Chapter 42
Atomic Physics
Importance of the
Hydrogen Atom


The hydrogen atom is the only atomic system
that can be solved exactly
Much of what was learned in the twentieth
century about the hydrogen atom, with its
single electron, can be extended to such
single-electron ions as He+ and Li2+
More Reasons the Hydrogen
Atom is Important

The hydrogen atom is an ideal system for
performing precision tests of theory against
experiment


Also for improving our understanding of atomic
structure
The quantum numbers that are used to
characterize the allowed states of hydrogen
can also be used to investigate more
complex atoms

This allows us to understand the periodic table
Final Reasons for the Importance
of the Hydrogen Atom


The basic ideas about atomic structure must
be well understood before we attempt to deal
with the complexities of molecular structures
and the electronic structure of solids
The full mathematical solution of the
Schrödinger equation applied to the hydrogen
atom gives a complete and beautiful
description of the atom’s properties
Atomic Spectra



A discrete line spectrum is observed when a
low-pressure gas is subjected to an electric
discharge
Observation and analysis of these spectral
lines is called emission spectroscopy
The simplest line spectrum is that for atomic
hydrogen
Emission Spectra Examples
Uniqueness of Atomic Spectra


Other atoms exhibit completely different line
spectra
Because no two elements have the same line
spectrum, the phenomena represents a
practical and sensitive technique for
identifying the elements present in unknown
samples
Absorption Spectroscopy


An absorption spectrum is obtained by
passing white light from a continuous source
through a gas or a dilute solution of the
element being analyzed
The absorption spectrum consists of a series
of dark lines superimposed on the continuous
spectrum of the light source
Absorption Spectrum,
Example


A practical example is the continuous spectrum
emitted by the sun
The radiation must pass through the cooler gases of
the solar atmosphere and through the Earth’s
atmosphere
Balmer Series

In 1885, Johann
Balmer found an
empirical equation that
correctly predicted the
four visible emission
lines of hydrogen




Hα is red, λ = 656.3 nm
Hβ is green, λ = 486.1
nm
Hγ is blue, λ = 434.1 nm
Hδ is violet, λ = 410.2 nm
Emission Spectrum of
Hydrogen – Equation

The wavelengths of hydrogen’s spectral lines
can be found from
1
1 
 1
 RH 

2
2 
λ
2
n



RH is the Rydberg constant



RH = 1.097 373 2 x 107 m-1
n is an integer, n = 3, 4, 5,…
The spectral lines correspond to different values
of n
Other Hydrogen Series




Other series were also discovered and their
wavelengths can be calculated
1
1

 RH  1  2  n  2, 3, 4,
Lyman series:
λ

n 
Paschen series:
1
 1 1
 RH  2  2  n  4 , 5 , 6 ,
λ
3 n 
Brackett series:
1
 1 1
 RH  2  2  n  5 , 6 , 7 ,
λ
4 n 
Joseph John Thomson





1856 – 1940
English physicist
Received Nobel Prize in
1906
Usually considered the
discoverer of the electron
Worked with the deflection
of cathode rays in an
electric field

Opened up the field of
subatomic particles
Early Models of the Atom,
Thomson’s


J. J. Thomson established
the charge to mass ratio for
electrons
His model of the atom


A volume of positive charge
Electrons embedded
throughout the volume
Rutherford’s Thin Foil
Experiment



Experiments done in 1911
A beam of positively
charged alpha particles hit
and are scattered from a
thin foil target
Large deflections could not
be explained by
Thomson’s model
Early Models of the Atom,
Rutherford’s

Rutherford




Planetary model
Based on results of thin
foil experiments
Positive charge is
concentrated in the
center of the atom, called
the nucleus
Electrons orbit the
nucleus like planets orbit
the sun
Difficulties with the
Rutherford Model

Atoms emit certain discrete characteristic
frequencies of electromagnetic radiation


The Rutherford model is unable to explain this phenomena
Rutherford’s electrons are undergoing a centripetal
acceleration



