3'1 A Free Particle

1
3.1 A Free Particle
Free particle experiences no forces so potential
energy independent of position (take as zero)
Linear ODE with constant coefficients so try
Time-independent Schrodinger equation
k is wave number of De Broglie wave
General solution
(3.2)
A C D
or
B i(C - D)
Combine with time dependence to get full wave
function
2
Notes

• Plane wave is a solution (just as well, since our
  plausibility argument for the Schrodinger
  equation was based on this being so)
• Note signs
• Sign of time term (-i?t) is fixed by sign adopted
  in time-dependent Schrodinger Equation
• Sign of position term (ikx) depends on
  propagation direction of wave
• There is no restriction on the allowed energies,
  so there is a continuum of states

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3.2 Infinite Square Well
V(x)
Consider a particle confined to a finite length
a lt x lt a by an infinitely high potential barrier
x
-a
a
No solution in barrier region (particle would
have infinite potential energy).
In well region
as for a free particle.
Boundary conditions
Continuity of ? at x a
Note discontinuity in d?/dx allowable, since
potential is infinite
Continuity of ? at x -a
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Infinite square well (2)
Add and subtract these conditions
Add (1) (2)
Even solution ?(x) ?(x)
Now, subtract (1) (2)
Odd solution ?(x) ?(x)
Energy
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Infinite well normalization and notes
Normalization
Need to choose constants so ? is normalised

• Notes on the solution
• Energy quantized to particular values
  (characteristic of bound-state problems in
  quantum mechanics, where a particle is localized
  in a finite region of space.
• Potential is even under reflection stationary
  state wavefunctions may be even or odd (we say
  they have even or odd parity)
• Compare notation in 1B23 and in books

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The infinite well and the Uncertainty Principle
Position uncertainty in well
Momentum uncertainty in lowest state from
classical argument (agrees with fully quantum
mechanical result, as we will see in 4)
Ground state close to minimum uncertainty
Compare with Uncertainty Principle
(worked out properly)
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3.3 Finite square well
Now make the potential well more realistic by
making the barriers a finite height V0
Region I
Region II
Region III
S. eq. as in region I.
as in I
Gen. sol.
General solution
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Finite square well (2)
Match value and derivative of wavefunction at
region boundaries
Match ?
(3.10)
(3.12)
Match d?/dx
(3.13)
(3.11)
Add and subtract
(3.14)
(3.16)
(3.15)
(3.17)
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Finite square well (3)
(3.17)/(3.14) (3.18)
Divide equations
(3.16)/(3.15) (3.19)
(3.18) and (3.19) must be satisfied
simultaneously!!
This is not possible unless we divide solutions
into 2 classes
B 0 C D so
even solutions
A 0 C D so odd
solutions
Cannot be solved algebraically. Convenient form
for graphical solution
Plot Against
Find intersection points.
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Graphical solution for finite well
K0 3, a 1
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Notes

• Penetration of particle into forbidden region
  where V gt E (particle cannot exist here
  classically)
• Number of bound states depends on depth of
  potential well, but there is always at least one
  (even) state
• Potential is even function, wavefunctions may be
  even or odd (we say they have even or odd parity)
• Limit as V0 ? 8

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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
Material A (e.g. AlGaAs)
Material B (e.g. GaAs)
Electron potential energy
Position
Now used in many electronic devices (some
transistors, diodes, solid-state lasers)
Esaki
Kroemer
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3.4 Particle Flux
Rae 9.1 BM 5.2, BJ 3.2
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
will 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
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Particle Flux (2)
More generally what is rate of change of
probability that a particle exists in some region
(say, between x a and x b)?
Use time-dependent Schrodinger equation
(since V is real )
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Particle Flux (3)
Probability in region
a ? x ? b
(A)
Integrate by parts
Flux entering at x a
Flux leaving at x b

