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Lectures:
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
V. Theory of Sommerfeld
V.1. RULES quantization WILSON - Sommerfeld
Wilson quantization rules - Sommerfeld relate to any physical quantity, which is a periodic function of time(when describing a system functions are functions that the system is periodycznymi kwantowalny).
where & tau - period
p q - momentum corresponding to the generalized coordinate q
n q - the number of quantum
Example: The point of mass m moving in a circular orbit.
(R, & phi) q 1 = r, q 2 = & phi
r = const & phi = & omega t
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(V.1.1)
Since L = P, then:, Namely:
Dependence (V.1.4) compatible (identical) to the second postulate of Bohr . It is the mathematicalsolution.
If (V.1.1) is applied to an oscillating particle of mass m, we can find the total energy:
Equation (V.1.5) is consistent with the fourth postulate of Bohr .
Sommerfeld generalized Bohr model and showed that the electron in the orbit can move the elliptical orbit (thewheel is a special case), the number of orbits depends on n is exactly equal to it. n - we'll call the main quantum number.
V.2. The fine structure of the hydrogen spectrum.
All spectral lines ranging from n = 2 are split - it's fine structure. subtle structure is explained on the basis of relativistic account.
(V.1.2)
(V.1.3)
(V.1.4)
(V.1.5)
12/9/12 Introduction to atomic and quantum physics - Sommerfeld Theory
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Rys.V.1. The fine structure of the hydrogen atom spectrum.
Sommerfeld was able to calculate the parameters of the orbits and their energies.
, N & phi = 1, 2, 3, ...
, N r = 0, 1, 2, ...
Equations (V.2.3), (V.2.3b), (V.2.3c) - define the orbit at which the electron moves.
- The main quantum number
- Azimuthal quantum number
- Define the shape of the orbit
→
V.3. EXAMPLES OF ORBIT Sommerfeld.
Rys.V.2. Examples Sommerfeld orbits.
In terms of energy all the orbits for the n are the same as the energy does not depend on n & phi - orbital
degeneracy. degeneration was abolished after the introduction of relativistic account.
(V.2.1)
(V.2.2)
(V.2.3)
(V.2.3b)
(V.2.3c)
12/9/12 Introduction to atomic and quantum physics - Sommerfeld Theory
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Energy - is characteristic of the orbit
Z (V.3.2) that explains the structure and spectral lines subtle effect After taking into
account the relativistic energy is given by:
where: E NR - part nonrelativistic (V.2.3c)
E R - Amendment relativistic
Amendment describes the shift of energy levels - the appearance of fine structure.
and - the fine structure constant, the characteristic size of the formulas of quantum physics There are many interpretations of the constant a, one of which is:
v 1 is the electron velocity on the first orbit
As a result, the existence of the fine structure appear more levels and creates a more complicated spectrum.
(V.3.1)
(V.3.2)
(V.3.3)
(V.3.4)
(V.3.5)
Rys.V.3. Allowed transitions are indicated by green arrows, red means go prohibited.
The fact that transitions are allowed and which are not, says the selection rule:
(V.3.6)