It should radiate electromagnetic waves of the same
frequency
The radius should steadily decrease as this radiation is
given off
The electron should eventually spiral into the nucleus
 It doesn’t
Niels Bohr





1885 – 1962
Danish physicist
An active participant in the
early development of
quantum mechanics
Headed the Institute for
Advanced Studies in
Copenhagen
Awarded the 1922 Nobel
Prize in physics

For structure of atoms and
the radiation emanating
from them
The Bohr Theory of Hydrogen



In 1913 Bohr provided an explanation of
atomic spectra that includes some features of
the currently accepted theory
His model includes both classical and nonclassical ideas
He applied Planck’s ideas of quantized
energy levels to orbiting electrons
Bohr’s Theory, cont.



This model is now considered obsolete
It has been replaced by a probabilistic
quantum-mechanical theory
The model can still be used to develop ideas
of energy quantization and angular
momentum quantization as applied to atomicsized systems
Bohr’s Assumptions for
Hydrogen, 1

The electron moves in
circular orbits around
the proton under the
electric force of
attraction

The Coulomb force
produces the centripetal
acceleration
Bohr’s Assumptions, 2

Only certain electron orbits are stable



These are the orbits in which the atom does not
emit energy in the form of electromagnetic
radiation
Therefore, the energy of the atom remains
constant and classical mechanics can be used to
describe the electron’s motion
This representation claims the centripetally
accelerated electron does not emit energy and
therefore does not eventually spiral into the
nucleus
Bohr’s Assumptions, 3

Radiation is emitted by the atom when the electron
makes a transition from a more energetic initial state
to a lower-energy orbit





The transition cannot be treated classically
The frequency emitted in the transition is related to the
change in the atom’s energy
The frequency is independent of frequency of the electron’s
orbital motion
The frequency of the emitted radiation is given by
Ei – Ef = hƒ
 If a photon is absorbed, the electron moves to a higher
energy level
Bohr’s Assumptions, 4


The size of the allowed electron orbits is
determined by a condition imposed on the
electron’s orbital angular momentum
The allowed orbits are those for which the
electron’s orbital angular momentum about
the nucleus is quantized and equal to an
integral multiple of h
Mathematics of Bohr’s
Assumptions and Results

Electron’s orbital angular momentum
mevr = nħ where n = 1, 2, 3,…

The total energy of the atom is

2
1
e
E  K  U  mev 2  ke
2
r
The total energy can also be expressed as
k ee 2
E
2r

Note, the total energy is negative, indicating a bound
electron-proton system
Bohr Radius

The radii of the Bohr orbits are quantized

n2 2
rn 
n  1, 2, 3,
2
mekee
This shows that the radii of the allowed orbits
have discrete values—they are quantized


When n = 1, the orbit has the smallest radius, called
the Bohr radius, ao
ao = 0.052 9 nm
Radii and Energy of Orbits

A general expression
for the radius of any
orbit in a hydrogen
atom is


rn = n 2 a o
The energy of any orbit
is
ke e 2  1 
En  
 2  n  1, 2,3,
2ao  n 

This becomes
En = - 13.606 eV / n2
Specific Energy Levels


Only energies satisfying the previous
equation are allowed
The lowest energy state is called the ground
state


This corresponds to n = 1 with E = –13.606 eV
The ionization energy is the energy needed
to completely remove the electron from the
ground state in the atom

The ionization energy for hydrogen is 13.6 eV
Energy Level Diagram


Quantum numbers are
given on the left and
energies on the right
The uppermost level,
E = 0, represents the
state for which the
electron is removed
from the atom

Adding more energy than
this amount ionizes the
atom
Active Figures 42.7 and 42.8


Use the active figure to
choose initial and final
energy levels
Observe the transition
in both figures
PLAY
ACTIVE FIGURE
Frequency of Emitted Photons

The frequency of the photon emitted when
the electron makes a transition from an outer
orbit to an inner orbit is
Ei  Ef kee  1 1 
ƒ

