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A Generalization of
Neils Bohr's Model of Quantization

Niels Bohr's Analysis of the Electron in an Atom

The analysis of Niels Bohr that is referred to here is not the highly simplified version that is usually associated with his name but a more sophisticated analysis that preceded that known as the planetary model of the atom. The version of Bohr's analysis that is utilized here is the one presented by David Bohm in his Quantum Theory. It is this analysis that gives a basis for the quantization of the angular momentum of an electron in an atom. First Bohr's analysis for an electron will be given, then a generalization will be presented that leads to the amazing result that the quantization found by Bohr under Coulombic attraction also applies for any central force formula or any potential function so long as the kinetic energy is the Newton formula of ½mv²

The electron of charge −e is attracted to the point charge of the nucleus of charge +Ze. The nucleus is so massive compared to the electron that it can be consider fixed in position while the electron revolves around it. The potential energy of an electron at a distance r from the nucleus is −αZe²/r, where α is the constant for the Coulomb force. The kinetic energy of the electron moving at a velocity of v is in classical physics ½mv², where m is the mass of the electron. Thus the total energy of the electron is

E = ½mv² −αZe²/r

In a circular orbit the balancing of the Coulombic attraction with the centrifugal force gives:

αZe²/r² = mv²/r
and thus
mv²r = αZe²
and hence
the potential energy
αZe²/r = mv²

This latter condition means that

E = ½mv² − mv² = −½mv²

The angular momentum pθ is equal to mvr; i.e. the linear momentum mv times the distance to the center of revolution. From the previous condition it follows that

vpθ = αZe²
and thus
v = αZe²/pθ

This means that the total energy is

E = −½m(αZe²)²/pθ²

If the electron goes from angular momentum pθ1 to angular momentum pθ2 there will be a change in energy ΔE which will go into a photon of frequency ν where

ΔE = hν = −½m(αZe²)²[1/pθ1² − 1/pθ2²]

For a change in angular momentum which is small the change in energy ΔE may be approximated by

ΔE = (dE/dpθ)Δpθ
= m[(αZe²)²/pθ³]Δpθ

According to classical physics the angular frequency associated with a charge revolving in a circular orbit of radius r with a tangential velocity v is ω=v/r. The regular frequency f and the angular frequency ω are related by f=ω/2π. This should be the frequency associated with the escape of an electron from the hydrogen atom. Therefore

(h/2π)(v/r) = = m[(αZe²)²/pθ³]Δpθ

The expression h/2π is usually denoted as h and called h-bar.

Since pθ=mvr the above equation reduces to

hv/r = [(αZe²)²/(m²v³r³)]Δpθ

Previously the relation αZe²/r = mv² was derived. Squaring both sides of the equation gives

(αZe²)²/r² = m²v4
and dividing both sides by m²v³r gives
(αZe²)²/(m²v³r³) = v/r

Comparing this equation with the equation

h(v/r) = [(αZe²)²/(m²v³r³)]Δpθ

reveals the stunning result that

Δpθ = h

In words, the angular momentum of an electron in an atom must change by an increment equal to Planck's constant divided by 2π. Or, stated differently the angular momentum of an electron is quantized.

The condition on the incremental change means that angular momentum must be of the form

pθ = lh + k

Imposing the condition that the physical relationships must be the same for left-handed and right-handed coordinate sytem requires that k must be either 0 or ½.

Bohr's analysis was superceded in the 1920's by quantum mechanics based upon Schroedinger's equation and Bohr's work came to be referred to as the Old Quantum Theory. The analysis based upon Schroedinger's equation provided a superior analysis for the case of hydrogen-like atoms, but Schroedinger's equation can be solved for only a very limited number of cases. For other cases, even ones only slightly different from the standard cases, the Schroedinger equation provides no insights. Bohr's analysis, on the other hand, does provide some insights into the non-standard cases.

The Generalization of Bohr's Analysis

Let v be the tangential velocity of a particle in circular orbit of radius r. Let K(v) be the kinetic energy function and V(r) the potential energy function. The total energy E is then

E(v,r) = K(v) + V(r)

For the Newtonian (non-relativistic) case K(v)=½mv², where m is the mass of the particle. For the relativistic case K(v)=mc²[(1−(v/c)²]−1/2−1], where c is the speed of light.

