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H  E or  K  V   E


 h 2   2

2
2 
 2  2  2  2   V ( x, y, z )  ( x, y, z )  E ( x, y, z )
z 
 8 m  x y

Solving Schrodinger
Eqn. for H atom
Prob(r, , )   (r, , )
n, l, m
Quantum numbers
n, l, m
Characteristics of the
Orbital: 3D Waves
2
Born’s Interpretaion of
Orbitals
Orbitals and Orbits:
Heisenberg’s
Uncertainty Principle An Analogy
Eu  E  E  h
Electronic Transitions
“In Chapter 3, 3.6 and 3.7 will NOT be delivered in the classes, and will be excluded
from midterm exam.”
Chapter 3. Wave Mechanics and the Hydrogen Atom:
Quantum Numbers, Energy Levels, and Orbitals
3.1 Solving the Schrödinger Equation for the H Atom (appendix B)
An operator can be defined in mathematics as follows:
df ( x)  d 
^
   f ( x)  D f ( x)
dx
 dx 
differential operator
Operator is an entity operating automatically on the factor to its right
The Schrodinger Equation can be simplified by defining Hamiltonian
operator.
 _ h2   2

2
2 
 2  2  2  2   V ( x, y, z )  ( x, y, z )  E ( x, y, z )
y
z 
 8 m  x

Hamiltonian operator, Ĥ
^
H  E or  K^  V   E


^
K
 kinetic energy operator
Solving the Schrödinger Eq: ĤΨ=EΨ

To find the wave function Ψ and E
 you need to solve the differential eq

The solutions depend on the potential energy V(x,y,z)

Demonstration by a particle in a box

The Coulomb potential for e – nucleus interaction

Spherical coordinate is more convenient than the
Cartesian coordinate for the central or radial Coulomb
potential
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Schrödinger Equation: Wave Equation for Particles
The classical wave eq for describing a standing wave function ψ
in one dimension: d2ψ/dx2 + (4π2/λ2)ψ = 0
From de Broglie: λ2 = h2/2mK = h2/[2m(E-V)]
d2ψ/dx2 + (4π2/λ2)ψ = d2ψ/dx2 + [8π2m(E-V)/h2]ψ = 0 or
-(h2/ 8π2m)(d2ψ/dx2) + Vψ = Eψ ………….(Eq 1)
Let’s define ‘momentum operator’, p^ = (h/2πi)(d/dx),
based on p = mv = mdx/dt (see p739).
And the kinetic energy operator:
K^ = p^2/2m = (1/2m)(h/2πi)2(d/dx)2 = - (h2/ 8π2m)(d2/dx2)…(Eq 2)
Eq 1 and Eq 2
K^ψ + Vψ = (K^ + V)ψ = Eψ or H^ψ = Eψ
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A Particle in a Box (Appendix B)
Solving the Schrödinger Eq: ĤΨ=EΨ
Boundary condition:
Boundary Condition yields quantum numbers
Energy
Wavefunction
A Particle in a Box (Appendix B)
A Particle in a Box (Appendix B)
Wavefunction
Normalization condition
Schrödinger Eq for Hydrogen Atom
V(x, y, z) – can be expressed in various coordinate systems.
r cosΘ= z
Spherical polar coordinates (r, θ, Ф) SPC
vs. Cartesian coordinates (x, y, z)
Coulomb potential, V → convenient to express
in
SPC (V = -e2/r) ; Cartesian, V = -e2/[x2 + y2 +z2]1/2
- reducing to that of an electron moving
around a fixed proton.
 h 2   2

2
2 
 2  2  2  2   V ( x, y, z )  ( x, y, z )  E ( x, y, z )
z 
 8 m  x y

Cartesian
Coordinate
 h 2   2

2
2 
e 2
 2  2  2  2 
 ( x, y, z )  E ( x, y, z )
2
2
2
z 
( x  y  z ) 
 8 m  x y
Spherical
Polar Coordn.
 h 2  1   2  
1
 
 
1
 2  e 2 

 2  2 r
 ( x, y, z )  E ( x, y, z )
 2
 sin 
 2 2
2 
8

m
r

r

r
r
sin





r
sin



r








Spherical Polar Coordinate
 h 2  1   2  
1
 
 
1
 2  e 2 

 2  2 r
 ( x, y, z )  E ( x, y, z )
 2
 sin 
 2 2
2 
8

m
r

r

r
r
sin





r
sin



r








  h 2  1   2    e 2

 1
 

r


E

(
x
,
y
,
z
)

sin

 2  2 


 r 2 sin   



 8 m  r r  r   r

1
2 

 ( x, y , z )  0
 2 2
2
 r sin   
  h 2    2    e 2 r 2
 1  
 
1 2 
2
 Er  ( x, y, z )  
 ( x, y , z )  0
 2  r
 
 sin 
 2
2
8

m

r

r
r
sin





sin






 




