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The modern atom
Chapter 23
Successes and failures of
the Bohr model
 Successes:
 Failures
- combining successfully
- a mixture of classical and quantum
Rutherford’s “solar system”
ideas (electrons move classically on
model, with the Planck
orbits, but their possible energy states
hypothesis on the quantified
are quantified)
energy states at atomic level + - postulates that on the allowed orbits
Einstein’s photons
electrons do not radiate
- explaining the atomic emission
(conflict with Maxwell’s theory)
and absorption spectra
- could not account for the maximal
- explaining the general features
electron numbers on one shell
of the periodic table
- could not explain splitting of the spectral
- a first “working” model for the
lines in magnetic fields
atom
- it is a non-relativistic theory although
the speed of the electrons is close to c
De’ Broglie’s hypothesis and the
birth of quantum mechanics
 Electromagnetic waves (photons) have both particle and wave properties
 D’Broglie proposed that all particles in the Universe could present this
dualism:
- each entity in the Universe exhibits both particle and wave properties
h
 Wavelength associated to the particles:

(h: Planck’s constant, m: mass, v: 2speed)
mv
rr  n
 electrons in atoms have also wave properties
 electrons form standing waves
in atoms --> explains Bohr’s 2rr  n ;  2rr  n h ;  L  mvr  n h
mv
2
quantum hypothesis:
 wave nature of the electron proved experimentally (Davisson & Gremer)
Wave-particle duality and
two-slit experiments
 Experimental results if electrons were normal particles
 Experimental results if electrons
behave like waves
 Conclusions: Electrons
interfere like photons
wave nature of particles
important when mv is
small (otherwise  is too
small to produce detectable effects)
The nature of the waves
associated to particles
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Waves associated to particles--> matter-waves
matter-waves characterized by their amplitude, called wave-function: 
Intensity of the wave ~ 2
What is oscillating? Intensity of what varies?
Intensity of matter waves represents the likelihood or probability
of finding an electron at that location and time
new view of physics --> quantum mechanics
new rules and equations which describe the dynamics of the matter waves
fundamental equation: Schrodinger’s wave equation --> gives 
Schrodinger’s equation a very complicated second-order differential
equation, can be solved analytically only in some simple case (Solving
Schrodinger’s equation = finding 
knowledge of  provides all possible information about atomic particles
Closing a particle in a
box
 Behavior of a classical particle closed in a
box
 Behavior of a quantum particle closed in a
box
 I. One dimensional box
- properties found by solving Schrodinger’s
equation
- possible : standing waves--> quantized
wavelength
- quantized values of the possible energies
- one quantum number characterizing the
states
 II. Three dimensional (real) box
- three dimensional standing waves for 
- quantized energy values
- three quantum numbers distinguish the
states
The Quantum-Mechanical Atom
 Schrodinger’s equation is analytically solvable for the H atom
 Schrodinger’s equation cannot be solved analytically for atoms with more
than one electron
 Very complicated standing waves, hard to visalize
 no classical electron orbits, replaced by probability clouds...
 discrete energy spectra, characterized by three quantum numbers
- n: associated with the electrons energy (n=1,2,3,4…..)
- l : associated the angular momentum’s magnitude (l=0,1,2…n-1)
- ml : characterizing the direction of the angular momenta (it’s
projection on a given direction) (ml=-l, -l+1, -l+2, ….l-1,l)
 an additional quantum number: ms not explained by the classical theory of
Schrodinger
 ms characterizes the spin of the electron
- two possible orientations: up and down (dependent on the spinning
direction)
- ms = +1/2 or -1/2
 four quantum number label the possible states of the electrons in atoms
The exclusion principle
and the periodic table
 Exclusion principle: No two electrons can have the same state (no
two electrons can have the same set of quantum numbers)
(Wolfgang Pauli, 1924)
 atomic shells: a group of allowed states, where the energy is very close
 chemical properties determined by the number of electrons in the outermost
incomplete shell
 first shell: possible states: n=1, l=0, ml=0, ms=+1/2 or -1/2 (total: 2)
 second shell: n=2, l=0, ml=0, ms=+1/2 or -1/2
n=2, l=1, ml=1,0,-1, ms=+1/2 or -1/2
(total: 8)
 third shell: n=3, l=0, ml=0, ms=+1/2 or -1/2
n=3, l=1, ml=1,0,-1, ms=+1/2 or -1/2
(total: 8)
 fourth shell:
n=4, l=0, ml=0, ms=+1/2 or -1/2
n=3, l=2, ml=2,1,0,-1,-2, ms=+1/2 or -1/2
n=4, l=1, ml=1,0,-1, ms=+1/2 or -1/2
(total: 18)
 ……..
 filling of the states with electrons, after increasing energy of the states
(electrons always occupy the most lower allowed energy state)
Quantum number for the
electrons (First 30 elements)
The Uncertainty Principle
 Schrodinger equation --> the atomic particles motion are governed by
probability laws (very hardly accepted by the physics community)
 Werner Heisenberg another description and interpretation of the quantum
world (more mathematical description with matrix algebra)
 Both description leads to the UNCERTAINTY PRINCIPLE
There are some quantities which cannot be determined together
exactly (independently how good and precise apparatus we use),
better we know one of them, worst is our knowledge about the
other one
p yaxis:
y  h
Examples: I. momentum and position in the direction of 
one
(better we know the momentum, of a particle less
we know about it’s energy)
II. Energy and time of the measurements
( longer the time for the measurement, smaller is the
uncertainty in the measured energy value)
Et  h
The complimentary principle and the
determinism of quantum physics
 Complimentary principle: all entities in our physical word exhibit both
wave and particle properties, these two aspects are complimentary.
Better we know one aspect of the particle, worst is our knowledge about
the other.
Example: If we know the position (particle aspect), we have no idea
about the momentum which determines the wavelength (wave aspect)
 classical Newtonian determinism (mechanistic view) does not hold!
- strong argument against predetermined Universe
 In quantum mechanics the future becomes a statistical (or
probabilistic) issue
 We have only probabilities for predicting each of the possible futures
(nature will select among them, with probabilities revealed by the theory
of quantum mechanics)
Home-Work assignment:
Part I. 603/1,3-4,7-18; 604/19-29; 606/1-12;
Part II. 604/35-36; 605/37-41, 44-54; 606/13-22; 607/26
Summary:
- Bohr’s model was not complete, since it combined classical and quantum
concepts in a non-rigorous manner
- De’ Broglie’s revolutionary hypothesis: waves associated to all particles
(wavelength inversely proportional to its momentum) matter-waves
- Atomic particles confined to finite spaces form standing-wave patterns that
quantize their physical properties
- Quantum particles are described by the wave-function
- Square of the wave-function: probabilistic interpretation
- Wave-function determined by the Schrodinger equation
- There are four quantum numbers associated with the allowed states of the
electrons in an atoms
- The electrons fill the available states, respecting Pauli’s exclusion principle
and the minimal energy principle
- There is an indeterminacy of knowledge resulting from the wave-particle
duality (Heisenberg’s uncertainty principle)
- New philosophical view on determinism: --> probabilistic determinism