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### Bohr's model worked for a simple system, but did not
completely explain additional detail associated with the emission
spectrum. To a point it worked. In these circumstances
modifications of the model are typical needed. The primary
modification of the Bohr model which resulted in our current view
of the electronic structure of the atom was to recognize the
electron exhibited wave properties as well as particle
properties. This rather simple statement was the result of
considerable labor. Let's go back to the origin of the ideas
which subsequently lead to this important modification of
describing the behavior of an electron in an atom.

### In his study of the Photoelectric effect Einstein recognized
that light exhibited particle behavior. When light of specific
frequency was incident on a metal surface electrons were ejected
from the metal. The incident light in causing electrons to be
ejected was behaving as a particle. Light, electromagnetic
radiation, could not only be thought of as having wave
properties, but also particle properties. A packet of light
(photon) had a frequency, but it also had an energy given by;

### Now we began paying a price for our search to elucidate the
structure of the atom and a gentleman by the name of Louis Victor
de Broglie is responsible. What he did was to make the atom an
abstract and and difficult to imagine. He made the universe
probabilistic and indeterminate. Now understand that the universe
has not changed, but our view of it has.

### The pivotal concept suggested by de Broglie, was that if
light could exhibit particle properties, and therefore, help us
explain phenomenon, then why could not matter have wave
properties. de Broglie was saying that matter, under certain
circumstances might also display appreciable wavelike properties.
He suggested that the wavelength of matter behaving in this
manner could be calculated using the relationship;

### where 'm' is the mass of the particle and 'v' is its
velocity.

### Called matter waves they should not be confused with the
waves of electromagnetic radiation. Matter waves are not radiated
into space or emitted by a particle; they are never dissociated
from the particle. The speed of matter waves is never the speed
of light, nor is it constant, but changes with the particle. de
Broglie suggested that electrons, because of their very small
mass and high velocity, could behave as waves. There existed
experimental evidence to support this idea. Even Thomson had
observed the beam of electrons in the cathode ray tube would cast
a shadow on the end of the tube when a piece of metal is placed
in the beam.

### For years after de Broglie's postulate Clinton Davidson and
L. Germer investigating the scattering of electrons by atoms in a
crystal showed that the electron beam was reflected (diffracted)
in an identical way that X-rays were diffracted. Exhibiting the
wave-like character of the electron. Erwin Schrodinger, an
Austrian theoretical physicist, assumed that the wavelike
character of the electron could be incorporated into a model of
the atom. And here is where our world falls apart, and it becomes
very difficult to describe electron behavior in atoms. Because
now the medium has turned form pictures to probability and
quantum mechanics or wave mechanics. The language is differential
calculus, the words are symbols and numbers. The results are
complex, but nonetheless dramatic.

### We will avoid the pit and try to describe the results. Try is
what I will do, because quantum mechanics changes the way we
think about the motion of particles as small as the electron.
Large objects moving slowly are easy to follow. Like a baseball.
A propeller turning slowly is also easy to follow, However, as we
speed up the blade it becomes difficult to know exactly where the
two ends of the propeller are located. Their movement describes a
region where, if we were to insert a pencil, we would expect to
encounter one end of the blade or the other. Outside its length
we would assume we would be safe. In the world of the atom the
motion of the electron is equally complex. It can be thought of
as smoothed out around the nucleus. We can not see an individual
electron or how fast it is going.

### Werner Heisenberg showed by using quantum mechanics that it
is impossible to know both the position and velocity of the
electron;

### Now this problem is only encountered in the realm of the
atom, we do not have such problem in the macroscopic world. (If
we did more of us would pay closer attention to the 'walk' and
'don't walk' sign at traffic signals). Heisenberg's
uncertainty principle says that because of the wavelike
character of the very small electron we can not know precisely
where the electron is when traveling along a path or trajectory.

### If we accept that an electron has wave character than an
electron can be described as a wave. If we use a single wave to
describe the velocity of the electron, as shown in the animation,
we do not know where the electron is located exactly. We have an
idea of its vertical location, between the peak maxima, but we do
not know where it is located horizontally. If we use a number of
different waves and add them together we can better locate the
position of the electron, but we have no idea what the velocity
of the electron is because of the different velocities of the
waves that contribute to the electron. So Heisenberg's uncertain
principle concludes that we can not know both the position and
the velocity of an electron simultaneously. This means we can not
know exactly where the electron is an any instant.

