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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
with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition
where p_{r} 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 n_{r} is generally different from the quantum number n_{θ} 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
The Schroedinger equation implies that n_{θ} is equal to (l(l+1)^{½} where l is an integer, rather than n_{θ} itself being an integer. In what follows it is not required that n_{θ} be an integer, only that is limited to discrete values.
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 r_{min} and a maximum distance r_{max}. From a value for (dr/dt) of 0 at r_{min} the value (dr/dt) becomes positive. However at r_{max} the value of (dr/dt) becomes zero again. Thereafter (dr/dt) becomes negative.
In Lagrangian dynamics the Lagrangian function is the kinetic energy function less the potential energy function; i.e.,
where v_{i}=(dq_{i}/dt).
The generalized momentum conjugate to q_{i} is given by
Likewise the generalized force is defined as
Thus the equations of motion for the system are of the form
The quantization condition
is then of the form
but dq_{i} may be expressed as (dq_{i}/dt)dt and (dq_{i}/dt) is just v_{i}.
Furthermore
If the kinetic energy function is of the form K=Σ(½J_{i}v_{i}²) and the potential energy function is independent of v_{i} then the quantization condition takes the form
where the integral is over a cycle. The coefficient J_{i} may be a mass or a moments of inertia for example.
If the quantization conditions are summed over all the variables the result is
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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