Introduction to Quantum
Mechanics
and Quantum Numbers
The Quantum Mechanical Model
quantum mechanics: the application of
quantum theory to explain the
properties of matter, particularly
electrons in atoms
Schrödinger’s Standing Wave
Erwin Schrödinger and Louis de Broglie
found that an electron bound to a
nucleus in an atom resembled a
standing wave, so they began research
on a description of the atom based on
wave behaviour instead of particle
behaviour.
Schrödinger and de Broglie took the
idea of standing waves and applied it to
the electron in a hydrogen atom.
In their model, the electron is a circular
standing wave around the nucleus. This
circular standing wave consists of
wavelengths that are multiples of whole
numbers (n = 1, 2, 3, 4, ...).
Only certain circular orbits have a
circumference into which a whole
number of wavelengths can fit.
Any other orbits of the standing electron
wave are not allowed because they
would cause the standing wave to
cancel out or collapse.
Orbitals and Probability Distributions
Schrödinger’s work on quantum
mechanics led to his development of a
mathematical equation, called the
Schrödinger wave equation, that could
be used to calculate electron energy
levels.
Orbital: the region around the nucleus
where an electron has a high probability
of being found
http://hyperphysics.phy-
astr.gsu.edu/%E2%80%8Chbase/quant
um/schr2.html
Werner Heisenberg came up with a
statistical approach for locating
electrons.
To measure the location and speed of
an object, you must be able to observe
it.
Heisenberg’s Uncertainty Principle: the
idea that it is impossible to know the
exact position and speed of an electron
at a given time
The best we can do is to describe the
probability of finding an electron in a
specific location.
wave function: the mathematical
probability of finding an electron in a
certain region of space
Quantum mechanics does not describe
how an electron moves or even if it
moves. It only tells us the statistical
probability of finding the electron in a
given location in an atom. The area or
region where we are likely to find an
electron is an orbital.
Using wave functions, physicists have
created a three-dimensional electron
probability density, which is a plot that
indicates regions around the nucleus
with the greatest probability of finding
an electron.
The electron probability density plot for
a hydrogen electron in the ground state
(lowest energy state) is spherical and is
called the 1s orbital.
The greatest probability of finding the
electron occurs at a distance rmax from
the nucleus. This distance is the same
as the distance Bohr calculated for the
radius of the first circular orbit of
hydrogen’s electron.
The two main ideas of the quantum
mechanical model of the atom are that
electrons can be in different orbitals by
absorbing or emitting quanta of energy,
and that the location of electrons is
given by a probability distribution.
Quantum Numbers
There are 4 quantum numbers
(numbers that describe the quantum
mechanical properties of orbitals; from
the solutions to Schrödinger’s wave
equation)
The Principal Quantum
Number (n)
The integer, n, that Bohr used to label
the orbits and energies describes a
main shell of electrons, and is referred
to today as the principal quantum
number. Bohr’s theory used only one
quantum number, which is the main
reason that it worked well for hydrogen
but not for other atoms.
The Secondary Quantum
Number, (l)
Arnold Sommerfeld (1915) boldly
employed elliptical orbits to extend the
Bohr theory and successfully explain
that the main lines of the bright-line
spectrum for hydrogen were actually
composed of more than one line.
He introduced the secondary quantum
number, l, to describe additional
electron energy sublevels, or
subshells, that formed part of a main
energy level.
Using the analogy of a staircase for an
energy level, this means that one of
Bohr’s main energy “steps” is actually a
group of several little “steps”.
Notice that the number of sublevels equals
the value of the principal quantum number.
The Magnetic Quantum
Number, ml
The scientific work of analyzing atomic
spectra was still not complete. If a gas
discharge tube is placed near a strong
magnet, some single lines split into new
lines that were not initially present. This
observation was first made by Pieter
Zeeman in 1897 and is called the
normal Zeeman Effect.
He observed, for example, triplets
where only one line existed without the
magnetic field. The Zeeman effect was
explained using another quantum
number, the magnetic quantum
number, ml , added by Arnold
Sommerfeld and Peter Debye (1916).
Their explanation was that orbits could
exist at various angles. The idea is that
if orbits are oriented in space in different
planes, the energies of the orbits are
different when the atom is near a strong
magnet.
Shapes and Orientations of Orbitals
The Spin Quantum Number,
ms
Paramagnetism is another kind of
magnetism of substances and is
recognized as a relatively weak
attraction to a strong magnet.
Paramagnetism refers to the magnetism
of individual atoms; ferromagnetism is
due to the magnetism of a collection of
atoms.
Samuel Goudsmit and George
Uhlenbeck, found that a fourth quantum
number was necessary to account for
the details of the emission spectra of
atoms due to paramagnetism.
Since they knew from classical physics
that a spinning charge produces a
magnetic moment, it seemed
reasonable to assume that the electron
could have two oppositely directed “spin
states”
In 1925, Wolfgang Pauli, suggested that
each electron spins on its axis. For an
electron, the two spins are equal in
magnitude but opposite in direction, and
these are the only choices; i.e., the spin
is quantized to two and only two values.
This fourth quantum number is called
the spin quantum number,ms, and is
given values of either +1/2 or -1/2.
Qualitatively, we refer to the spin as
either clockwise or counterclockwise or
as up or down.
Pauli’s Exclusion Principle
In a given atom, no two
electrons can have the same
set of four quantum numbers
(n, l, ml, and ms).
Summary of Quantum Numbers