### We can not know how fast an electron moves in an atom (about
the nucleus). However, quantum
mechanics does allow us to describe the behavior of the atom
in a statistical sense. We talk about the probability of finding
the electron at a certain point in space in the vicinity of the
atom. We can not describe the electron as orbiting the nucleus.
Because to do so suggests the electron has a trajectory which
suggests we can know both the position and velocity of the
electron. So in quantum mechanics we describe the behavior of the
electron in three-dimensional terms. Instead of finding an
electron in an orbit, as Bohr's model suggested, we describe the
electron in an orbital, which is a 3-dimensional orbit. But now
the question becomes what 3-dimensional shape or shapes does an
electron in a hydrogen atom have?

### The information that we are seeking can be obtained by
solving a complex mathematical relationship, referred to as
Schrodinger's equation. It is a complex differential equation,
the unknown in the equation is , a mathematical expression called a wavefunction.
The square of is a
measure of the relative probability density of the electron at a
position (x,y,z). In optics the square of the amplitude of a wave
is called the intensity of radiation. We also recall the
intensity is interpreted as the number of photons present. Bright
light suggests many photons. For wavefunctions the square of the
amplitude indicates the probability of finding the particle at
each point in space. When the Schrodinger equation is solved we
obtain a general expression for .

### Boundary conditions are then applied that are associated with
the particular physical condition. For an electron in an atom,
the boundary conditions are that ^{2} must be continuous, single valued
and finite everywhere. This makes sense as probability functions
do not fluctuate radically from one place to another; the
probability of finding the electron a few thousandths of a
nanometer from a given location will not be radically different
from the probability at the original position. Second the
probability of finding an electron in a given place cannot have
two different values simultaneously. Third, since the probability
of finding an electron somewhere must be 100% or 1.000, if the
electron really exists, the probability at any one point cannot
be infinite.

### According to quantum mechanics the electron is spread out
like a wave, and because of Heisenberg's uncertainty principle we
can only speak of the probability of finding the electron at some
point in space. The wave which describes how the electron is
distributed, spacially, is called a wave function. And like any
other wave, the wave which describes an electron has regions
where the value of the wave is large and regions where the value
of the wave is small. These wave functions are also called atomic
orbitals and can be thought of as regions of space where the
probability of finding the electron is high. Recall that for the
Bohr atom the region of space where the probability of finding
the electron was defined as an orbit of a particular radius. This
was a two dimensional description which was inadequate. In
quantum mechanics we must describe the electron in three
dimensional shapes. In the the Bohr model the principle quantum
number, *n*, was all that was needed to calculate the energy
and the location of the electron because all of the shapes of the
orbits were identical..a circle. In quantum mechanics the atomic
orbitals require three quantum numbers to complete describe the
energy and the shape the electron occupies. In the Bohr model we
could describe the elctron as in an orbit. But in the quantum
mechanical model the electron occupies an orbital.

### But since we are talking about additional quantum numbers we
need to take a minute to discuss the relationship of these new
quantum numbers. Remember a quantum number is a whole number
which is used to label the of an electron in an atom.

### Since an orbital is 3-dimensional, the solutions to the
Schrodinger equation for the hydrogen atom are characterized by
three integer quantum numbers: *n*, *l*, and *m*_{l}.

### The three quantum numbers that are obtained from the
wavefunction are referred to as '*n*' the principal quantum
number, '*l*' the azimuthal quantum number and '*m*_{l}'
the magnetic quantum number.

### The three quantum numbers *n*, *l*, and *m*_{l}
are mathematically related to one another. The principal quantum
number ('*n*') can have whole number values equal to 1, 2,
3, 4, .....

### The principle quantum number (*n*) establishes the
energy of the electron and the size of the orbital (distance from
the nucleus). We also use the term shell in reference to '*n*'
value of an electron. The azimuthal quantum number ('*l*')
can have values that are related to '*n*'. '*l*' can
have integer values that range from 0 to *n*-1.

