Lecture 12' Quantum Harmonic Oscillator

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Lecture 12. Quantum Harmonic Oscillator

• Outline
• Sketching a Wavefunction
• Finite Potential Well
• Superposition of Eigenstates Particle Motion
  in a Box
• Quantum Harmonic Oscillator
• - Energy Spectrum
• - Eigenfunctions

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Sketching a wavefunction
This equation shows how the second derivative of
the energy eigenfunction ?, or the curvature of
the function ?(x), is linked to the
magnitude/sign of the eigenfunction. ( E-U 0
the curvature 0).
Consider
and

• In a classically allowed region, an energy
  eigenfunction always curves toward the
  horizontal axis (wavelike), in a classically
  forbidden region away from the horizontal
  axis (exponential-like).
• Absolute magnitude of E-U(x)
• in a classically allowed region, greater
  IE-U(x)I implies shorter wavelength, in a
  classically forbidden region steeper
  exponential tails
• the amplitude of oscillations in the
  classically allowed region is bigger when
  IE-U(x)I is smaller (compare with the probability
  of finding a slow-moving particle)

• Also
• a wavefunction of energy level n should have
  (n-1) nodes
• if the potential is symmetric with respect to
  some x0, the wavefunction should be either
  symmetric or anti-symmetric with respect to this
  x0.

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Sketching a wavefunction (contd)
Sketch the wavefunction with n5
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(No Transcript)
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Finite Potential Well (bound states)
Outside the well (regions 1 and 3)
1
2
3
E
In these regions, the solutions are combinations
of exponential functions
Within the well (region 2)
Boundary conditions at work
There are six unknowns (A-G). We request that
(i) the wavefunction to be finite at x ?? (2
equations)
(ii) both ? and d?/dx must be continuous at x0
and xL (4 equations).
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Finite Potential Well (contd)
Stitching of the solutions in regions 1-2 and
2-3 leads to the spectrum quantization for E lt U0

• the wavefunction leaks under the barrier
• (the closer E to U0, the longer the tails)

Does this violate the energy conservation? NO!
To observe a particle in classically forbidden
region means to locate it within

• For E gt U0, the spectrum is continuous
  (unbound states), see Lecture 13.

U(x)
E
U0
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Superposition of Eigenstates
The general solution of the t-dependent S. Eq.
is a linear combination of energy eigenfunctions
The superposition of energy eigenfunctions is
NOT a solution (i.e. an eigenfunction) of the
t-independent S.Eq. since it is not associated
with a single energy.
Also, it is NOT a stationary state
superpositions of eigenfunctions evolve with
time!
At the last lecture, we illustrated this by
considering a particles state which is a linear
combination of just two stationary states
The wave function at subsequent times
If one measures the energy for an ensemble of
identical quantum systems in such a state, the
results of measurements will provide either E1 or
E2 (with probabilities that depend on c1 and c2)
- the probability density oscillates, this is
certainly not a stationary state!
- interference of probabilities
So the measured probability density oscillates
with a frequency that depends on the difference
in energies
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Particle Motion in a Box
Lets consider the superposition of two
eigenfunctions of a particle in an infinite
potential well
- ground state
- 1st excited state
This probability density oscillates in time
between two values
This superposition describes the particle moving
back and forth in the well!
particle localized on the left side of the well
particle localized on the right side of the well
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Example
An electron in the infinite potential well is
initially (at t0) confined to the left side of
the well, and is described by the following
wavefunction
If the well width is L0.5nm, determine the time
t0 it takes for the particle to move to the
right side of the well.
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Classical Harmonic Oscillator (no friction)
Harmonic motion near a stable equilibrium
position under the influence of a restoring force
F -kx

• x(t) the instantaneous displacement of the
  particle at time t from the equilibrium position
• A the amplitude of oscillations
• the angular frequency ? ?t ?0 is the phase

the velocity
the potential energy
the kinetic energy
the total energy
The total energy is conserved if all forces are
conservative (no friction).
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Quantum Harmonic Oscillator
Solving the equation beyond our scope. The
spectrum
- note that the count starts at n 0 !
The inter-level distance depends on how wide the
well is greater k (stronger spring) ?
greater ?E
Quantum or Classical? When do we need to take
the spectrum discreteness into account?
When we deal with macroscopic masses, this is
rarely the case the separation between the
levels of a macroscopic pendulum with ?1s-1 and
m10-3kg is tiny compared to kBT
However, recall nanomechanics (relatively)
small masses and large k result in ?1010 s-1
at T0.1K
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Spectra for Different Potential Wells
The spectra depend on how quickly the potential
broadens.
Energy intervals between adjacent levels increase
proportional to n (at large n)
Equidistant energy levels (borderline between
expanding ?E and shrinking ?E spectra)
Sketch the spectra for the van der Waals
potential well
van-der Waals
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Example Diatomic Molecules
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Example
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Infrared Spectroscopy of Molecules
By measuring the vibrational spectrum of a
molecule, the force constant k of the molecule
can be determined. This provides important
information about molecular bonding.
? - the reduced mass
Example The frequency that corresponds to the
transitions between vibrational energy levels for
carbon monoxide is 6.4?1013Hz (i.e. ?E0.26eV).
Determine the harmonic oscillator force constant
for this molecule.
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Energy Eigenfunctions

• Applying the general rules of sketching
  wavefunctions
• E-U lt 0 ?(x) and d2 ?/dx2 have the same sign
• E-U gt 0 ?(x) and d2 ?/dx2 have opposite
  signs
• (n-1) nodes for a wavefunction of energy level
  n
• symmetric potential - either symmetric or
  anti-symmetric wavefunctions

1
2
3
Because of a weird numbering of states (n 0,
1, 2,...), the number of nodes is n, not n-1.
The ground state (n0)
- Gaussian
The energy eigenfunctions
The 1st excited state (n1)
Hermite polynomials
The 2st excited state (n2)
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Example
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Correspondence Principle
The predictions of quantum mechanics approach
those of classical physics in the limit of large
quantum numbers.
In particular, the spectrum discreteness becomes
insignificant if ?E/Eltlt1.
Though the energy intervals increase with n as
2n1, the energy itself increases faster (n2)
The overall shape of the probability density
(neglect fast oscillations) approaches the
classical one at large n.
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Problem (Normalization, expectation value,
sketching wavefunctions)
The eigenfunction of the 1st excited state of a
harmonic oscillator has the form
where m is the mass of oscillator,
(a) Find the expectation value of the potential
energy of a harmonic oscillator in the 1st
excited state (express the result in terms of the
frequency ?). (b) Sketch the eigenfunction ?7 of
the seventh excited state of a harmonic
oscillator and the corresponding probability
density. Provide necessary explanations (in
particular, explain the x-dependence of the
amplitude and period of eigenfunction
oscillations).
(a)
(b)