Bohr Model of Hydrogen atom
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• In 1913, the Danish physicist Niels H. D. Bohr
proposed a model of the hydrogen atom that
combined the work of Planck, Einstein, and
Rutherford and was remarkably successful in
predicting the observed spectrum of hydrogen
• The Rutherford model assigned charge and mass
to the nucleus but was silent regarding the
distribution of the charge and mass of the
electrons.
• Bohr made an assumption that the electron in
the hydrogen atom moved in an orbit about the
positive nucleus, bound by the electrostatic
attraction of the nucleus
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• Such a model is mechanically stable because
the Coulomb potential V = -kZe2/r provides
the centripetal force
kZe 2 mv 2
F 2 
r
r
(# # )
necessary for the electron to move in a circle of
radius r at speed v
The total energy of the electron is the sum of
the kinetic and the potential energies
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• From eqn (##)
• We see
So, the total energy can be written as
•
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Bohr’s postulates
1. Electrons could move in certain orbits without
radiating. He called these orbits stationary states.
2. The atom radiates when the electron makes a
transition from one stationary state to another
(Fig below) and that the frequency f of the
emitted radiation is not the frequency of motion
in either stable orbit but is related to the energies
of the orbits by Planck’s theory
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•
• where h is Planck’s constant
and Ei and Ef are the
energies of the initial and
final states.
• This equation is referred to
as the Bohr frequency
condition.
• (Fig aside) The electron
orbits without radiating
until it jumps to another
allowed radius of lower
energy, at which time
radiation is emitted
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• In order to determine the energies of the allowed,
nonradiating orbits, Bohr made a third assumption,
now known as the correspondence principle, which
had profound implications:
3. In the limit of large orbits and large energies,
quantum calculations must agree with classical
calculations
• Thus, the correspondence principle says that,
whatever modifications of classical physics are made
to describe matter at the submicroscopic level,
when the results are extended to the macroscopic
world, they must agree with those from the classical
laws of physics that have been so abundantly
verified in the everyday world
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• Solving Equation (##) for the speed of the
orbiting electron yields,
• Bohr’s quantization of the angular momentum
L is
• where the integer n is called a quantum
number and
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• The circular orbits r becomes;
Then,
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is called the Bohr radius
• The stationary orbits of Bohr’s first postulate
have quantized radii denoted by the subscript
on rn
• Notice that the Bohr radius a0 for hydrogen (Z =
1) corresponds to the orbit radius with n = 1,
the smallest Bohr orbit possible for the
electron in a hydrogen atom.
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• The total energy of the electron (eqn ###) then
becomes
• Thus, the energy of the electron is also
quantized; that is, the stationary states
correspond to specific values of the total energy
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• This means that energies Ei and Ef that appear
in the frequency condition of Bohr’s second
postulate must be from the allowed set En
• Then,
which can be written in the form of the Rydberg equation
by substituting f = c/λ and dividing by c to obtain
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is Bohr’s prediction for the value of the
Rydberg constant
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• Using the values of m, e, c, and  , Bohr
calculated R and found his result to agree
(within the limits of uncertainties of the
constants) with the value obtained from
spectroscopy,
• The possible values of the energy of the
hydrogen atom predicted by Bohr’s model are
given by
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is the magnitude of En with n = 1.
E1 = -E0 is called the ground state
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Energy-level diagram for hydrogen
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The spectral lines corresponding to the transitions shown
for the three series (Lyman, Balmer, and Paschen series)
The energy required to remove the electron
from the atom, 13.6 eV, is called the
ionization energy, or binding energy, of the
electron.
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Example:
Compute the wavelength of the Hβ spectral line,
that is, the second line of the Balmer series
predicted by Bohr’s model.
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Reduced Mass Correction
• The assumption by Bohr that the nucleus is
fixed is equivalent to the assumption that it has
infinite mass
• Note: the Rydberg constant is normally written
as R
• If the nucleus has mass M, its kinetic energy
will be
• P = Mv is the momentum.
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• The total kinetic energy of the nucleus and
electron is
• Is the reduced mass
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• Using this reduced mass, the Rydberg constant
is then written
Example:
Compute the Rydberg constants for H and He+
applying the reduced mass correction
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Correspondence Principle
• According to the correspondence principle, when
the energy levels are closely spaced, quantization
should have little effect; classical and quantum
calculations should give the same results
• From the energy-level diagram, the energy levels
are close together when the quantum number n
is large
• This leads us to a slightly different statement of
Bohr’s correspondence principle: In the region of
very large quantum numbers, quantum
calculation and classical calculation must yield
the same results
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• If we compare the frequency of a transition
between level ni = n and level nf = n-1 for
large n with the classical frequency, which is
the frequency of revolution of the electron.
