3'1 The Free Particle

1
Outline of Section 3

• Solve the TISE for various 1D potentials
• Free particle
• Infinite square well
• Finite square well
• Particle flux
• Potential step
• Transmission and reflection coefficients
• The barrier potential
• Quantum tunnelling
• Examples of tunnelling
• The harmonic oscillator

2
A Free Particle
Free particle no forces so potential energy
independent of position (take as zero)
Linear ODE with constant coefficients so try
Time-independent Schrödinger equation
General solution
Combine with time dependence to get full wave
function
3
Notes

• Plane wave is a solution (just as well, since our
  plausibility argument for the Schrödinger
  equation was based on this assumption).
• Note signs in exponentials
• Sign of time term (-i?t) is fixed by sign adopted
  in time-dependent Schrödinger Equation
• Sign of position term (ikx) depends on
  propagation direction of wave. ikx propagates
  towards 8 while -ikx propagates towards 8
• There is no restriction on k and hence on the
  allowed energies. The states form a continuum.

4
Particle in a constant potential
General solutions we will use over and over again
Time-independent Schrödinger equation
Case 1 E gt V (includes free particle with V 0
and K k)
Solution
Case 2 E lt V (classically particle can not be
here)
Solution
5
Infinite Square Well
Consider a particle confined to a finite length
altxlta by an infinitely high potential barrier
No solution in barrier region (particle would
have infinite potential energy).
x
In the well V 0 so equation is the same as
before
General solution
Boundary conditions
Continuity of ? at x a
Note discontinuity in d?/dx allowable, since
potential is infinite
Continuity of ? at x -a
6
Infinite Square Well (2)
Add and subtract these conditions
Even solution ?(x) ?(-x)
Odd solution ?(x) -?(-x)
Energy
We have discrete states labelled by an integer
quantum number
7
Infinite Square Well (3) Normalization
Normalize the solutions
Calculate the normalization integral
Normalized solutions are
8
Infinite Square Well (4)
Sketch solutions
Wavefunctions
Probability density
Note discontinuity of gradient of ? at edge of
well. OK because potential is infinite there.
9
Infinite Square Well (5)
Relation to classical probability distribution
Classically particle is equally likely to be
anywhere in the box
Quantum probability distribution is
But
so the high energy quantum states are consistent
with the classical result when we cant resolve
the rapid oscillations. This is an example of the
CORRESPONDENCE PRINCIPLE.
10
Infinite Square Well (5) notes

• Energy can only have discrete values there is no
  continuum of states anymore. The energy is said
  to be quantized. This is characteristic of
  bound-state problems in quantum mechanics, where
  a particle is localized in a finite region of
  space.
• The discrete energy states are associated with an
  integer quantum number.
• Energy of the lowest state (ground state) comes
  close to bounds set by the Uncertainty Principle
• The stationary state wavefunctions are even or
  odd under reflection. This is generally true for
  potentials that are even under reflection. Even
  solutions are said to have even parity, and odd
  solutions have odd parity.
• Recover classical probability distribution at
  high energy by spatial averaging.
• Warning! Different books differ on definition of
  well. E.g.
• BM well extends from x -a/2 to x a/2.
• Our results can be adapted to this case easily
  (replace a with a/2).
• May also have asymmetric well from x 0 to x
  a.
• Again can adapt our results here using
  appropriate transformations.