 2  2
h
2aoh  nf ni 
2

It is convenient to look at the wavelength
instead
Wavelength of Emitted
Photons

The wavelengths are found by
 1 1
1 ƒ k ee 2  1 1 
 
 2  2   RH  2  2 
λ c 2aohc  nf ni 
 nf ni 

The value of RH from Bohr’s analysis is in
excellent agreement with the experimental
value
Extension to Other Atoms

Bohr extended his model for hydrogen to
other elements in which all but one electron
had been removed
ao
rn   n 
Z
k ee 2  Z 2 
En  
 2  n  1, 2, 3,
2ao  n 
2

Z is the atomic number of the element and is the
number of protons in the nucleus
Difficulties with the Bohr
Model

Improved spectroscopic techniques found
that many of the spectral lines of hydrogen
were not single lines


Each “line” was actually a group of lines spaced
very close together
Certain single spectral lines split into three
closely spaced lines when the atoms were
placed in a magnetic field
Bohr’s Correspondence
Principle

Bohr’s correspondence principle states that
quantum physics agrees with classical
physics when the differences between
quantized levels become vanishingly small

Similar to having Newtonian mechanics be a
special case of relativistic mechanics when v << c
The Quantum Model of the
Hydrogen Atom

The potential energy function for the
hydrogen atom is
e2
U (r )   k e
r


ke is the Coulomb constant
r is the radial distance from the proton to the
electron

The proton is situated at r = 0
Quantum Model, cont.


The formal procedure to solve the hydrogen
atom is to substitute U(r) into the Schrödinger
equation, find the appropriate solutions to the
equations, and apply boundary conditions
Because it is a three-dimensional problem, it
is easier to solve if the rectangular
coordinates are converted to spherical polar
coordinates
Quantum Model, final



ψ(x, y, z) is converted to
ψ(r, θ, φ)
Then, the space variables
can be separated:
ψ(r, θ, φ) = R(r), ƒ(θ), g(φ)
When the full set of
boundary conditions are
applied, we are led to three
different quantum numbers
for each allowed state
Quantum Numbers, General


The three different quantum numbers are
restricted to integer values
They correspond to three degrees of freedom

Three space dimensions
Principal Quantum Number

The first quantum number is associated with
the radial function R(r)




It is called the principal quantum number
It is symbolized by n
The potential energy function depends only
on the radial coordinate r
The energies of the allowed states in the
hydrogen atom are the same En values found
from the Bohr theory
Orbital and Orbital Magnetic
Quantum Numbers

The orbital quantum number is symbolized
by ℓ



It is associated with the orbital angular momentum
of the electron
It is an integer
The orbital magnetic quantum number is
symbolized by mℓ

It is also associated with the angular orbital
momentum of the electron and is an integer
Quantum Numbers, Summary
of Allowed Values




The values of n can range from 1 to 
The values of ℓ can range from 0 to n - 1
The values of mℓ can range from –ℓ to ℓ
Example:


If n = 1, then only ℓ = 0 and mℓ = 0 are permitted
If n = 2, then ℓ = 0 or 1
 If ℓ = 0 then mℓ = 0
 If ℓ = 1 then mℓ may be –1, 0, or 1
Quantum Numbers,
Summary Table
Shells

Historically, all states having the same
principle quantum number are said to form a
shell


Shells are identified by letters K, L, M,…
All states having the same values of n and ℓ
are said to form a subshell

The letters s, p, d, f, g, h, .. are used to designate
the subshells for which ℓ = 0, 1, 2, 3,…
Shell and Subshell Notation,
Summary Table
Wave Functions for Hydrogen

The simplest wave function for hydrogen is
the one that describes the 1s state and is
designated ψ1s(r)
ψ1s (r ) 


1
πa
3
o
e
r ao
As ψ1s(r) approaches zero, r approaches 
and is normalized as presented
ψ1s(r) is also spherically symmetric

This symmetry exists for all s states
Probability Density

The probability density for the 1s state is

 1  2r ao
ψ1s   3  e
 πao 
The radial probability density function P(r) is
the probability per unit radial length of finding
the electron in a spherical shell of radius r
and thickness dr
2
Radial Probability Density


A spherical shell of
radius r and thickness
dr has a volume of 4πr2
dr
The radial probability
function is
P(r) = 4πr2 |ψ|2
P(r) for 1s State of Hydrogen

The radial probability
density function for the
hydrogen atom in its
ground state is
 4r 2  2r ao
P1s (r )   3  e
 ao 


The peak indicates the
most probable location
The peak occurs at the
Bohr radius
P(r) for 1s State of Hydrogen,
cont.

