Quantum Mechanics
Wavefunctions and Quantum Wells
Quantum Mechanics
(Wave Mechanics)
• The probability of finding a particle in a particular
region within a particular time interval is found by
integrating the square of the wave function:
• P (x,t) =  |Y(x,t)|2 dx =  |c(x)|2 dx
• |c(x)|2 dx is called the “probability density; the
area under a curve of probability density yields the
probability the particle is in that region
• When a measurement is made, we say the wave
function “collapses” to a point, and a particle is
detected at some particular location
Quantum Mechanics
Schroedinger’s Equation:

 2 Y( x )  C ( E  U ) Y( x )
x
2
where
C
16  2 m
h
2
 m   2 1
 40 nm eV
 me 
Application: particle in a box
• If a particle is confined to a region by infinitelyhigh walls, the probability of finding it outside
that region is zero.
• Since nature is generally continuous (no
instantaneous changes), the probability of finding
it at the edges of the region is zero.
• The position-dependent solution to the
Schrödinger equation for this case has the form of
a sine function:
c(x) = B sin (nx/a) (a=well width)
More particle in a box
c(x) = B sin (nx/a)
n=3
c(x)
n=2
|c(x)|2
certain wavelengths l = 2a/n are allowed
 Only certain momenta p = h/l = hn/2a are allowed
 Only certain energies E = p2/2m = h2n2/8ma2 are
allowed - energy is QUANTIZED
 Allowed energies depend on well width
 Only
What about the real world?
•
•
•
•
So confinement yields quantized energies
In the real world, infinitely high wells don’t exist
Finite wells, however, are quite common
Schrödinger equation is slightly more
complicated, since Ep is finite outside well
• Solution has non-trivial form (“trust me”)
|c(x)|2
n=2
n=1
What about the real world?
• Solution has non-trivial form, but only certain
states (integer n) are solutions
• Each state has one allowed energy, so energy is
again quantized
• Energy depends on well width a
• Can pick energies for electron by adjusting a
|c(x)|2
n=2
n=1
x
Putting Several Wells Together
• How does the number of energy bands
compare with the number of energy levels
in a single well?
• As atom spacing decreases, what happens to
energy bands?
• What happens when impurities are added?
Quantum wells
• An electron is trapped since no empty energy
states exist on either side of the well
Escaping quantum wells
• Classically, an electron could gain thermal energy and
escape
• For a deep well, this is not very probable
Escaping quantum wells
• Thanks to quantum mechanics, an electron has a non-zero
probability of appearing outside of the well
• This happens more often than thermal escape
What have we learned today?
 Quantum mechanics challenges our
physical intuition but it is the way things
really work.
 Particles are described with a wave function
Y(x,t) which describes the propagation
through space and time (when unobserved).
What have we learned today?
 Integrating the square of the wave function over a
region gives us the probability of finding the
object in that region
 A “particle” confined to an infinitely-high box is
described by a wave function of a sine.
 “particles” in finite wells or in atoms are described
by more complicated wave functions
 All three situations result in quantized (only
certain values allowed) energies
Before the Next Class
• Do Activity 23 Evaluation before next class
• Finish Homework 24
• Start Reading Chapters 9 and 10 (3rd hour
exam will cover)
• Do Reading Quiz 25 (due before class 25)