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The Bohr-Sommerfeld Model
and Nuclear Quantization

Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or Bohr-Sommerfeld model, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's strictly circular orbits. This model supplemented the quantized angular momentum condition of the Bohr model

p = nθ = nθ(h/2π) = nθh

with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition

0T(prdqr) = nrh

where pr is the radial momentum, m(dr/dt). This is the momentum which is canonically conjugate to the radial position. T is one full orbital period. The quantum number nr is generally different from the quantum number nθ for angular momentum L.

The integral is of action-angle coordinates. This condition, suggested by Bohr's correspondence principle, is the only one possible, since the quantum numbers do not change as the system evolves. Another way of saying this is that the quantum numbers are adiabatic invariants.

The Sommerfeld condition implies the quantization conditions for angular momentum and energy

L = nθh
E = −K/(nh
where n=nr+nθ

The Schroedinger equation implies that nθ is equal to (l(l+1)½ where l is an integer, rather than nθ itself being an integer. In what follows it is not required that nθ be an integer, only that is limited to discrete values.

History of the Quantization Condition

The quantization conditions of both Bohr and Sommerfeld did not arise just as idle conjectures. Both were arrived at by analysis and refined by debate among the noted physicists of the time.

The first version was that the integral over the phase space of the appropriate coordinates. This is shown below for a single degree of freedom.

For the Bohr atom's circular orbits this would be

The angular momentum for an orbit is constant. The integral of dθ, ∫dθ, from 0 to 2π is just 2π.

Arnold Sommerfeld argued that "What is true for the orbit angle is also true for the radial distance." The radial distance for an orbit cycles between a minimum distance rmin and a maximum distance rmax. From a value for (dr/dt) of 0 at rmin the value (dr/dt) becomes positive. However at rmax the value of (dr/dt) becomes zero again. Thereafter (dr/dt) becomes negative.

Lagrangian Dynamics and the
Bohr-Sommerfeld Quantization Condition

In Lagrangian dynamics the Lagrangian function is the kinetic energy function less the potential energy function; i.e.,

L(qi, vi, t) = K(vi, qi, t) − V(qi, t)

where vi=(dqi/dt).

The generalized momentum conjugate to qi is given by

pi = ∂L/∂vi

Likewise the generalized force is defined as

Fi = ∂L/∂qi.

Thus the equations of motion for the system are of the form

dpi/dt = Fi

The quantization condition

∫pidqi = nih

is then of the form

∫(∂L/∂vi)dqi = nih

but dqi may be expressed as (dqi/dt)dt and (dqi/dt) is just vi.


∂L/∂vi = (∂L/∂(½vi²)(∂(½vi²)/∂vi)
= (∂L/∂(½vi²))vi
and hence
(∂L/∂vi)vi = (∂L/∂(½vi²))vi²

If the kinetic energy function is of the form K=Σ(½Jivi²) and the potential energy function is independent of vi then the quantization condition takes the form

∫ (Jivi²)dt = nih

where the integral is over a cycle. The coefficient Ji may be a mass or a moments of inertia for example.

If the quantization conditions are summed over all the variables the result is

2∫K(t)dt = Nh
or, equivalently
∫K(t)dt = (N/2)h

where N is the sum of the quantum numbers. The coefficient of h then can be either an integer or a half integer.

(To be continued.)

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