ)∇²Ψ − ((K/2)/r)Ψ = EΨ
or, equivalently
∇²Ψ + (2μ/h²)(E + ((K/2)/r)Ψ = 0

where E is a constant, identified as energy.

The Laplacian operator ∇² in spherical polar coordinates applied to a wave function Ψ is

(1/r²)(∂/∂r(r²(∂Ψ/∂r)) + (1/(r²sin(θ)))(∂/∂θ)(sin(θ)(∂Ψ/∂θ)) + (1/r²sin²(θ))(∂²Ψ/∂φ²)

Separability

Separability means that the wave function Ψ(r, θ, φ) is of the form R(r)(Θ(θ)(Φ). Since the wave function Ψ can be complex, so can any of the factors R, Θ and Φ. Under the separability assumption the Schroeding equation for positronium becomes

[(ΘΦ/r²)d/dr(r²(dR/dr))
+ (RΦ/(r²sin(θ))d/dθ(sin(θ)(dΘ/dθ))
+ (RΘ/r²)(d²Φ/dφ²)
+ (2μ/h²)(E − (K/2)/r) = 0
or, upon division by RΘΦ
and multiplication by r²sin²(θ)
 
(sin²(θ)/R)d/dr(r²(dR/dr))
+ (sin(θ)/Θ))d/dθ(sin(θ)(dΘ/dθ))
+ (1/Φ)(d²Φ/dφ²)
+ (2μ/h²)r²(E + (K/2)/r) = 0

This latter equation can be rearranged into the form

(1/Φ)(d²Φ/dφ²) = −[(sin²(θ)/R)d/dr(r²(dR/dr))
+ (sin(θ)/Θ))d/dθ(sin(θ)(dΘ/dθ)) + (2μ/h²)r²(E + (K/2)/r)]

The left-hand side (LHS) is a function only of φ and the right-hand side (RHS) is not a function of φ, therefore the common value of the LHS and the RHS is a constant. For the moment let that constant be denoted as α. This means that Φ satisfied the equation

d²Φ/dφ² = αΦ

This equation had the solution

Φ(φ) = exp(α½φ)

The variable φ represents an angle; and hence φ+2π is the same as φ. Therefore Φ(φ+2π)=Φ(φ). This cannot happen if α½ is a real number; it can happen only if it is imaginary. Consequently α must be negative. Therefore let α be equal to −ν². Therefore

Φ(φ) = exp(iνφ)

The condition that Φ(φ+2π)=Φ(φ) is satisfied only if ν is an integer.

The RHS of the equation whose LHS was α is also equal to α, now denoted as −ν². This means that

−[(sin²(θ)/R)d/dr(r²(dR/dr)) + (sin(θ)/Θ))d/dθ(sin(θ)(dΘ/dθ))
+ (2μ/h²)r²(E + (K/2)/r)] = −ν²

This equation can, by division by −sin²(θ), be put into the form

(1/R)d/dr(r²(dR/dr)) + (2μ/h²)r²(E + (K/2)/r) =
ν²/sin²(θ) − (1/(Θsin(θ))d/dθ(sin(θ)(dΘ/dθ))

The LHS of the above equation is a function only of R and the RHS a function only of θ. Let the common value be denoted as Λ. Thus there are two equations to be satisfied; i.e.,

ν²/sin²(θ) − (1/sin(θ)d/dθ(sin(θ)(dΘ/dθ)) = ΛΦ
 
d/dr(r²(dR/dr)) + (2μ/h²)r²(E + (K/2)/r) = ΛR

Elsewhere it is shown that in order for there to be an acceptable solution to the first of the above two equation Λ must be equal to l(l+1) for some integer l. The solution for R(r) from the second of the above two equation then depends up the quantum numbers ν and l. It is found that each solutions for R(r) is associated with an integer n, called the principal quantum number. For example, the solution for ν=0, l=0 and n=1 is of the form

Ψ(r,θ,φ) = N*exp(−r/a0)

where N and a0 are constants. The value of a0 for positronium is given by

a0 = h²/(μK) = 1.058×10-10 meters = 1.058 Å.

The above solution for Ψ is a spherically symmetric solution centered at the origin and declining exponentially as a function of the distance from that origin. In the case of positronium origin of the coordinate system is the center of mass of the positron and electron. The positron and electron have the same wave function. However the conditional probability distribution is such that if the positron is at (r, θ, φ) then the electron is at (r, −θ, −φ).

The wave functions for other values of the quantum numbers are not spherically symmetric. For example, the solution for ν=0, l=1 and n=2 is of the form

Ψ(r,θ,φ) = N(r/a0)exp(-r/(2a0))cos(θ)

For ν=1, l=1 and n=2 it is of the form

Ψ(r,θ,φ) = N(r/a0)exp(-r/(2a0))sin(θ)(cos(φ)+i*sin(φ))

(To be continued.)

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