11
Finite Square Well
Now make the potential well more realistic by
making the barriers a finite height V0
Region I
Region II
Region III
12
Finite Square Well (2)
Boundary conditions match value and derivative
of wavefunction at region boundaries
Match ?
Match d?/dx
Now have five unknowns (including energy) and
five equations (including normalization condition)
Solve
13
Finite Square Well (3)
Even solutions when
Cannot be solved algebraically. Solve
graphically or on computer
Odd solutions when
14
Finite Square Well (4) Graphical solution
k0 4 a 1
Even solutions at intersections of blue and red
curves (always at least one) Odd solutions at
intersections of blue and green curves
15
Finite Square Well (5)
Sketch solutions
Wavefunctions
Probability density
Note exponential decay of solutions outside well
16
Finite Square Well (6) Notes
17
Example the quantum well
Quantum well is a sandwich made of two
different semiconductors in which the energy of
the electrons is different, and whose atomic
spacings are so similar that they can be grown
together without an appreciable density of
defects
Now used in many electronic devices (some
transistors, diodes, solid-state lasers)
18
Summary of Infinite and Finite Wells
Infinite well Infinitely many solutions
Even parity Odd parity
Finite well Finite number of solutions At
least one solution (even parity) Evanescent
wave outside well.
Odd parity solutions
Even parity solutions
19
Particle Flux
In order to analyse problems involving scattering
of free particles, need to understand
normalization of free-particle plane-wave
solutions.
Conclude that if we try to normalize so that
we get A 0.
This problem is related to Uncertainty Principle
Position completely undefined single particle
can be anywhere from -8 to 8, so probability of
finding it in any finite region is zero
Momentum is completely defined
Solutions Normalize in a finite box Use
wavepackets (later) Use a flux interpretation
20
Particle Flux (2)
More generally what is the rate of change of
probability that a particle is in some region
(say, between xa and xb)?
Use time-dependent Schrödinger equation
21
Particle Flux (3)
Flux entering at xa
Flux leaving at xb
minus
Interpretation
Note a wavefunction that is real carries no
current
22
Particle Flux (4)
Check apply to free-particle plane wave.
Makes sense
particles passing x per unit time particles
per unit length velocity
So plane wave wavefunction describes a beam of
particles.
23
Particle Flux (5) Notes
24
Potential Step
Consider a potential which rises suddenly at x
0
x
Boundary condition particles only incident from
left
Case 1 E gt V0 (above step)
x lt 0, V 0
25
Potential Step (2)
Continuity of ? at x 0
Solve for reflection and transmission amplitudes
26
Potential Step (3) Transmission and Reflection
Fluxes
Calculate transmitted and reflected fluxes
x lt 0
x gt 0
(cf classical case no reflected flux)
Check conservation of particles
27
Potential Step (4)
Case 2 E lt V0 (below step)
Solution for x lt 0 same as before
Solution for x gt 0 is now evanescent wave
Matching boundary conditions
Transmission and reflection amplitudes
Transmission and reflection fluxes
This time we have total reflected flux.
28
Potential Step (5) Notes
29
Rectangular Potential Barrier
Now consider a potential barrier of finite
thickness
x
Boundary condition particles only incident from
left
Region I
Region II
Region III
30
Rectangular Barrier (2)
Match value and derivative of wavefunction at
boundaries
Match ?
Match d?/dx
Eliminate wavefunction in central region
31
Rectangular Barrier (3)
Transmission and reflection amplitudes
For very thick or high barrier
Non-zero transmission (tunnelling) through
classically forbidden barrier region.
Exponentially sensitive to height and width of
barrier.
32
Examples of Tunnelling
Tunnelling occurs in many situations in physics
and astronomy
1. Nuclear fusion (in stars and fusion reactors)
Fusion (and life) occurs because nuclei tunnel
through the barrier
33
Examples of Tunnelling
2. Alpha-decay
a-particle must overcome Coulomb repulsion
barrier.
Tunnelling rate depends sensitively on barrier
width and height. Explains enormous range of
a-decay rates, e.g. 232Th, t1/2 1010 yrs,
218Th, t1/2 10-7s. Difference of 24 orders of
magnitude comes from factor of 2 change in
a-particle energy!
34
Examples of Tunnelling
3. Scanning tunnelling microscope
A conducting probe with a very sharp tip is
brought close to a metal. Electrons tunnel
through the empty space to the tip. Tunnelling
current is so sensitive to the metal/probe
distance (barrier width) that even individual
atoms can be mapped.
Tunnelling current proportional to
and
so
If a changes by 0.01A (1/100th of the atomic
size) then current changes by a factor of
0.98, i.e. a 2 change, which is detectable
STM image of Iodine atoms on platinum. The
yellow pocket is a missing Iodine atom
35
Summary of Flux and Tunnelling
The particle flux density is
Particles can tunnel through classically
forbidden regions. Transmitted flux decreases
exponentially with barrier height and width
We get transmission and reflection at potential
steps. There is reflection even when E gt V0.
Only recover classical limit for E gtgt V0
(correspondence principle)
36
Simple Harmonic Oscillator
Example particle on a spring, Hookes law
restoring force with spring constant k
Time-independent Schrödinger equation
Problem still a linear differential equation but
coefficients are not constant.
Simplify change to dimensionless variable
37
Simple Harmonic Oscillator (2)
Asymptotic solution in the limit of very large y
Try it
Equation for H(y)
38
Simple Harmonic Oscillator (3)
Solve this ODE by the power-series method
(Frobenius method)
Find that series for H(y) must terminate for a
normalizable solution
Can make this happen after n terms for either
even or odd terms in series (but not both) by
choosing
Hence solutions are either even or odd functions
(expected on parity considerations)
Label normalizable functions H by the values of n
(the quantum number)
Hn is known as the nth Hermite polynomial.
39
Simple Harmonic Oscillator (4)
EXAMPLES OF HERMITE POLYNOMIALS AND SHO
WAVEFUNCTIONS
are normalization constants
40
Simple Harmonic Oscillator (5) Wavefunctions
High n state (n30)
wavefunction
Probability density

• Decaying wavefunction tunnels into classically
  forbidden region
• Spatial average for high energy wavefunction
  gives classical result
• another example of the CORRESPONDENCE PRINCIPLE

41
Summary of Harmonic Oscillator
42
Example of SHOs in Atomic Physics Bose-Einstein
Condensation
87Rb atoms are cooled to nanokelvin temperatures
in a harmonic trap. de Broglie waves of atoms
overlap and form a giant matter wave known as a
BEC. All the atoms go into the ground state of
the trap and there is only zero point energy (at
T0). This is a superfluid gas with macroscopic
coherence and interference properties. Signature
of BEC phase transition The velocity
distribution goes from classical
Maxwell-Boltzmann form to the distribution of the
quantum mechanical SHO ground state.
43
Example of SHOs Molecular vibrations
VIBRATIONAL SPECTRA OF MOLECULES Useful in
chemical analysis and in astronomy (studies of
atmospheres of cool stars and interstellar
clouds).
SHO very useful because any potential is
approximately parabolic near a minimum