Some potential energy function that would be of interest are:

Hideki Yukawa in 1933 hypothesized that the potential energy function for the strong force which hold the nucleons in nuclei together is the same as for the Coulombic force except that it is multiplied by an exponential factor which is a function of distance. Yukawa established that the coefficient of distance in this exponential factor depends upon the mass of the particle carrying the force. Yukawa predicted that the strong force is carried by a particle about 200 times the mass of an electron. Later such a particle with a mass 270 times that of an electron was found. It was called the pi meson, now shortened to pion. Yukawa was awarded the Nobel Prize in Physics in 1946 for his analysis.

The third potential mentioned above stems from a perception that the exponential factor should be applied to the force rather than the potential. The strong force falls off with distance more than according to the inverse square function because the meson carrying the force are decaying with time and hence distance. For more on this topic see Yukawa.

The Conditions Which Prevail for Circular Orbits

The central force based upon a potential function of V(r) is −V'(r). This force must be sufficient to keep the particle in a circular orbit. The usual way this is expressed is that the attractive force must balance the centrifugal force; i.e.,

mv²/r = V'(r)
which is equivalent to
mv² = rV'(r)

This means that v² = rV'(r)/m and thus v is a function of r, and vice versa. It also means that

v²/r² = (v/r)² = V'(r)/(rm)
and hence
(v/r) = [V'(r)/(rm)]1/2

The relationship v² = rV'(r)/m also means that

2v(dv/dr) = (V'(r) + rV"(r))/m
and hence
dv/dr = ½(V'(r) + rV"(r))/(mv)

The angular momentum pθ is equal to mvr, the linear momentum mv times the distance to the center of the orbit. Since v is a function of r and pθ=mvr this means pθ is a function of r. Therefore

dpθ/dr = mr(dv/dr) + mv
or, factoring out (1/v) from the RHS
dpθ/dr = (1/v)[mvr(dv/dr) + mv²]

It was previously established that mv²=rV'(r) so

dpθ/dr = (1/v)[mvr(dv/dr) + rV'(r)]
or, equivalently
dpθ/dr = (r/v)[mv(dv/dr) + V'(r)]

Changes in Energy

Since total energy for a particle in a circular orbit is a function of r and angular momentum pθ is a monotonic function of r, total energy can be expressed in terms of angular momentum pθ. For changes in angular momentum there will be changes in total energy:

ΔE = (dE/dpθ)Δpθ

The derivative evaluates to

dE/dpθ = [K'(v)(dv/dr) + V'(r)](dr/dpθ)
but dr/dpθ is the reciprocal of dpθ/dr, so
dE/dpθ = [K'(v)(dv/dr) + V'(r)]/(dpθ/dr)

and upon substitution of the expression derived for (dpθ/dr)

dE/dpθ = [K'(v)(dv/dr) + V'(r)]/[(r/v)(mv(dv/dr) + V'(r))]

The change in energy for a particle goes into a photon of frequency ω where

hω = ΔE

where h is Planck's constant divided by 2π.

For a particle in a circular orbit the classical angular frequency associated with it is (v/r) and this should be the frequency observed when a particle is captured. Thus

h(v/r) = (dE/dpθ)Δpθ

Substituting in the expession found for dE/dpθ and solving for Δpθ one obtains

Δpθ =
h[mv(dv/dr) + V'(r)]/[K'(v)(dv/dr) + V'(r)]

For Newtonian mechanics K(v)=½mv² and thus K'(v)=mv. When this is substituted into the above equation the numerator and the denominator of the fraction are identical. Thus

Δpθ = h

This holds for any potential function V(r). Therefore in any particle system based upon a central force the angular momentum is quantized. This means that

pθ = lh + k

where l and k are integers. The requirement that results not be affected by a change from right-handed coordinates to left-handed coordinates requires that k be equal to 0 or ½.

This is a remarkable conclusion; i.e., that angular momentum can change only by an increment equal to Planck's constant, regardless of the potential function so long as the kinetic energy is the standard Newtonian form.

It also follows that if angular momentum is quantized then so are the orbital radii, tangential velocities and energy levels, although not in such a simple form.

For more see The Relativistic Case

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