Electron in a Box
A wave packet in a square well (an electron in a box)
changing with time.
Last Time: Light
Wave model: Interference pattern is in terms of wave intensity
Photon model: Interference in terms of probability
The probability of detecting a photon within a narrow region of
width δx at position x is directly proportional to the square of
the light wave amplitude function at that point.
Prob(in δ x at x) ∝ A(x) δ x
2
Probability Density Function:
P ( x) ∝ A(x)
2
The probability density function is independent of the width, δx , and depends
only on x. SI units are m-1.
Double Slit: Electrons
A light analogy…..
There is no electron wave so we
assume an analogy to the electric
wave and call it the wave function,
psi, :
Ψ ( x)
The intensity at a point on the screen
is proportional to the square of the
wave function at that point.
P( x) = Ψ ( x)
2
The Probability Density Function is
the “Reality”!
Probability: Electrons
The probability of detecting an electron within a narrow region
of width δx at position x is directly proportional to the square of
the wave function at that point:
Prob(in δ x at x) =
Ψ (x) δ x
2
Probability Density Function:
P( x) =
Ψ (x)
2
The probability density function is independent of the width, δx , and depends
only on x. SI units are m-1. Note: The above is an equality, not a proportionality
as with photons. This is because we are defining psi this way. Also note, P(x) is
unique but psi in not since –psi is also a solution. DEGENERACY.
If the strip isn’t narrow, then we
integrate the probability density
function so that the probability that
an electron lands somewhere
between xL and xR is:
xR
xR
xL
xL
( x)dx
∫ P=
Prob( xL ≤ x=
≤ xR )
∫ ψ ( x) dx
dP( x)
=0
dx
Most Probable:
Normalization
∞
=
∫ P( x)dx
−∞
∞
=
∫ ψ ( x) dx 1
2
−∞
2
Electron Waves leads to
Quantum Theory
2L
Waves:
, n 1, 2,3.....
=
λn =
n
2
1 2 p
h
De Broglie: λ =
→=
E
mv=
2
2m
p
2
2
hn
Combine: En =
2
8mL
Energy is Quantized!
Wave Packet:
Making Particles
out of Waves
p=
h
λ
c=λf
p = hf / c
Superposition of waves to make a defined wave packet. The more
waves used of different frequencies, the more localized.
However, the more frequencies used, the less the momentum is known.
Heisenberg Uncertainty Principle
You make a wave
packet by wave
superposition and
interference.
The more waves you
use, the more defined
your packet and the
more defined the
position of the particle.
However, the more
waves you use of
different frequencies
(energy or momentum)
to specify the position,
the less you specify the
momentum!
∆E ∆t > h / 4π
∆x∆p > h / 4π
Little h bar!
Which of these particles, A or B, can
you locate more precisely?
A. A
B. B
C. Both can be located with same precision.
Which of these particles, A or B, can
you locate more precisely?
A. A
B. B
C. Both can be located with same precision.
Heisenberg Microscope
Small wavelength (gamma) of light
must be used to find the electron
because it is too small. But small
wavelength means high energy. That
energy is transferred to the electron in
an unpredictable way and the motion
(momentum) becomes uncertain. If
you use long wavelength light
(infrared), the motion is not as
disturbed but the position is uncertain
because the wavelength is too long to
see the electron. This results in the
Uncertainty Principle.
∆x ~ λ
∆p =
h/λ
∆x∆p > h
Electron in a Box
The possible
wavelengths for an
electron in a box of
length L.
∆x ~ L
h/ L
∆p ~ p ~ h / λ =
If you squeeze the walls to decrease ∆x, you increase ∆p!
∆x∆p ~ L ⋅ h / L > h
Improved technology will not save
us from Quantum Uncertainty!
Quantum Uncertainty comes from
the particle-wave nature of matter
and the mathematics
(wave functions)
used to describe them!
Heisenberg Uncertainty
Trying to see what slit an electron goes
through destroys the interference pattern.
Electrons act like waves
going through the slits but
arrive at the detector like a
particle.
Which Hole Did the Electron
Go Through?
If you make a very dim beam of
electrons you can essentially send
one electron at a time. If you try
to set up a way to detect which
hole it goes through you destroy
the wave interference pattern.
Conclusions:
• Trying to detect the electron, destroys the interference pattern.
• The electron and apparatus are in a quantum superposition of states.
• There is no objective reality.
Feynman’s version of the
Uncertainty Principle
It is impossible to design an
apparatus to determine
which hole the electron
passes through, that will not
at the same time disturb the
electrons enough to destroy
the interference pattern.
General Principles
Where do the Wave Functions come
from???
Solutions to the time-independent Schrödinger equation:
 2 d 2ψ
−
+ Uψ =
Eψ
2
2m dx
OR
d 2ψ 2m (U − E )
=
ψ
2
2

