More Quantum Mechanics

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More Quantum Mechanics
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Wavefunction requirements
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Particle in a box
Quantum mechanical version- the particle is
confined by an infinite potential on either side.
The boundary condition- the probability of
finding the particle outside of the box is ZERO!
For the quantum mechanical case
here, we assumed the walls were infinitethere
was no possibility the particle could escape.
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The Finite Walled Box
This begins to look like the familiar undamped
sho equation
Since EltU
Setting
Generally, solutions are then
But remember the conditions imposed on
wavefunctions so that they make physical sense
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I
II
III
U
C1 must be 0 in Region III and C2 must be zero in
Region I, otherwise, the probabilities would be
infinite in those regions.
E
-L/2
L/2
Note that the wavefunction is not necessarily 0
in Regions I and II. (It is 0 in the limit of an
infinite well.) How is this possible when UgtE??
The uncertainty principle.
Now we must use the condition of continuity (the
wavefunction must be continuous at the boundary,
and so must its first derivative).
Suppose we had a discontinuous function
dy/dx
Here, the acceleration would be infinite. Uh-oh!
y(x)
x
x
L/2
L/2
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Even solutions
Odd solutions
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Finite box allowed energies and wavefunctions
Note that the particle has negative kinetic
energy outside of the well KE-U
The wavefunction decreases rapidly- 1/e in a
space of 1/a
1/a is a penetration depth. To observe the
particle in this region, you must measure the the
position with an accuracy of less than 1/a
From the uncertainty principle
The uncertainty in the measurement is larger than
the negative kinetic energy.
An real world example of a finite box would be
a neutron in a nucleus.
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The Finite vs. the infinite box
Note that the allowed energies are inversely
proportional to the length of the box. The
energy levels of a finite box are lower than for
an infinite well because the box is effectively
larger. There is less confinement energy.
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Charge Coupled Devices
an application
As close as one gets to a potential well in real
life.
CCD must perform 4 tasks to generate an image
Generate Charge --gt Photoelectric Effect
Collect Charge --gt pixels an array of
electrodes (called gates) Transfer Charge
--gt Apply a differential voltage across gates.
Signal electrons move down vertical registers
(columns) to horizontal register. Each line is
serially read out by an on-chip amplifier.
Detect Charge --gt individual charge packets are
converted to an output voltage and then digitally
encoded.
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Schematic of the Operation of a CCD
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Low noise high efficiency for weak signals
uniform response large dynamic range
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The Quantum Harmonic Oscillator
At x0, all of a particles energy is potential
energy, as it approaches the boundary, its
kinetic energy becomes less and less until it all
of the particles energy is potential energy- it
stops and is reflected back. An example would be
a vibrating diatomic molecule.
This is analogous to a classical system, such as
a spring, where potential energy is being
exchanged for kinetic energy.
In contrast to the square well, where the
particle moves with constant kinetic energy until
it hits a wall and is reflected back, in the
parabolic potential well of the harmonic
oscillator, the kinetic energy decreases
(wavelength increases) as the boundary is
approached.
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The Quantum Harmonic Oscillator
Kinetic energy
Making an approximation-assuming small
penetration depth and high frequency, the
condition for an infinite number of half
wavelengths as in an infinite well must be recast
as an integral to account for a variable
wavelength
The wavelength is position dependent
Note that the width of the well is greater for
higher energies. As the energy increases, the
confinement energy decreases. The levels are
evenly spaced. We have Plancks quantization
condition!
Boundary conditions
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Quantum harmonic oscillator wavefunctions and
allowed energies
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The correspondence principle
In the limit of large n, the probabilities start
to resemble each other more closely.
In classical physics, the block on a spring has
the greatest probability of being observed near
the endpoints of its motion where it has the
least kinetic energy. (It is moving slowly here.)
This is in sharp contrast to the quantum case for
small n.
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What have we learned from the Schrodinger
Equation?

• We can find allowed wavefunctions.
• We can find allowed energy levels by plugging
  those wavefunctions into the Schrodinger equation
  and solving for the energy.
• We know that the particles position cannot be
  determined precisely, but that the probability of
  a particle being found at a particular point can
  be calculated from the wavefunction.
• Okay, we cant calculate the position (or other
  position dependent variables) precisely but given
  a large number of events, can we predict what the
  average value will be? (If you roll a dice once,
  you can only guess that the number rolled will be
  between 1 and 6, but if you roll a dice many
  times, you can say with certainty that the
  fraction of times you rolled a three will
  converge on 1 in 6)

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Probability
Alternatively, you can count the number of times
you rolled a particular number and weight each
number by the the number of times it was rolled,
divided by the total number of rolls of the dice
If you roll a dice 600 times, you can average the
results as follows
After a large number of rolls, these ratios
converge on the probability for rolling a
particular value, and the average value can then
be written
This works any time you have discreet values.
What do you do if you have a continuous variable,
such as the probability density for you particle?
It becomes an integral.
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Expectation Values
The expectation value can be interpreted as the
average value of x that we would expect to obtain
from a large number of measurements.
Alternatively it could be viewed as the average
value of position for a large number of particles
which are described by the same wavefunction.
We have calculated the expectation value for the
position x, but this can be extended to any
function of positions, f(x). For example, if the
potential is a function of x, then
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Expectation Values and Operators
expression for kinetic energy
kinetic plus potential energy gives the total
energy
the potential
position x x
momentum p
potential energy U U(x)
kinetic energy K
total energy E
observable
operator
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Calculating an observable from an operator
In general to calaculate the expectation value of
some observable quantity
Weve learned how to calculate the observable of
a value that is simply a function of x
But in general, the operator operates on the
wavefunction and the exact order of the
expression becomes important