Download Chapter 41. One-Dimensional Quantum Mechanics

Chapter 41. One-Dimensional Quantum
Mechanics
Quantum effects are
important in nanostructures
such as this tiny sign built by
scientists at IBM’s research
laboratory by moving xenon
atoms around on a metal
surface.
Chapter Goal: To
understand and apply the
essential ideas of quantum
mechanics.
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Consider a particle with mass m and mechanical energy E
in an environment characterized by a potential energy
function U(x). The Schrödinger equation for the particle’s
wave function is
Conditions the wave function must obey are
1.  ψ(x) is a continuous normalizable function.
2.  ψ(x) = 0 if U(x) is infinite.
3.  ψ(x) → 0 as x → +∞ and x → −∞.
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Consider a complex wave with wave vector k associated
with momentum by de Broglie and energy associated with
frequency by Einstein’s expression E=hf for photons
If E=p2/2m it is a solution to a complex wave free particle equation
Here we identify operators:
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To describe a wave interacting with a source of potential,
add a potential energy term to the “energy”
Look for a solution harmonic in time of the form
We arrive at the time-independent Schrodinger equation
We must find E and the spatial wave function simultaneously.
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The complex wave function can be written in terms of real
and imaginary parts linked through two equations
Like E and B, the two parts go hand in hand.
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Consider a particle of mass m confined in a rigid, onedimensional box. The boundaries of the box are at x = 0 and
x = L.
1.  The particle can move freely between 0 and L at
constant speed and thus with constant kinetic energy.
2.  No matter how much kinetic energy the particle has,
its turning points are at x = 0 and x = L.
3.  The regions x < 0 and x > L are forbidden. The
particle cannot leave the box.
A potential-energy function that describes the particle in
this situation is
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The solutions to the Schrödinger equation for a particle in a rigid
box are standing waves
What “waves” is an oscillation between real and imaginary parts
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The normalization condition, which we found in Chapter
40, is
This condition determines the constants A:
The normalized wave function for the particle in quantum
state n is
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.
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.
Quantum dot: particle in 3D box
CdSe quantum dots
dispersed in hexane
(Bawendi group, MIT)
Color from photon
absorption
Decreasing particle size
 
 
Energy level spacing increases
as particle size decreases.
i.e
2
2
n
+
1
h
n 2h 2
(
)
E n +1 − E n =
−
2
8mL
8mL2
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Determined by energylevel spacing
•  Niels Bohr put forward the idea that the average behavior
of a quantum system should begin to look like the
classical solution in the limit that the quantum number
becomes very large—that is, as n → ∞.
•  Because the radius of the Bohr hydrogen atom is r =
n2aB, the atom becomes a macroscopic object as n
becomes very large.
•  Bohr’s idea, that the quantum world should blend
smoothly into the classical world for high quantum
numbers, is today known as the correspondence
principle.
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As n gets even bigger and the number of oscillations
increases, the probability of finding the particle in an
interval δx will be the same for both the quantum and the
classical particles as long as δx is large enough to include
several oscillations of the wave function. This is in
agreement with Bohr’s correspondence principle.
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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The quantum-mechanical solution for a particle in a finite
potential well has some important properties:
•  The particle’s energy is quantized.
•  There are only a finite number of bound states. There
are no stationary states with E > U0 because such a
particle would not remain in the well.
•  The wave functions are qualitatively similar to those
of a particle in a rigid box, but the energies are
somewhat lower.
•  The wave functions extend into the classically
forbidden regions.
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The wave function in the classically forbidden region of a
finite potential well is
The wave function oscillates until it reaches the classical
turning point at x = L, then it decays exponentially
within the classically forbidden region. A similar analysis
can be done for x ≤ 0.
We can define a parameter η defined as the distance into the
classically forbidden region at which the wave function has
decreased to e–1 or 0.37 times its value at the edge:
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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The wave functions of the first three states are
Where ω = (k/m)–½ is the classical angular frequency, and n
is the quantum number
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The solutions are
similar to waves in
a finite potential
well, oscillating in
the classical region,
exponentially
damped outside.
The harmonic
oscillator is an
important model in
QM just as it is in
classical physics.
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QUESTION:
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The level spacing of
typical bonds corresponds
to emission and
absorption of infrared
radiation.
Heat is associated with
quantum oscillations of
bond lengths.
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Like sound passing through
a wall, a matter wave can
pass through a classically
impenetrable barrier.
In general a particle is partly
reflected and partly
transmitted so in a sense in
two places at once.
Even a potential well
reflects a matter wave.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Once the penetration distance
η is calculated using
Equation 41.41, the
probability that a particle
striking the barrier from the
left will emerge on the right
is found to be
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Tunneling between conductors
•  Make one well deeper:
particle tunnels, then stays in other well.
•  Well made deeper by applying electric field.
•  This is the principle of scanning tunneling microscope.
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Scanning Tunneling Microscopy
Tip, sample are quantum ‘boxes’
Tip
Potential difference induces
tunneling
Tunneling exponentially
sensitive to tip-sample spacing
Sample
•  Control position tip and sample at nm scale with peiezoelectric
effect – distortion induced by electric field in a crystal
•  Scan and record current or scan at constant current.
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Surface steps on Si
Images courtesy
M. Lagally,
Univ. Wisconsin
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Manipulation of atoms
•  Take advantage of tip-atom interactions to
physically move atoms around on the surface
 
 
This shows the assembly
of a circular ‘corral’ by
moving individual Iron
atoms on the surface of
Copper (111).
The (111) orientation
supports an electron
surface state which can
be ‘trapped’ in the corral
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D. Eigler (IBM)