Application of Quantum Mechanics to Atomic Physics
A large portion of the chapter focuses on the hydrogen atom.
Reasons for the importance of the hydrogen atom include.
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+
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.
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.
Other Ideas in Atomic Physics
The full mathematical solution of the Schrödinger equation applied to the
hydrogen atom gives a complete and beautiful description of the atom’s
Quantum numbers are used to characterize various allowed states in the atom.
The quantum numbers have physical significance.
Certain quantum states are affected by a magnetic field.
The exclusion principle is important for understanding the properties of
multielectron atoms and the arrangement of the elements in the periodic table.
All objects emit thermal radiation characterized by a continuous distribution of
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
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.
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.
In 1885, Johann Balmer found an
empirical equation that correctly
predicted the four visible emission lines
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
RH 2 2
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.
Values of n from 3 to 6 give the four visible lines.
Values of n > 6 give the ultraviolet lines in the Balmer series.
The series limit is the shortest wavelength in the series and corresponds to n
Other Hydrogen Series
Other series were also discovered and their wavelengths can be calculated:
RH 1 2 n 2, 3, 4,
RH 2 2 n 4 , 5 , 6 ,
RH 2 2 n 5 , 6 , 7 ,
Joseph John Thomson
1856 – 1940
Received Nobel Prize in 1906
Usually considered the discoverer of
Worked with the deflection of cathode
rays in an electric field
Opened up the field of subatomic
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 atom as a whole would be
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
Based on results of thin foil
Positive charge is concentrated in
the center of the atom, called the
Electrons orbit the nucleus like
planets orbit the sun
Difficulties with the Rutherford Model
Atoms emit certain discrete characteristic frequencies of electromagnetic
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.
1885 – 1962
An active participant in the early
development of quantum mechanics
Headed the Institute for Advanced
Studies in Copenhagen
Awarded the 1922 Nobel Prize in
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 non-classical ideas.
He applied Planck’s ideas of quantized energy levels to Rutherford’s orbiting
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 atomic-sized systems.
Bohr’s Postulates for Hydrogen, 1
The electron moves in circular orbits
around the proton under the electric
force of attraction.
The Coulomb force produces the
Bohr’s Postulates, 2
Only certain electron orbits are stable.
Bohr called these stationary states.
These are the orbits in which the atom does not emit energy in the form of
electromagnetic radiation, even though it is accelerating.
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
continuously emit energy and therefore does not eventually spiral into the
Bohr’s Postulates, 3
Radiation is emitted by the atom when the electron makes a transition from a
more energetic initial stationary state to a lower-energy stationary state.
The transition cannot be treated classically.
The frequency emitted in the transition is related to the change in the atom’s
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 Postulates, 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
Bohr’s Postulates, Notes
These postulates were a mixture of established principles and completely new
Postulate 1 – from classical mechanics
Treats the electron in orbit around the nucleus in the same way we treat a
planet in orbit around a star.
Postulate 2 – new idea
It was completely at odds with the understanding of electromagnetism at the
Postulate 3 – Principle of Conservation of Energy
Postulate 4 – new idea
Had no basis in classical physics
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
E K U mev ke
The electron is modeled as a particle in uniform circular motion, so the electrical
force on the electron equals the product of its mass and its centripetal
The total energy can also be expressed as E
The total energy is negative, indicating a bound electron-proton system.
The radii of the Bohr orbits are quantized
n 1, 2, 3,
This shows that the radii of the allowed orbits have discrete values—they are
This result is based on the assumption that the electron can exist only in
certain allowed orbits determined by n (Bohr’s postulate 4).
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 = n2ao
The energy of any orbit is
ke e 2 1
n 1, 2,3,
This becomes En = - 13.606 eV / n2
Specific Energy Levels
Only energies satisfying the energy 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.
Frequency and Wavelength 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 2 1 1
2aoh nf ni
It is convenient to look at the wavelength instead.
The wavelengths are found by
1 ƒ k ee 2 1 1
H 2 2
λ c 2aohc nf ni
The value of RH from Bohr’s analysis is in excellent agreement with the
Extension to Other Atoms
Bohr extended his model for hydrogen to other elements in which all but one
electron had been removed.
rn n 2
k ee 2 Z 2
2 n 1, 2, 3,
Z is the atomic number of the element and is the number of protons in the
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.
The Bohr model cannot account for the spectra of more complex atoms.
Scattering experiments show that the electron in a hydrogen atom does not move
in a flat circle, but rather that the atom is spherical.
The ground-state angular momentum of the atom is zero.
These deviations from the model led to modifications in the theory and ultimately
to a replacement theory.
Bohr’s Correspondence Principle
Bohr’s correspondence principle states that quantum physics agrees with
classical physics when the differences between quantized levels become
Similar to having Newtonian mechanics be a special case of relativistic
mechanics when v << c.