Interpretation
Note a wavefunction that is real carries no
current
Note for a stationary state can use either ?(x)
or ?(x,t)
(3.20)
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Particle Flux (4)
Sanity check apply to free-particle plane wave.
so, flux
Makes sense
particles passing x per unit time particles
per unit length velocity
Wavefunction describes a beam of particles.
implies one particle per unit
length
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3.5 Potential Step
Rae 9.1 BJ 4.3
V(x)
Consider a potential which rises suddenly at x
0
V0
Case 1
x
0
Boundary condition particles only incident from
left
Case 1 E lt V0 (below step)
x lt 0
x gt 0
Free particle S.E.
(as for Region II offinite well)
with
Choose oncoming wave from left ? A 1 (1
particle/unit length)
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Potential Step (2)
Continuity of ? at x 0
(3.22)
(3.23)
Solve for reflection and transmission
Take
and take
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Transmission and reflection coefficients
Incident particle flux from left (see
eq 3.21)
Reflected particle flux
Hence Probability of reflection is
Transmitted flux is
inevitable since is real in right hand
region and real alwaysgives zero current.
? probability of transmission 0
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Potential Step (3)
Case 2 E gt V0 (above step)
Solution for x gt 0 is now
continuous
(3.24)
Matching conditions
continuous
(3.25)
Take
Take
Transmission and reflection coefficients
By our previous argument, reflection probability
is
(3.26)
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Summary of transmission through potential step
For case transmitted flux is now
? Transmission probability
(3.27)
check
as required

• Notes
• Some penetration of particles into forbidden
  region even for energies below step height (case
  1, E lt V0)
• No transmitted particle flux, 100 reflection
  (case 1, E lt V0)
• Reflection probability does not fall to zero for
  energies above barrier (case 2, E gt V0).
• Contrast classical expectations
• 100 reflection for E lt V0, with no penetration
  into barrier
• 100 transmission for E gt V0

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3.6 Rectangular Potential Barrier
Rae 2.5 BJ 4.4 BM 5.9
V(x)
III
I
II
Now consider a potential barrier of finite
thickness
V0
x
a
0
Boundary condition particles only incident from
left
Region I
Region II
Region III
NB since the barrier isof finite width the
solution does not causeproblems with
normalization
No incident wave fromright
Take A 1 (one particleper unit length)
take F 0
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Rectangular Barrier (3)
Transmission and reflection coefficients
Transmission coeff.
Reflection coeff.
For very thick or high barrier
Non-zero transmission (tunnelling) through
classically forbidden barrier region
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Examples of tunnelling
Tunnelling occurs in many situations in physics
and astronomy
1. Nuclear fusion (in stars and fusion reactors)
V
Coulomb interaction (repulsive)
Incident particles
Internuclear distance x
Strong nuclear force (attractive)
V
Distance x of electron from surface
Work function W
Material
3. Field emission of electrons from surfaces
(e.g. in plasma displays)
Vacuum
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3.7 Simple Harmonic Oscillator
Rae 2.6 BM 5.5 BJ 4.7
Mass m
Example particle on a spring, Hookes law
restoring force with spring constant k
x
Time-independent Schrodinger equation
Problem still a linear differential equation but
coefficients are not constant.
Simplify change variable to
where
(3.34a)
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Simple Harmonic Oscillator (2)
Asymptotic solution in the limit of very large y
Check
as
as expected.
So Schrodinger eq. in limit of large y
but only
solution is normalizable.
Equation for H
Substitute in (3.34a)
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Simple Harmonic Oscillator (3)
Must solve this ODE by the power-series method
(Frobenius method) this is done as an example in
2246.

• We find
• The series for H(y) must terminate in order to
  obtain a normalisable solution
• Can make this happen after n terms for either
  even or odd terms in series (but not both) by
  choosing

For some integer n
Hn is known as the nth Hermite polynomial.
Label resulting functions of H by the values of n
that we choose.
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The Hermite polynomials
For reference, first few Hermite polynomials are
NOTE for n even ? even soln
for n odd ? odd solution
Full wavefunction requires normalization
NOTE Hn contains yn as the highest power. Each H
is either an odd or an even function, according
to whether n is even or odd.
NOTE
but
the energy of the quantumn H.O. is quantised
Minimum energy (Zero-point energy)
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Simple Harmonic Oscillator (4)
Transforming back to the original variable x, the
wavefunction becomes
Probability per unit length of finding the
particle is
Compare classical result probability of finding
particle in a length dx is proportional to the
time dt spent in that region
v velocity
For a classical particle with total energy E,
velocity is given by
so
large where
Turning points
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Notes

• Zero-point energy
• Quanta of energy
• Even and odd solutions
• Applies to any simple harmonic oscillator,
  including
• Molecular vibrations
• Vibrations in a solid (hence phonons)
• Electromagnetic field modes (hence photons), even
  though this field does not obey exactly the same
  Schrodinger equation
• You will do another, more elegant, solution
  method (no series or Hermite polynomials!) next
  year
• For high-energy states, probability density peaks
  at classical turning points (correspondence
  principle)