The average value of r for the ground state of
hydrogen is 3/2 ao


The graph shows asymmetry, with much more
area to the right of the peak
According to quantum mechanics, the atom
has no sharply defined boundary as
suggested by the Bohr theory
Electron Clouds



The charge of the electron
is extended throughout a
diffuse region of space,
commonly called an
electron cloud
This shows the probability
density as a function of
position in the xy plane
The darkest area, r = ao,
corresponds to the most
probable region
Wave Function of the 2s state

The next-simplest wave function for the
hydrogen atom is for the 2s state


n = 2; ℓ = 0
The normalized wave function is
 1 
1
ψ2s (r ) 


4 2π  ao 

3
2

r  r
2
e
ao 

ψ2s depends only on r and is spherically
symmetric
2 ao
Comparison of 1s and 2s
States


The plot of the radial
probability density for
the 2s state has two
peaks
The highest value of P
corresponds to the
most probable value

In this case, r 5ao
Active Figure 42.12


Use the active figure to
choose values of r/ao
Find the probability that
the electron is located
between two values
PLAY
ACTIVE FIGURE
Physical Interpretation of ℓ


The magnitude of the angular momentum of
an electron moving in a circle of radius r is
L = mevr
The direction of L is perpendicular to the
plane of the circle


The direction is given by the right hand rule
In the Bohr model, the angular momentum of
the electron is restricted to multiples of 
Physical Interpretation
of ℓ, cont.

According to quantum mechanics, an atom in
a state whose principle quantum number is n
can take on the following discrete values of
the magnitude of the orbital angular
momentum:
1
1 
 1
 RH  2  2 
λ
n 
2

L can equal zero, which causes great difficulty
when attempting to apply classical mechanics to
this system
Physical Interpretation of mℓ



The atom possesses an orbital angular
momentum
There is a sense of rotation of the electron
around the nucleus, so that a magnetic
moment is present due to this angular
momentum
There are distinct directions allowed for the
magnetic moment vector μ with respect to the
magnetic field vector B
Physical Interpretation of mℓ, 2


Because the magnetic moment μ of the atom
can be related to the angular momentum
vector, L, the discrete direction of μ translates
into the fact that the direction of L is
quantized
Therefore, Lz, the projection of L along the z
axis, can have only discrete values
Physical Interpretation of mℓ, 3



The orbital magnetic quantum number mℓ
specifies the allowed values of the z
component of orbital angular momentum
Lz = mℓ
The quantization of the possible orientations
of L with respect to an external magnetic field
is often referred to as space quantization
Physical Interpretation of mℓ, 4

L does not point in a specific direction



Even though its z-component is fixed
Knowing all the components is inconsistent with
the uncertainty principle
Imagine that L must lie anywhere on the
surface of a cone that makes an angle θ with
the z axis
Physical Interpretation of mℓ,
final


θ is also quantized
Its values are specified
through
Lz
cos θ 

L

m

 1
mℓ is never greater than
ℓ, therefore θ can never
be zero
Zeeman Effect


The Zeeman effect is the
splitting of spectral lines in a
strong magnetic field
In this case the upper level,
with ℓ = 1, splits into three
different levels
corresponding to the three
different directions of µ
Spin Quantum Number ms



Electron spin does not come from the
Schrödinger equation
Additional quantum states can be explained
by requiring a fourth quantum number for
each state
This fourth quantum number is the spin
magnetic quantum number ms
Electron Spins