The Quantum Model of the Hydrogen Atom
The difficulties with the Bohr model are removed when a full quantum model
involving the Schrödinger equation is used to describe the hydrogen atom.
The potential energy function for the hydrogen atom is
U (r ) k e
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
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
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.
The three different quantum numbers
are restricted to integer values.
They correspond to three degrees of
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
Quantum Numbers, Summary of Allowed Values
The values of n are integers that can range from 1 to
The values of ℓ are integers that can range from 0 to n - 1.
The values of mℓ are integers that can range from –ℓ to ℓ.
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
Historically, all states having the same principle quantum number are said to form
Shells are identified by letters K, L, M,… for which n = 1, 2, 3, …
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 Notation, Summary Table
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 )
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.
The probability density for the 1s state is
1 2r ao
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.
Radial Probability Density
A spherical shell of radius r and
thickness dr has a volume of
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
4r 2 2r ao
P1s (r ) 3 e
The peak indicates the most probable
The peak occurs at the Bohr radius.
The average value of r for the ground
state of hydrogen is 3/2 ao.
According to quantum mechanics, the
atom has no sharply defined boundary
as suggested by the Bohr theory.
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
ψ2s (r )
4 2π ao
ψ2s depends only on r and is spherically symmetric.
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
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 the angular momentum vector is perpendicular to the plane of
The direction is given by the right hand rule.
In the Bohr model, the angular momentum of the electron is restricted to
multiples of .
This incorrectly predicts that the ground state of hydrogen has one unit of
If L is taken to be zero in the Bohr model, the electron must be pictured as a
particle oscillating along a straight line through the nucleus.
of ℓ, cont.
Quantum mechanics resolves these difficulties found in the Bohr model.
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:
, n 1
L can equal zero, which causes great difficulty when attempting to apply
classical mechanics to this system.
In the quantum mechanical representation, the electron cloud for the L = 0
state is spherically symmetric and has no fundamental rotation axis.
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 with respect
to the magnetic field vector .
Because the magnetic moment μ of the atom can be related to the angular
momentum vector, L, the discrete direction of the magnetic moment vector
translates into the fact that the direction of the angular momentum is quantized.
Therefore, Lz, the projection of L along the z axis, can have only discrete values.
Physical Interpretation of mℓ, cont.
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.
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
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.
Only two directions exist for electron
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
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.
The magnitude of the spin angular momentum is
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, cont.
The z component of spin angular
momentum is Sz = msh = ± ½ h
Spin angular moment is quantized.
Spin Magnetic Moment
The spin magnetic moment µspin is related to the spin angular momentum by
The z component of the spin magnetic moment can have values
μspin , z
There are eight quantum states corresponding to n = 2.
These states depend on the addition of the possible values of ms.
Table 42.4 summarizes these states.
Quantum Numbers for n = 2 State of Hydrogen
1900 – 1958
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
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
Therefore, no two electrons in the same atom can have the same set of
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.
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
If the atom were not in the lowest-energy state available to it, it would radiate
energy until it reached this state.
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 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
The filling of electron
states must obey
both the exclusion
principle and Hund’s
Dmitri Mendeleev made an early attempt at finding some order among the
He arranged the elements according to their atomic masses and chemical
The first table contained many blank spaces and he stated that the gaps were
there only because the elements had not yet been discovered.
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
For example, the inert (or noble) 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
Transitions in which ℓ does not change
are very unlikely to occur and are called
Such transitions actually can occur,
but their probability is very low
compared to allowed transitions.
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
A photon has energy, linear momentum and angular momentum.
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
Allowed energies are
13.6 eV Zeff
Zeff depends on n and ℓ
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
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.
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.
Henry G. J. Moseley plotted the values
of atoms as shown.
λ is the wavelength of the Kα line of
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.
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.
At ordinary temperatures, most of the
atoms in a sample are in the ground
The absorption of the photon causes
some of the atoms to be raised to
Once an atom is in an excited state, the
excited atom can make a transition to a
lower energy level.
This process is known as spontaneous
Because it happens naturally
In addition to spontaneous emission,
stimulated emission occurs.
Stimulated emission may occur when
the excited state is a metastable state.
A metastable state is a state whose
lifetime is much longer than the typical
An incident photon can cause the atom
to return to the ground state without
Therefore, you have two photons with
identical energy, the emitted photon
and the incident photon.
They both are in phase and travel in the
Lasers – Properties of Laser Light
Laser light is coherent.
The individual rays in a laser beam maintain a fixed phase relationship with
Laser light is monochromatic.
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.
The photons can stimulate other atoms to emit photons in a chain of similar
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
Energy-Level Diagram for Neon in a Helium-Neon Laser
The atoms emit 632.8-nm photons
through stimulated emission.
The transition is E3* to E2
* indicates a metastable state
Medical and surgical procedures
Precision surveying and length measurements
Precision cutting of metals and other materials
Biological and medical research