### For example, if* n*=1, then '*l*' can only have one
value and it is *l*=0. If *n*=2, then '*l*' can
have two values, 0 and 1. The '*l*' quantum numbers each
have a distinctive shape which is characteristic of the value of
'*l*'. All the '*l*' values associated with a
particular '*n*' value are called subshells. When *n*=1,
'*l*' can have only one value, therefore there is one
subshell associated with the *n*=1 shell. If* n*=2
there are two subshells. The magnetic quantum number can have
values which depend on '*l*'. They are related to '*l*'
in the following way;

*m*_{l} = -'*l*', -('*l*'+1),
... 0 ..., +('*l*'-1), +'*l*'

### The magnetic quantum number defines the number of orbitals in
a subshell. In the *l*=0 subshell there is only one *m*_{l}
value, therefore one orbital is associated with the *l*=0
subshell. If *l*=1 then there are three orbitals.

### So the set of quantum numbers define the state of an
electron. Taken together the three quantum number specify the
orbital the electron occupies. They also specify the energy of
the orbital, the shape of the orbital and the orientation of the
orbital.

### It becomes combersome to have to attach labels with three
quantum numbers to each possible state for an electron. To
simplify the notation a variation is used to label the orbital
the electron can occupy. In this variation the numeric value of
the principal quantum number is retained. However, the azimuthal
quantum number is identified with a letter: s (*l*=0), p (*l*=1),
d (*l*=2), f (*l*=3). We use these letters to denote
the subshells and we speak of a 1s subshell (*n*=1,*l*=0)
a 2s subshell (*n*=2,*l*=0) a 2p subshell (*n*=2,*l*=1)
etc. The magnetic quantum number and the spin quantum number are
combined. The *m*_{l} values are what we
referred to as orbitals and each orbital can have a maximum of
two electrons. There for an '*s*' subshell can only have one
orbital (*m*_{l}=0) which contains a maximum
of two electrons. A '*p*' subshell can have three orbitals (*m*_{l}
= -1,0,+1) each of which can hold a maximum of two electrons.
Therefore the '*p*' subshell has three orbitals which can
hold, together, six electrons. The '*d*' subshell has five
orbitals for a total of ten electrons. We can place all of these
orbitals on an energy diagram to show the orbital energies in the
hydrogen atom.

### Each of the orbitals for the quantum states differentiated by
*n*, *l*, and *m*_{l} corresponds
to a different probability distribution function for the electron
in space. The simplest such probability functions, for '*s*'
orbitals (*l* = 0), are spherically symmetrical. The
probability of finding the electron is the same in all directions
but varies with the distance from the nucleus. The electron is no
longer in an Bohr orbit of fixed distance from the nucleus,
rather, it is an electron probability cloud. These probability
clouds are called hydrogenlike atomic orbitals. The 2*s*
orbital is also spherically symmetrical, but its radial
distribution function has a node, that is, zero probability, at r
= .1058 nm from the nucleus. The highest density corresponds to
the Bohr radius of .212 nm. There is also a high probability of
finding the electron closer to and further from the nucleus than
the most probable of .212 nm. The 3*s* orbital is larger in
size than both the 1*s* and 2*s* orbitals. There are
three 2*p* orbitals, 2*p*_{x}, 2*p*_{y},
and 2*p*_{z}. Each orbital is cylinderically
symmetrical with respect to rotation around one of the three
principal axes x, y, or z, as identified by the subscript. Each *p*
orbital has two lobes of high probability density separated by a
nodal plane of zero density. The sign of the wavefunction,
&fnof, is positive in one lobe and negative in the other. The
3*p*, 4*p* and higher *p* orbitals have one, two
or more additional nodal shells around the nucleus (details of
secondary importance). The important feature is that the orbitals
get larger as the principal quantum number increases are mudally
perpendicular and strongly directional.

### The five *d* orbitals first appear for *n* = 3. The
*d*_{xz}, *d*_{yz}, and *d*_{xy}
are identical in shapebut differ in orientation. Each has four
lobes of electron density bisecting the angles between the
principal axes. The remaining two are unusal: the *d*_{x}*2*_{-y}*2*
orbital has lobes of density along the x and y axes, and the *d*_{z}*2*
orbital has lobes along the z axes, with a small doughnut or ring
in the xy plane.

### So the azimuthal quantum number can also be called the
orbital-shape quantum number: *s* orbitals are spherical, *p*
orbitals are cylinderical extensions of two lobes along the
principal axes and d orbitals have extensions along two mutually
perpendicular directions. The third quantum number describes the
orientation of the orbital in space. It is also called the
magnetic quantum number because in a magnetic field orbitals with
with different spatial orientations have different energies.

### Here is an animation with all
of this stuff put together.