Only two directions exist for
electron spins
The electron can have spin
up (a) or spin down (b)
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
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
Dirac showed this results from the relativistic
properties of the electron
Spin Angular Momentum



The total angular momentum of a particular
electron state contains both an orbital
contribution L and a spin contribution S
Electron spin can be described by a single
quantum number s, whose value can only be
s=½
The spin angular momentum of the electron
never changes
Spin Angular Momentum, cont

The magnitude of the spin angular
momentum is
S

3
s(s  1) 
2
The spin angular momentum can have two
orientations relative to a z axis, specified by
the spin quantum number ms = ± ½


ms = + ½ corresponds to the spin up case
ms = - ½ corresponds to the spin down case
Spin Angular Momentum, final


The z component of
spin angular
momentum is Sz = msh
=½h
Spin angular moment
is quantized
Spin Magnetic Moment


The spin magnetic moment µspin is related to
the spin angular momentum by
e
μspin   S
me
The z component of the spin magnetic
moment can have values
μspin , z
e

2me
Quantum States

There are eight quantum states
corresponding to n = 2


These states depend on the addition of the
possible values of ms
Table 42.3 summarizes these states
Quantum Numbers for n = 2 State
of Hydrogen
Wolfgang Pauli



1900 – 1958
Austrian physicist
Important review article on
relativity






At age 21
Discovery of the exclusion
principle
Explanation of the
connection between
particle spin and statistics
Relativistic quantum
electrodynamics
Neutrino hypothesis
Hypotheses of nuclear spin
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 can ever be in the same quantum state


Therefore, no two electrons in the same atom can have the
same set of quantum numbers
If the exclusion principle was not valid, an atom
could radiate energy until every electron was in the
lowest possible energy state and the chemical
nature of the elements would be modified
Filling Subshells


The electronic structure of complex atoms
can be viewed as a succession of filled levels
increasing in energy
Once a subshell is filled, the next electron
goes into the lowest-energy vacant state

If the atom were not in the lowest-energy state
available to it, it would radiate energy until it
reached this state
Orbitals


An orbital is defined as the atomic state
characterized by the quantum numbers n, ℓ and mℓ
From the exclusion principle, it can be seen that only
two electrons can be present in any orbital


One electron will have spin up and one spin down
Each orbital is limited to two electrons, the number
of electrons that can occupy the various shells is
also limited
Allowed Quantum States,
Example with n = 3

In general, each shell can accommodate up to
2n2 electrons
Hund’s Rule

Hund’s Rule states that when an atom has
orbitals of equal energy, the order in which
they are filled by electrons is such that a
maximum number of electrons have unpaired
spins

Some exceptions to the rule occur in elements
having subshells that are close to being filled or
half-filled
Configuration of Some
Electron States
Periodic Table



Dmitri Mendeleev made an early attempt at
finding some order among the chemical
elements
He arranged the elements according to their
atomic masses and chemical similarities
The first table contained many blank spaces
and he stated that the gaps were there only
because the elements had not yet been
discovered
Periodic Table, cont.



By noting the columns in which some missing
elements should be located, he was able to
make rough predictions about their chemical
properties
Within 20 years of the predictions, most of
the elements were discovered
The elements in the periodic table are
arranged so that all those in a column have
similar chemical properties
Periodic Table, Explained


The chemical behavior of an element
depends on the outermost shell that contains
electrons
For example, the inert gases (last column)
have filled subshells and a wide energy gap
occurs between the filled shell and the next
available shell
Hydrogen Energy Level
Diagram Revisited


The allowed values of ℓ are
separated horizontally
Transitions in which ℓ does
not change are very unlikely
to occur and are called
forbidden transitions

Such transitions actually
can occur, but their
probability is very low
compared to allowed
transitions
Selection Rules

The selection rules for allowed transitions are




Δℓ = ±1
Δmℓ = 0, ±1
The angular momentum of the atom-photon system
must be conserved
Therefore, the photon involved in the process must
carry angular momentum


The photon has angular momentum equivalent to that of a
particle with spin 1
A photon has energy, linear momentum and angular
momentum
Multielectron Atoms

