Download Nobel Prize in Physics 1945 "for the discovery of the Exclusion

Pauli Exclusion Principle
- Consider many electron atoms. What determines the quantum state (characterized by its
quantum numbers) that the electrons in such an atom occupy?
- Are all the electrons in the same quantum state? This is unlikely as atoms with different
atomic number Z have very different chemical properties hinting at different electron
configurations in the atom, e.g. the chemically active halogen fluorine Fz=9, the inert noble
gas neon NeZ=10, the metal sodium NaZ=11
- exclusion principle: No two electrons in an atom can exist in the same quantum state. Each
electron must have a different set of quantum numbers n, l, ml, ms.
- fundamental principle governing the electronic configuration in atoms discovered by
Wolfgang Pauli in 1925
- principle found through the study of spectra and their relation to quantum numbers
- e.g. in Helium no transitions are observed to and from the ground state with both electron
spins pointing in the same direction; transitions are only observed to and from ground states
with electron spins pointing in different directions.
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Pauli's observation: every unobserved state or unobserved transition involves two or more
electrons with identical quantum numbers.
Nobel Prize in Physics 1945
Wolfgang Pauli
Austria
Princeton University
Princeton, NJ, USA
b. 1900
d. 1958
"for the discovery of the Exclusion
Principle, also called the Pauli Principle"
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Symmetries of wave functions: Fermions and Bosons
- the wave function ψ of a quantum system that is composed of n non-interacting
indistinguishable particles can be written as a (tensor) product of its component wave
functions ψI
example: for two indistinguishable particles in two quantum states a and b
the labeling of the particle should make no difference whatsoever for the probability density
|ψ|2 of the combined system.
the probability density is independent of the symmetry of the wave function
symmetric 2-particle wave function
anti symmetric 2-particle w.f.
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possible wave functions for two particles
- for indistinguishable particles the cases I and II can not be distinguished. Thus they are
equivalent and can appear with equal probability.
symmetric wave function:
anti-symmetric wave function:
- the pre factor reflects the equal probability with which the two possibilities occur
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Consequences:
- two particles in the same state in a symmetric wave function are allowed
- two particles in the same state in an antisymmetric wave function are not allowed
We find that such an anti symmetric wave function would be consistent with the
Pauli exclusion principle for electrons where no two particles can be in the same
quantum state, i.e. the probability of finding such a state would vanish. This is
realized by an anti symmetric wave function.
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Particles with a half integer (1/2, 3/2, ...) spin (electrons, protons, neutrons) are described
by an antisymmetric wave function and are called Fermions. They obey the exclusion
principle.
5 fermions at T = 0
Particles with an integer (0, 1, 2, 3, ...) spin (photons, alpha particles, Helium atoms, ...)
are described by symmetric wave functions and are called Bosons. They do not obey the
exclusion principle, i.e. an arbitrary number of particles can occupy a quantum state with
the same quantum number.
5 Bosons at T = 0
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Periodic Table of Elements:
Dimitri Mendeleev (1869): When the elements are listed in order of atomic number Z,
elements with similar chemical and physical properties recur at regular intervals.
- this observation was made long before quantum theory was developed and even before the
concept of atomic number Z and atomic masses was known
- nevertheless he managed to set up a table of the then known 63 elements sorted by their
chemical properties
- gaps in his table were indicating at the existence of elements that were not yet discovered
at that time
In the periodic table elements are arranged in sequence of their atomic number Z forming
the rows of the table. Elements with similar chemical properties are arranged in the same
column.
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groups: elements with similar chemical and structural
properties are arranged in groups which form the columns of
the periodic table
periods: rows of elements in the periodic table. Along the
period transition from chemically active metals, to less
active metals, to active non-metals and inert gases is
observed.
The majority of the elements are metals. There are a number
of nonmetals and few the inert gases
variation of chemical activity
across the periodic table
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some important groups
group I: hydrogen (H) and the alkali metals lithium (Li), sodium (Na), potassium (K) etc.