For large n,
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• The classical frequency of revolution of the
electron is
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which is the same as Equation (####)
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Quantum Mechanical description of
One-electron atoms
• Hydrogen is used
because it is a one
electron atom and the
simplest bound system
in nature
• The proton (+ve Charge)
is at the center and the
electron (-ve charge) is
moving around the
center
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• The attractive force between them is
Coloumbic in nature and the potential is
• Where Z is the charge of the nucleus
• Z = 1 for hydrogen
• Since it is a two body problem, we convert
it to one body problem by introducing
reduced mass
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Where m and M are the masses of the
electron and proton, respectively.
• So, the total energy of the system =
kinetic energy + potential energy
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• For the quantum mechanical treatment, we
will convert the classical dynamical quantities
(px, py, and pz ) to its corresponding quantum
mechanical operators
Substituting in Equation (1), we get
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• We introduce the WAVEFUNTION to
represent the electron
• The wavefunction contains the
information about the position and the
time evolution of the electronic motion
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• So operating eqn (2), on the wavefuntion,
we obtain
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• Eqn (3) is known as Time dependent
Schrodinger Equation
• Since the potential V( z, y, x) does not
depend on time t, and we are interested to
evaluate the energy of the stationary (time
independent) states, we can take this
wavefunction as the explicit dependence of
time such as
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• Substituting this to Eqn (3), we get
• This is the Time Independent Schrӧdinger
Equation
• Now we have to evaluate the stationary
state energies (E) of the electron in this
system by solving this equation to explain
the observed spectra of hydrogen
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• Since the potential is spherically symmetric,
• Converting Eqn (4) to its spherical polar
coordinate form (Figure below),
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• Radial distance ⇒ r,
Polar angle ⇒ θ and
Azimuthal angle ⇒ φ
• The form of the
Laplacian operator in
Spherical polar
coordinates is shown
below
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• Using the separation of the variables, we
can write the wavefunction as the product
form of the independent variables
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Substituting in eqn (5), we get,
• Carrying out the partial differentiation,
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• Multiplying the this equation by
and rearranging we get,
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• Separating the radial (R(r)) and angular (Θ(θ),
Φ(φ)) part
• Note here that, the partial derivative
forms are converted to total derivative
form.
• Now each equation should be equal to a
constant, let’s take as λ
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• Now, separating the polar and the
azimuthal part
• Here, we have taken that both sides should
be equal to a constant m2
• Now let us first consider the azimuthal
part
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• The general solution of this equation
may be written in the form
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• Which satisfies the orthonormality condition
• As the eigen functions must be single valued,
• And using Euler’s formula,
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• This is satisfied only if ml = 0, ±1, ±2, ..
• Therefore, acceptable solutions only exist
when ml can only have certain integer
values, i.e. it is a quantum number.
• Thus, ml is called the magnetic quantum
number in spectroscopy because it plays
role when atom interacts with magnetic
fields
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• For the polar part,
• Rearranging, we get
Let us introduce new variable ω=Cosθ, then,
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• The Eqn (5) has singularities at ω =±1 , which
may be eliminated by having a solution Θ in
the form of
• Then Eqn (5) becomes
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• Which can be written as
• The last singular term in Eqn (6) can be
removed by taking s = ± ml ≥ 0, and then we
have
Which is a regular equation and hence its series
solution may be written as
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Substituting in eqn (7) yields,
Requiring that the series (Eqn 8) be limited by a
certain power r = q, i.e. requiring that it be a
polynomial of order q, we have to introduce the
condition, ar+2 = 0, ar≠ 0 which requires
Where q = 0, 1, 2, 3, ….
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• Now we introduce l=q+ml = 0,1,2,3,.....
(orbital quantum number) such that
• Then Eqn (5b) becomes
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• This is well known associated Legendre differential
equation with its solution as associated Legendre
polynomials
ml
C
• Where l
is the normalization factor and
is the associated Legendre polynomial defined by
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• Where Pl(ω) is the ordinary Legendre
Polynomial.
• This expression holds for negative values of ml
also
• From this we can establish the range of
variation of the azimuthal (magnetic)
quantum number ml = 0, ±1, ±2, ±3, ……, ±l
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• The associated Legendre Polynomials satisfy
the orthonormilization property
• Then Equation 10 becomes
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• So, we have now two quantum numbers,
namely, Orbital (l) and magnetic (ml)
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• Now, we can have the total angular part, from
Equation-10 and -11
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