Download Quantum mechanics provides us with an understanding of atomic

Quantum mechanics
provides us with an
understanding of atomic
structure and atomic
properties. Lasers are one of
the most important
applications of the quantummechanical properties of
atoms and light.
Chapter Goal: To
understand the structure and
properties of atoms.
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Topics today:
•  Two dimensional waves
•  The Hydrogen Atom: Angular Momentum
and Energy
•  The Hydrogen Atom: Wave Functions and
Probabilities
•  The rest on Wednesday
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D. Eigler (IBM)
•  48 Iron atoms assembled into a circular ring.
•  The ripples inside the ring reflect the electron quantum states of a circular
ring (interference effects).
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Particle in box: 2 dimensions
Motion in x direction
Motion in y direction
For a rectangular box, the x and y motions are
independent. For a general box, the classical
motion may be chaotic.
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Particle in box: 2 dimensions
Schrodinger’s equation:
Solution is a product of standing wave solutions, one for each
dimension:
The energy is the sum of energies along each dimension:
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Quantum Wave Functions for 2-D box
Probability
(2D)
Wavefunction
Ground state: same
wavelength (longest) in
both x and y
Need two quantum #’s,
one for x-motion
one for y-motion
Denote a q# pair (nx, ny)
Ground state: (1,1)
Probability = (Wavefunction)2
One-dimensional (1D) case
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2D excited states
(nx, ny) = (2,1)
(nx, ny) = (1,2)
These have exactly the same energy for a square
box, but the probabilities look different.
The different states correspond to ball bouncing
in x or in y direction.
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Particle in a box
What quantum state could this be?
A. nx=2, ny=2
B. nx=3, ny=2
C. nx=1, ny=2
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Waves confined by a circular
box vanish at the edge radius
and are products of a radial
wave and an azimuthal wave:
The two quantum numbers n and m count nodes in the radial
and azimuthal probability densities.
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Three dimensions
•  Object can have different velocity (hence
wavelength) in x, y, or z directions.
–  Need three quantum numbers to label state
•  (nx, ny , nz) labels each quantum state
(a triplet of integers)
•  Each point in three-dimensional space has a
probability associated with it.
•  Not enough dimensions to plot probability
•  But can plot a surface of constant probability.
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Particle in 3D box
•  Ground state
surface of constant
probability
•  (nx, ny, nz)=(1,1,1)
2D case
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(121)
(211)
All these states have the same
energy, but different probabilities
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(112)
(222)
(221)
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3-D particle in box: summary
•  Three quantum numbers (nx,ny,nz) label each state
–  nx,y,z=1, 2, 3 … (integers starting at 1)
•  Each state has different motion in x, y, z
•  Quantum numbers determine
–  Momentum in each direction: e.g.
–  Energy: for a cube
px =
h
h
= nx
λn x
2L
2
2
p
p
p
E=
+ y + z = E o ( n x2 + n y2 + n z2 )
2m 2m 2m
2
x
•  Some quantum states have same energy for a cube. The
degeneracy is related to the symmetry of the cube.
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Use spherical coordinates matched to symmetry
Must write the Laplacian in spherical coordinates and solve.
Solutions are products of waves in each coordinate
Solutions specified by three quantum numbers n, l, m with
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Use spherical coordinates matched to symmetry
Solutions are products of waves in each coordinate
Solutions specified by three quantum numbers n, l, m with
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Solutions to the Schrödinger equation for the hydrogen
atom potential energy exist only if three conditions are
satisfied:
1.  The atom’s energy must be one of the values
where aB is the Bohr radius. The integer n is called
the principal quantum number. These energies are
the same as those in the Bohr hydrogen atom. Note
that the energy does not depend on l and m. Thi is a
quirk of the Coulomb potential.
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2.  The angular momentum L of the electron’s orbit must
be one of the values
The integer l is called the orbital
quantum number.
3.  The z-component of the angular
momentum must be one of the values
The integer m is called the magnetic quantum number.
Each stationary state of the hydrogen atom is identified by a
triplet of quantum numbers (n, l, m).
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The quantum number ‘n’ is the number
of nodes in radial direction.
The quantum number ‘l’ is the number
of nodes in theta.
The quantum number ‘m’ is the number
of nodes in azimuth.
Subtle classical correlation:
Orbital angular momentum is conserved
and can be related to l and m. The wave
factor exp(-im phi) implies spatial
variation around the z axis and current
density proportional to m.
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Wave representing
electron
 
 
Electron wave extends around
circumference of orbit.
Only integer number of
wavelengths around orbit
allowed.
Wave representing
electron
Bohr’s wave is the
azimuthal factor in the
complete wave mechanics
solution.
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The Bohr model computes
energy independent of orbit
orientation. The full wave theory
of spherically symmetric bound
states in 3-d and finds an
increasing number of states as
the “radius” (n) increases. It is a
peculiarity of the Coulomb
potential that states with different
angular momentum may have the
same energy. This is NOT true
for a multielectron atom for
which the effective potential is
not 1/r.
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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Hydrogen atom:
Lowest energy (ground) state
1s-state
n = 1,  = 0, m = 0
•  Spherically symmetric.
•  Probability decreases
exponentially with
radius.
•  Shown here is a surface
of constant probability
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n=2: next highest energy
2s-state
2p-state
n = 2,  = 0, m = 0
n = 2,  = 1, m = 0
Same energy, but different probabilities
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2p-state
n = 2,  = 1, m = ±1
n=3: two s-states, six p-states and…
3p-state
3s-state
3p-state
n = 3,  = 0, m = 0
n = 3,  = 1, m = 0
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n = 3,  = 1, m = ±1
…ten d-states
3d-state
n = 3,  = 2, m = 0
3d-state
n = 3,  = 2, m = ±1
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3d-state
n = 3,  = 2, m = ±2
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The probability of finding an electron within a shell of
radius r and thickness δr around a proton is
where the first three radial wave functions of the electron in
a neutral hydrogen atom are
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Note that the total
wavefunction
amplitude is
complex and both
the real and
imaginary parts can
be positive or
negative. None of
these four
components is
readily visualized in
3 dimensions!
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Notice how the mean
radius increases with n
as in the Bohr model.
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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.