- properties: soft metals, low melting points, chemically very active
group VII: fluorine (F), chlorine (Cl), bromine (Br), iodine (I) etc. are the halogens
- properties: chemically active, oxidants, form diatomic molecules
group VIII: the nobel gases helium (He), neon (Ne), argon (Ar) etc.
-properties: chemically inactive, form almost no compounds, do not form molecules
transition elements: elements between group 2 and three in periods 4 and higher
- properties: metals, hard, brittle, high melting points, similar chemical behavior
lanthanides (rare earths): transition elements in period 6
actinides: transition elements in period 7
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Atomic Structure
principles determining the structure of atoms:
- only one electron can exist in a particular quantum state in an atom (Pauli principle)
- a system of particles is stable when its total energy is at minimum
note:
- these principles determine the distribution
of all electrons in an atom to the different
states
- electrons interact with each other in a
many electron atom
- it is useful to consider an electron in the
electrostatic potential of the nucleus with
charge Ze reduced by the other electrons
screening this charge (see example)
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Atomic Shell:
Electrons with the same principal quantum number n have a similar average distance from
the nucleus interacting with a similar electric field and thus also have a similar energy En.
Such electrons are said to occupy the same atomic shell.
labeling convention for atomic shells:
principal quantum number
shell label
- In many electron atoms the energy of a particular electron strongly depends on its principal
quantum number n but also depends on its angular momentum quantum number l.
- The probability distribution |ψ|2 of finding the electron at a certain distance from the
nucleus depends on n and l.
- For small l the electron is closer to the nucleus (where the screeing by the other electrons is
less effective) and thus is more strongly bound, i.e. it has a more negative energy.
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Electron binding energy versus atomic number Z
- increasing binding energy with Z
- decreasing binding energy with increasing n
- decreasing binding energy with increasing l
Subshells:
- Electrons with the same angular momentum
quantum number l are said to be part of the same
subshell.
- The dependence of their energy on magnetic
quantum number ml and spin quantum number ms
is small in comparison to the n and l dependence.
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Occupancy Electronic States in Many Electron Atoms
example: sodium (Na)
1s2 2s2 2p6 3s1
- principal quantum number n
- followed by angular quantum number l represented by
the associated letter
- superscript indicating the occupation of the subshell with
electrons
Capacity of shells and subshells:
total number of electrons
M shell (n = 3) capacity:
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The Periodic Table
closed shell: A shell in which all possible electron states are filled is called closed.
- closed s shell: 2 electrons
- closed p shell: 6 electrons
- closed d shell: 10 electrons
- closed f shell: 14 electrons
properties:
- closed subshells have zero total angular momentum
- closed subshells have zero spin momentum
- electrons in a closed shell are strongly bound because of the large only weakly screened
charge of the nucleus
- an atom with closed shell has no dipole moment, it interacts only weakly with other atoms
- closed shell atoms such as the noble gases are chemically relatively inert
closed l = 1 subshell
closed l = 2 subshell
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Ionization Energy
variation of the ionization energy with
atomic number Z
notes:
- inert gases have the highs ionization energy
- alkali metals have the lowest ionization energy, i.e. they are easy to ionize and form
positively charged ions
- decrease of ionization energy with in group for increasing Z
- increase in ionization energy with increasing Z within period due to increasing nuclear
charge
- halogens can lower their energy by adding an electron to their incomplete shell forming
negatively charges ions
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Size of Many Electron Atoms
variation of atomic radii with atomic
number Z
extracted from atom-atom spacing in
crystal lattices of element in a solid
configuration
note:
- alkali metals have larger atomic radii due to weakly bound outer electron
- with increasing charge Z in each period the effective radius is reduced due to imperfect
screening
- relatively small variation in atomic radius with Z
- atoms with Z ~ 90 have only a radius of 3 times that of the hydrogen atom
- the largest atom is cesium (Cs) with a radius of 4.4 times that of the hydrogen (H) atom
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