Download - probabilities: - initial state n: - final state m:

General properties of radiative transitions:
- Bohr's postulate for the transition frequency:
- consider a general solution ψn to a time-independent Schrödinger equation corresponding
to the energy eigenvalue En
- position expectation value:
- <x> is time independent and thus the average position of the electron does not change in a
given state. Therefore, the electron does not radiate as long as it remains in a specific
eigenstate ψn of the Hamiltonian describing it.
- Consider an electron changing its quantum mechanical state from ψn to ψm, i.e. changing
its energy from En to Em. During the transition the electron is in a state
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- probabilities:
- initial state n:
- final state m:
- radiation is generated during the transition from n to m
- calculate the position expectation value <x> in an intermediate superposition state
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- the expectation value <x>mn oscillates at the transition frequency
- <x>mn is also called the transition matrix element
- the strength of the transitions, i.e. the brightness of a spectral line in emission or the
strength of absorption, is proportional to the transition matrix element
- transitions for which <x>mn ≠ 0 are called allowed transitions, transitions for which
<x>mn = 0 are called forbidden transitions
- the conditions on two states with quantum numbers n and m that lead to allowed
transitions are called selection rules
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Selection rules for transitions in the Hydrogen
atom:
- w.f. with characteristic quantum numbers
- allowed transitions
- for u = x the dipole emitting the radiation would be
oriented along the x-direction
- solve integral for all transitions
- allowed transitions
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interpretation of selection rules:
- angular momentum selection rule
- the photon emitted carries an angular momentum ± h/2π so that the total angular
momentum is conserved in a transition
- magnetic quantum number selection rule
- right circularly polarized light (σ+)
- linearly polarized light (π)
- left circularly polarized light (σ-)
- classically the analog of a photon with angular momentum ± h/2π would correspond to a
left or right circularly polarized electromagnetic wave
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Zeeman Effect
- consider the potential energy of a magnetic moment μ in an
external field B
- the minimum energy configuration U = - μ B occurs when μ is aligned parallel to B, the
maximum energy U = + μ B occurs when μ is anti parallel to B.
- a current I flowing in a loop enclosing the area A generated by a single electron with a orbit
period T = 1/f generates a magnetic moment along the direction n normal to the area
- this is related to the electrons angular
momentum
- thus
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- with the gyromagnetic ratio of the electron given by:
- thus the potential energy of an electron with an angular momentum of L in an external
field B is given by
- quantum mechanically the z component lz of L is given
by
- there fore in the hydrogen atom
- with the Bohr magneton
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Spectrum of Hydrogen in a Magnetic Field
- the splitting of degenerate spectral lines (at the same energy) into individual lines by
magnetic field is called Zeeman effect after Pieter Zeeman who discovered it in 1896
- 2 l + 1 different values for ml
result in the same number of levels
with different energy
- number of transition frequencies
is limited by selection rules
Δl =± 1, Δml = 0, ± 1
- in the normal Zeeman effect
three different lines occur
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Electron Spin:
The solution of the spatial part of the hydrogen Schrödinger equation is not sufficient to
describe all properties of the hydrogen spectrum. An additional property of the electron, i.e. the
electron spin, needs to be considered.
- example: splitting of spectral lines into closely spaced pairs (doublets)
- e.g. first line in Balmer series (n=3 -> n=2) at λ = 656.3 nm is split into two lines with
separation Δλ = 0.14 nm in absence of external field
- such a splitting in the transition frequency is called fine structure
normal Zeeman effect:
transition frequency split
into three lines
anomalous Zeeman effect:
splitting into 4 or 6 or even
more lines with separation
different from Zeeman
prediction
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Goudsmit and Uhlenbeck: Every electron has an intrinsic angular momentum S, called
spin. The electron spin is associated with a magnetic moment μ.
- their idea was motivated by considering a spinning electron with
charge e giving rise to a magnetic moment and a spin angular
momentum
- considering the size of the electron (re < 10-16 m) the rotation rate would need to be so high
that the equatorial velocity would be much greater than the velocity of light
- no classical explanation for the phenomenon of spin
Relativistic Quantum Mechanics (Paul Dirac): A particle with mass and charge of an
electron must have the observed intrinsic angular momentum and magnetic moment.
Spin quantum number: the electron spin quantum number is s = 1/2.
Spin angular momentum:
- note that the relation between the spin angular momentum and its quantum number is
identical to the one for the orbital angular momentum L and its quantum number l
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Spin and space quantization:
spin:
z-component of spin:
- spin magnetic quantum number ms
- i.e. 2 s + 1 = 2 different orientations of spin with respect to
quantization axis (e.g. an magnetic field Bz along z-direction)
- spin magnetic moment of the electron
- magnetic moment of the electron along z-direction
- Bohr magneton
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- gyromagnetic ratio for orbital motion of electron:
- gyromagnetic ratio for electron spin:
- the electron g-factor
- g-factor is predicted to be g = 2 from relativistic quantum mechanics
- the small correction is due to quantum electrodynamics (QED) effects (vacuum fluctuations)
- full set of electron quantum numbers:
energy
orbital angular momentum
orbital angular momentum direction
spin direction
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Stern-Gerlach experiment:
- neutral silver (Ag)
atoms: magnetic moment
determined by single
electron
- in an inhomogeneous
magnetic field a force
acts on a magnetic
moment depending on its
orientation
- a homogeneous field
generates only a torque
- potential energy of atom magnetic moment
- force on magnetic moment of atom
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- classically all magnetic moment (spin) orientations should occur, but only two
orientations corresponding to the two possible spin states of the electron are observed
- first direct evidence for directional quantization (1921), there are only discrete
possibilities for the orientation of the magnetic moment with respect to the external field.
- quantitative analysis allows to measure value of the Bohr magneton
- general method to directly measure magnetic moments of atoms
- All atoms with one s electron show the same displacement. This indicates that the
magnetic moment of all electrons in closed inner shells is canceled out.
- For an outermost s electron there is no orbital angular momentum. Therefore the spin
angular momentum is directly measured.
- the magnitude and direction of angular and spin moments is conserved when atoms
move in space
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