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A question. Now can we figure out where everything comes from? Atoms, molecules, germs, bugs, you and me, the earth, the planets, the galaxy, the universe? Lets Try Start with the easiest – Then build Lets build A hydrogen atom More complicated atoms Molecules. Our tools. Schrödinger - PowerPoint PPT Presentation
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A question Now can we figure out where everything
comes from? Atoms, molecules, germs, bugs, you and me, the earth, the planets, the galaxy, the universe?Lets TryStart with the easiest – Then buildLets build
A hydrogen atom More complicated atoms Molecules
Our tools2
2 2
( ) 2 [ ( )] ( )d x m E U x xdx
Schrödinger(a 3D version)U(x)=Ze2/r
A “classical” picture (for our human brains”Electrons orbiting around a nucleus with charge Z
Thinking “classically” Electron going around nucleus of charge Z Planet going around sun
Different planets have different orbits [nn] N=1 = mercury, earth might be n=3
Planets have angular momentum [LL] Planet orbits have a tilt [mmLL]
Planets spin around their axis [SS] Planet spin has a tilt! [mmSS]
Use a 3-D version of this U(x)=Ze2/r
2
2 2
( ) 2 [ ( )] ( )d x m E U x xdx
2
2
B
REn
r a n
Rydberg’s Constant=R=13.6eV
We Get 3 Quantum numbers!n ~ Energy levelL ~ Orbital Angular momentummL ~ z-component of L
Does the electron have spin??
2 2, , /n l mE Z R n
Here it is: and its sort of like planets
Stern-Gerlach Experiment
A beam of neutral silver atoms is split into two components by a nonuniform magnetic field
The atoms experienced a force due to their magnetic moments
The beam had two distinct components in contrast to the classical prediction
If the electron had spin, it might explain this
Electron Spins Spin quantized
The electron can have spin S= up down
In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases
2
Intrinsic Spin Violates our intuition:
How can an elementary particle such as the e¯ be point like,wavelike and have perpetual angular momentum?
SZ
2
SZ
2
2
SZ=ms2 ms = +1/2 or -1/2
Electron Spins, cont The concept of a spinning electron is
conceptually useful The electron is a point particle, without any
spatial extent Therefore the electron cannot be considered to be
actually spinning The experimental evidence supports the electron
having some intrinsic angular momentum that can be described by ms
Sommerfeld and Dirac showed this results from the relativistic properties of the electron
Spin was first discovered in the context of the emission spectrum of alkali metals - "two-valued quantum degree of freedom" associated with the electron in the outermost shell.
In trying to understand splitting patterns and separations of line spectra, the concept of spin appeared
"it is indeed very clever but of course has nothing to do with reality". W. Pauli
A year later Goudsmit and Uhlenbeck, published a paper on this same idea.
Pauli finally formalized the theory of Spin in 1927
The Exclusion Principle The four quantum numbers discussed so far can
be used to describe all the electronic states of an atom regardless of the number of electrons in its structure
The Exclusion Principle states that no two electrons in an atom can ever be in the same quantum state Therefore, no two electrons in the same atom can
have the same set of quantum numbers
Orbitals An orbital is defined as the atomic state
characterized by the quantum numbers n, and m
From the Exclusion Principle, it can be seen that only two electrons can be present in any orbitalOne electron will have ms = ½ and one will
have ms = -½
Allowed Quantum States, Example
The arrows represent spin The n=1 shell can accommodate only two electrons,
since only one orbital is allowed In general, each shell can accommodate up to 2n2
electrons
The fact that two electrons could be in any energy level corresponding to a given set of quantum numbers (n,l,m) led eventually to the discovery of the spin of the electron.
Many-electron atoms
The energy levels corresponding to n = 1, 2, 3, … are called shells and each can hold 2n2 electrons.
The shells are labeled K, L, M, … for n = 1, 2, 3, ….
Fig 42-19, p.1376
Covalent bonds Model of H atom
particle in a box E=-13.6 ev (n=1 state) top of the box be at zero width of the box is ~ 2aB~0.1 nm
2 H atoms for a bond Separation =.12 nm match solutions at
boundary
Ge-κx
outside
solution
MatchsolutionsAsin(kx)
inside
solution
2
2
2 ( )
24.2
2 (0 )
m E Uk
where U eV
m E
Now solve for the new energies
E=-17.5 eVE=-9 eV
n=1 n=2
prob density prob density
Total energy – the covalent bond If the total energy is negative, then the
particles are bound. pp repulsive energy =
n=1 E=12 eV-17.5eV = -5.5eV covalent bond!!
n=2 E=12 eV – 9 eV=+3eV doesn’t bind
2
0
1 12 .124
e eV r nmr
Lasers
Excitation
Emmission A laser
Simulated Emmision
Lasers CD Telecommunications Surgery Etc etc
Fig 42-27, p.1386Optical Tweezers, DNA making an R
If the exclusion principle were not valid,a. every electron in an atom would have a
different mass.b. every atom would be in its state of greatest
ionization.c. the quantum numbers and would not exist.d. every pair of electrons in a state would
have to have opposite spins.e. every electron in an atom would end up in
the atom’s lowest energy state.