For multielectron atoms, the positive nuclear
charge Ze is largely shielded by the negative
charge of the inner shell electrons


The outer electrons interact with a net charge that
is smaller than the nuclear charge
Allowed energies are
En

2
13.6 Zeff

eV
2
n
Zeff depends on n and ℓ
X-Ray Spectra



These x-rays are a result of
the slowing down of high
energy electrons as they
strike a metal target
The kinetic energy lost can
be anywhere from 0 to all of
the kinetic energy of the
electron
The continuous spectrum is
called bremsstrahlung, the
German word for “braking
radiation”
X-Ray Spectra, cont.


The discrete lines are called characteristic
x-rays
These are created when



A bombarding electron collides with a target atom
The electron removes an inner-shell electron from
orbit
An electron from a higher orbit drops down to fill
the vacancy
X-Ray Spectra, final



The photon emitted during this transition has
an energy equal to the energy difference
between the levels
Typically, the energy is greater than 1000 eV
The emitted photons have wavelengths in the
range of 0.01 nm to 1 nm
Moseley Plot



Henry G. J. Moseley plotted
the values of atoms as shown
λ is the wavelength of the Kα
line of each element
 The Kα line refers to the
photon emitted when an
electron falls from the L to
the K shell
From this plot, Moseley
developed a periodic table in
agreement with the one
based on chemical properties
Stimulated Absorption


When a photon has energy hƒ equal to the
difference in energy levels, it can be absorbed by
the atom
This is called stimulated absorption because the
photon stimulates the atom to make the upward
transition
Active Figure 42.25


Use the active figure to
adjust the energy
difference between the
states
Observe stimulated
absorption
PLAY
ACTIVE FIGURE
Spontaneous Emission


Once an atom is in an
excited state, the
excited atom can make
a transition to a lower
energy level
Because this process
happens naturally, it is
known as
spontaneous
emission
Stimulated Emission


In addition to
spontaneous emission,
stimulated emission
occurs
Stimulated emission
may occur when the
excited state is a
metastable state
Stimulated Emission, cont.



A metastable state is a state whose lifetime is much
longer than the typical 10-8 s
An incident photon can cause the atom to return to
the ground state without being absorbed
Therefore, you have two photons with identical
energy, the emitted photon and the incident photon

They both are in phase and travel in the same direction
Active Figure 42.27


Use the active figure to
adjust the energy
difference between
states
Observe the stimulated
emission
PLAY
ACTIVE FIGURE
Lasers – Properties of
Laser Light

Laser light is coherent


Laser light is monochromatic


The individual rays in a laser beam maintain a
fixed phase relationship with each other
The light has a very narrow range of wavelengths
Laser light has a small angle of divergence

The beam spreads out very little, even over long
distances
Lasers – Operation

It is equally probable that an incident photon
would cause atomic transitions upward or
downward


Stimulated absorption or stimulated emission
If a situation can be caused where there are
more electrons in excited states than in the
ground state, a net emission of photons can
result

This condition is called population inversion
Lasers – Operation, cont.


The photons can stimulate other atoms to
emit photons in a chain of similar processes
The many photons produced in this manner
are the source of the intense, coherent light
in a laser
Conditions for Build-Up of
Photons


The system must be in a state of population
inversion
The excited state of the system must be a
metastable state


In this case, the population inversion can be established
and stimulated emission is likely to occur before
spontaneous emission
The emitted photons must be confined in the system
long enough to enable them to stimulate further
emission

This is achieved by using reflecting mirrors
Laser Design – Schematic



The tube contains the atoms that are the active medium
An external source of energy pumps the atoms to excited
states
The mirrors confine the photons to the tube

Mirror 2 is only partially reflective
Energy-Level Diagram for
Neon in a Helium-Neon Laser


The atoms emit 632.8nm photons through
stimulated emission
The transition is E3* to
E2

* indicates a metastable
state
Laser Applications

Applications include:




Medical and surgical procedures
Precision surveying and length measurements
Precision cutting of metals and other materials
Telephone communications