Depends only on r
Depends only on , 
Schrödinger wave function
→ factored into radial (R) and angular (Y) functions
 (r , , )  R(r )Y ( , )
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Boundary Condition
constraint that reality places on the solutions to a
physically relevant equation (e.g., a quadratic equation having + &
- two solutions for concentration)
* Ψ is just the embodiment of de Broglie’s hypothesis of matter wave)
 Ψ must be smooth, single-valued, and finite everywhere in space
 Ψ must become small at large distances r from the nucleus (proton)
Boundary Condition yields quantum numbers
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3.2 Quantum Numbers n, ℓ, and m
Wave function Ψ as the standing-wave for motions
Characterized by 3 integers, n, ℓ, m → quantum numbers
Why 3 quantum numbers? -> Because we have three dimensions
(x, y, z) or (r, ,  ) one quantum number for each dimension is
required.)
Principal quantum number, n
Energy from Schrödinger equation depends only on n
n = 1, 2, 3, ····
Azimuthal quantum no, ℓ (Angular momentum q. no.)
Associated with variation of an angle, θ
ℓ = 0, 1, 2, 3, ···· , n-1
Magnetic quantum no, m (Projection quantum no.)
behavior of H-atom in a magnetic field
m = - ℓ, - ℓ+1, -ℓ +2, ···,-1, 0, 1, 2, ··· ℓ-1, ℓ
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Degeneracy
“Normally” the energy should depend on all three quantum
numbers.
Hydrogen atom is special in that the energy depends only on
principal quantum number n.
Two or more sets of quantum numbers corresponds to the
same energy: referred as “degenerate”
for example) 200, 211, 210, 21-1 states have the same
energy.
Each n: given total energy → n2 possible combinations of
quantum numbers → degeneracy
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Specification of the Energy: Ionization
Ionization energy, IE
Energy levels of hydrogen atom
Ionization continuum:
Electron’s energy is no longer quantized !
IE in excited state n is -En
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Bohr – de Broglie Model
Specification of the Wave Functions: Orbitals
Schrödinger’s Wave Function
Ψ should be single valued, continuous as a function of Θ and Φ:
the wave meets itself as Θ and Φ cycle around the origin
← with quantum numbers
Ex: For n = 3 (compare the one orbit in the Bohr-de Broglie model)
32 (nine) different ways the electron can vibrate to form
a standing wave → 9 degeneracy
Orbitals
Mulliken: Ψ = a 3D Schrödinger’s wave function. He suggested the
term, ‘orbital’, to refer to Ψ (= a 3D Schrödinger’s wave function) as a
replacement of Bohr’s orbit
Radial part, Rnℓ(r)
Laguerre polynomial of (n-1) degree
(highest power = rn-1) multiplied by e-r/(na0 )
Here, * a0 ← Bohr’s radius
Angular part, Yℓm(θ,Ф) Legendre function (sin θ & cos θ) x eimФ
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Department of Chemistry, KAIST
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3.3 Characteristics of the Orbitals:
Three-Dimensional Waves
H-atom (the only atom for which the Schrödinger Eq can be solved
exactly  a model for bigger and many-electron atoms)
n = 1, ℓ = 0, m = 0 → R10(r) and Y00(θ,Ф)
1s orbital of E1
a function of r only
* spherically symmetric
* exponential decaying
* no nodes
a) 1D Ψ vs. r
b) 2D mountain by rotating
graph (a)
c) 2D contour map of
the amplitude
d) 3D cloud, proportional to
the amplitude
e) 3D boundary surface
diagram
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Nodes: places where the wave passes through zero amplitude
“You can start to see nodes in the excited states, which brings us to n = 2”
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n = 2, nℓm sets = 200, 210, 211, 21-1
2s (n = 2, ℓ = 0)
2p (n = 2, ℓ = 1)
Ψ200((r,θ,Ф) = (R20)(Y00)
← 2s orbital
zero at
r = 2a0 = 1.06Å nodal sphere or radial node
[r<2ao: Ψ>0, positive] [r>2ao: Ψ<0, negative]
<node: due to a + - oscillation in the wave>
Spectroscopic symbols
ℓ = 0 → s (sharp), ℓ = 1 → p (principal: most intense)
ℓ = 2 → d (diffuse), ℓ = 3 → f (fundamental)
ℓ=4 → g
ℓ = 5 → h, etc
n=2 orbitals for H-Atom
(a) 1D radial wave fn.
vs. r
(b) 2D cut through 3D
clouds
(c) 3D boundary
surface
shade-negative
(right)
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n = 2, ℓ =1, m = 0 → 2p0 orbital : R21 Y10
· Ф = 0 → cylindrical symmetry about the z-axis
· R21(r) → r/a0 no radial nodes except at the origin
· cos θ → angular node at θ = 90o, x-y nodal plane (positive/negative)
· r cos θ → z-axis 2p0 → labeled as 2pz
n = 2, ℓ = 1, m = ±1 → 2p+1 and 2p-1 (complex functions containing
both real and imaginary parts: a problem in graphing!)
Take their linear combinations to obtain
two real orbitals → constructed 2px and 2py
· Use Y11(θ,Ф) → e±iФ = cosФ ± i sinФ ← Euler’s formula was used
2px and 2py (real functions using Euler’s formula)
Px and py differ from pz only in the angular factors (orientations)
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n = 3, ℓ = 0(3s), 1(3p), 2(3d)
Generalization
· ns orbital → n -1 radial nodes,
________________
the same as the degree of R(r) polynomial (n-1
real roots)
· ℓ increases from 0 to n -1 → number of nodes stays same
· but, ℓ of the radial nodes → are exchanged for angular
nodes, in the form of nodal plane or nodal cone
· Algebraically, R(r) → loses one root for an increase in ℓ
by one and the degree of Y(θ,Ф) in cosθ sinθ
→ increases by one
Fig. 3.5
n = 3 orbitals for H
(a) Radial parts
(b) 2D cuts through
3D amplitude cloud
(c) Boundary surface
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Fig. 3.6
Boundary surface
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 h2   2

2
2 
 2  2  2  2   V ( x, y, z )  ( x, y, z )  E ( x, y, z )
z 
 8 m  x y

The kinetic and potential energies are transformed into the
Hamiltonian which acts upon the wavefunction to give the
quantized energies of the system and the form of the
wavefunction so that other properties may be calculated.
The wave nature of the electron has been clearly shown in
experiments like the Davisson-Germer experiment. This raises
the question "What is the nature of the wave?“:
The wave is the wavefunction for the electron.
Then, what is the nature of the wavefunction??
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3.4 Born’s Interpretations of Orbitals

Schrödinger wavefunction: Not much physical significance, just a
tool for predicting atomic properties

Mulliken: Orbitals as the replacement of the planet-like orbits of
the Bohr model

Bohr: Anything that could create a pulse in one of Geiger’s
electron detector tubes couldn’t be a smooth, continuous wave
Is there anything more we can say about the orbitals of the hydrogen atom?

Born: Quantum version of the orbit of the electron. In those
regions of space where the wave function is large, the electron is
more likely to be found in orbit. Intensity of light ∝ amplitude 2
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3.4 Born’s Interpretations of Orbitals
Probability of finding an electron
Prob(r,θ,Ф) ∝ ┃Ψ(r,θ,Ф)┃2
(3.13)
probability probability density probability amplitude (Ψ)
Normalization constant → a proportionality const. in Eq.(3.13)
to meet the total probability = 1.
┃Ψ┃2 x r2 sinθ dr dθ dФ → the probability of finding an
electron within that tiny volume
volume element
┃Ψ┃2 → electron density
-e┃Ψ┃2 → charge density
: ”wave function must be the quantum version of the orbit of the electrons”
: “in space where the wave function is large, the electron is more likely to
be found in orbit”
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In 1926 he collaborated with his student
Werner Heisenberg to develop the
mathematical formulation of the wave
function in the new quantum theory.
He later showed, in the work for which he
is perhaps best known, that the solution of
the Schrödinger equation has a statistical
meaning of physical significance.
“Probability cloud” is a strange sort of mist
made up of a single electron !
Max Born (1882-1970, Germany)
1926, Probability density function
1954 Nobel Prize in Physics
“The theory yields a lot, but it hardly brings
us any closer to the secret of the Old One.
In any case I am convinced that He does
not throw dice”
-Einstein-
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Radial Distribution Functions: RDFs
Prob(r) = r2[R(r)]2
2

 
0
 r 2 sin  d d
2
0
 4 r   4 r R (r )Y ( , )
2
2
2
 4 r 2 R (r ) Y ( , )
2
 4 r R (r ) 1/ 2 
2
2
2
2
 r 2 R(r )
2
2
┃Ψ┃2 x 4πr2 dr
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Schrödinger’s waves vs Bohr’s orbits

Bohr’s orbits: Each orbit has a fixed radius,
rn = n2 ao ; the electron is always found at r
= rn ; Bohr’s RDF is a sharp spike at rn.

Schrödinger’s RDF is a smooth curve with
one or more peaks
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Probability amplitude and
Probability density
┃Ψ┃2 : electron density
-e┃Ψ┃2 : charge density
Ψ vs ┃Ψ┃2
1) nodal surfaces are not changed
2) negative parts of orbitals
become positive
3) boundary surfaces are same
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RDF for H
RDF for H
Smooth with one or more peaks
Nodes appear at radii of zero
probability
Falls to zero smoothly at large r
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Bohr’s Model for Hydrogen Atom
1s
electron density
ao
r
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2s RDF:
the electrons spend most of its time in the range of outer peak,
and much less around the inner peak. How ??
How the electron gets from the outer to inner region
without trapping into the node at the nodal space ??
Key is in the Wave Property of Electrons: the standing wave
has amplitude everywhere simultaneously
Outer turning point – In Newton’s mechanics, a bound particle
cannot stray beyond a fixed radius.
Quantum tunneling – An electron in the orbital has a small but
finite probability beyond the outer turning point.
3.5 Heisenberg’s Uncertainty Principle
Heisenberg’s uncertainty principle
Position uncertainty x momentum uncertainty ≥ h/4π, at the same time
( = 5.27 x 10-35J s )
(The principle is valid between any pair of the conjugate variables)
ΔE Δ t ≥ h / 4
(In effect, the electron’s energy fluctuates within narrow bounds, and what is
supposed as the electron’s energy is, in fact, an average value over the very
narrow time parameter)
(This fluctuating electron energy might suggest a violation of the conservation
of energy, but not if the electron is exchanging energy at the Planck level with
other electrons or particles)
The basis for the initial realization of fundamental uncertainties in the ability of an
experimenter to measure more than one quantum variable at a time.
Figure 3.10.
Two views of
Heisenberg’s
uncertainty principle
λ ~ 0.1 Å
or 105 eV
IE (H) = 13.6 eV
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Why does a hydrogen atom have a ground state, and
what determines the energy of that state?

To require the electron to be roughly a distance ao from the
nucleus implies an uncertainty in momentum ∆p ~ h/ao

p can not be much less than ∆p

p ~ h/ao and K ~ p2/m ~ h2/mao2

Since ao ~ h2/me2, K ~ me4/h2 ~ -E1 (page 59, Eq. 2-13)

Confining a particle more tightly increases its kinetic energy →
The electron cannot collapse into the proton even though the
Coulomb attraction drawing it in → Small particle in bound
states show a lowest state where K is nonzero; the particle is
said to possess zero point energy.
Werner Heisenberg (1901-1976)
Born’s Assistant
1927, Uncertainty Principle
1932 Nobel Prize in Physics
“The battle with Born grew so intense in the early
months of 1927 that Heisenberg reportedly burst into
tears at one point,…”
“The theory yields a lot, but it hardly brings us any
closer to the secret of the Old One. In any case I am
convinced that He does not throw dice”
–Einstein-
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3.6 Orbitals and Orbits: An Analogy
Orbitals – analysis in terms of radial and angular motion
meaning of n, ℓ, and m.
Electronic motion → in-and-out, ① vibrational (r), and
round-and-round, ② rotational (Θ,Φ)motions.
i) Vibrational motion
· distance r changes with the motion
· determined by n - ℓ- 1, specifies the no. of nodes in the
radial function
· smaller ℓ for a given n → more vibration
ii) Rotational motion
· angles θ and Ф change with the motion
· determined directly by ℓ
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iii) Orientations of Rotational Plane
· z-axis projection of the angular momentum → determines
the possible orientations of the plane of rotation.
For fixed ℓ
m = 0 → The rotational plane is contained in the z-axis.
m = /ℓ/ → The plane is fixed in the x-y plane, perpendicular
to the z-axis.
m = 0-ℓ →The plane is “tilted”.
m = -ℓ → The rotation is clockwise.
m = +ℓ → The rotation is counter clockwise.
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Classical analogous orbits of the electron in 1s, 2s and 2p orbitals
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Bohr’s correspondence principle
As the quantum numbers get larger and larger, the
classical analogy is found to get better and better.
In the limit of large quantum numbers, the behavior of
any mechanical system becomes classical.
Quantum mechanics contains Newtonian mechanics
as a special case.
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3.7 A Qualitative Description of Electronic
Transition

Imagine a hydrogen atom in a stream of photons (in a
ray of light) along the x-axis: The oscillating electric
field Ex will exert an electrical force on the electron by
–eEx and the potential energy of eExx.

e·x : electric dipole

Electric dipole transition

Selection rule for H: ∆ ℓ ≡ ± 1
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3.7 Qualitative Description of Electronic Transition
Electric dipole transition
Selection rules
En state
Electric dipole transition
hν
Emission
Δl ≡ l2 - l1 = ±1
1s to 2p
NOT 1s to 2s
Eℓ state
Energy uncertainty
xpx  xpx
energy uncertainty
t  x

  mv  t   mvv  t
t  t

time uncertainty (Δt  τ- radiative lifetime)
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