dx
Where does that come from???
The Schrödinger Equation
Consider an atomic particle with mass m and mechanical
energy E in an environment characterized by a potential
energy function U(x).
The Schrödinger equation for the particle’s wave function is
Conditions the wave function must obey are
1. ψ(x) is a continuous function.
2. ψ(x) = 0 if x is in a region where it is physically
impossible for the particle to be.
3. ψ(x) → 0 as x → +∞ and x → −∞.
4. ψ(x) is a normalized function.
Wavefunction Fun
ψ ( x)
Wave Function: Probability Amplitude:
P( x) = ψ ( x)
Probability Density:
Probability:
Prob( xL ≤ x=
≤ xR )
∞
Normalization
=
∫ P( x)dx
−∞
2
xR
xR
xL
xL
( x)dx
∫ P=
=
∫ ψ ( x) dx 1
2
−∞
dP( x)
=0
dx
∞
〈 x 〉 = ∫ xP ( x)dx = ∫ x ψ ( x) dx
2
−∞
Most Probable:
2
∞
∞
Expectation “average” value:
∫ ψ ( x) dx
−∞
Quantum Cases
•
•
•
•
•
•
Free Particle
Particle in a rigid box
Particle in a finite box “quantum well’
Quantum Tunneling
Harmonic Oscillator
Hydrogen Atom (Chapter 42)
Wave Function of a Free Particle
• The wave function of a free particle moving along
the x-axis can be written as ψ(x) = Aei(kx-ωt)
– A is the constant amplitude
– k = 2π/λ is the angular wave number of the wave
representing the particle
– A free particle must have a sinusoidal wavefunction
because it is not confined.
• Although the wave function is often associated
with the particle, it is more properly determined
by the particle and its interaction with its
environment
– Think of the system wave function instead of the
particle wave function
Free Particle Problem
A free electron has a wave function at t=0
(
ψ (x ) = Ae
i 5.00 × 1010 x
where x is in meters.
(a) Show that it satisfies the SE.
(b) Find its de Broglie wavelength
(c) Find its momentum
)
A Particle in a Rigid Box
Consider a particle of mass m confined in a rigid, onedimensional box. The boundaries of the box are at x = 0 and
x = L.
1. The particle can move freely between 0 and L at
constant speed and thus with constant kinetic energy.
2. No matter how much kinetic energy the particle has,
its turning points are at x = 0 and x = L.
3. The regions x < 0 and x > L are forbidden. The
particle cannot leave the box.
A potential-energy function that describes the particle in
this situation is
A Particle in a Rigid Box
The solutions to the Schrödinger equation for a particle in a rigid
box are
For a particle in a box, these energies are the only values of E
for which there are physically meaningful solutions
to the Schrödinger equation. The particle’s energy is
quantized.
A Particle in a Rigid Box
The normalization condition, which we found in Chapter
40, is
This condition determines the constants A:
The normalized wave function for the particle in quantum
state n is
A Particle in a Rigid Box
Schrödinger Equation Applied to
a Particle in a Box
• Solving for the allowed energies gives
 h2  2
En = 
n
2 
 8mL 
• The allowed wave functions are given by
 nπx 
=
ψn (x ) A=
sin 

 L 
2
 nπx 
sin 

L
 L 
– The second expression is the normalized wave function
– These match the original results for the particle in a box
HW P.17
Graphical Representations for a
Particle in a Box
Energy of a Particle in a Box
• We chose the potential energy of the
particle to be zero inside the box
• Therefore, the energy of the particle is just
its kinetic energy
 h2  2
=
En =
n n 1, 2, 3 ,
2 
 8mL 
• The energy of the particle is quantized
The Quantum Jump
Quantum Energy States
2
2
hn
En =
2
8mL
Energy is Quantized!
Only discrete energy
states are allowed.
Where is the electron between jumps?
EXAMPLE :Energy Levels and
Quantum jumps
QUESTIONS:
Finite Potential Wells
The wave function in the classically forbidden region of a
finite potential well is
The wave function oscillates until it reaches the classical
turning point at x = L, then it decays exponentially
within the classically forbidden region. A similar analysis
can be done for x ≤ 0.
We can define a parameter η defined as the distance into the
classically forbidden region at which the wave function has
decreased to e–1 or 0.37 times its value at the edge:
Finite Potential Wells
The quantum-mechanical solution for a particle in a finite
potential well has some important properties:
• The particle’s energy is quantized.
• There are only a finite number of bound states. There
are no stationary states with E > U0 because such a
particle would not remain in the well.
• The wave functions are qualitatively similar to those
of a particle in a rigid box, but the energies are
somewhat lower because the wave functions are
spread out which means lower kinetic energy.
• The wave functions extend into the classically
forbidden regions.
Penetration distance of an electron
Quantum-Mechanical Tunneling
The probability that a particle
striking the barrier from the
left will emerge on the right
is found to be
Applications of Tunneling
• Alpha decay
– In order for the alpha particle to escape from
the nucleus, it must penetrate a barrier whose
energy is several times greater than the energy
of the nucleus-alpha particle system
• Nuclear fusion
– Protons can tunnel through the barrier caused
by their mutual electrostatic repulsion
More Applications of Tunneling –
Scanning Tunneling Microscope
• An electrically conducting
probe with a very sharp
edge is brought near the
surface to be studied
• The empty space between
the tip and the surface
represents the “barrier”
• The tip and the surface are
two walls of the “potential
well”
Scanning Tunneling Microscope
• The STM allows
highly detailed images
of surfaces with
resolutions
comparable to the size
of a single atom
• At right is the surface
of graphite “viewed”
with the STM
Quantum Tunneling Cosmology
BIG BANG!
The Universe Tunneled in from
Nothing
The Quantum Harmonic Oscillator
The potential-energy function
of a harmonic oscillator:
where we’ll assume the
equilibrium position is xe = 0.
The Schrödinger equation for a
quantum harmonic oscillator is
then
The Quantum Harmonic Oscillator
The wave functions of the first three states are
Where ω = (k/m)–½ is the classical angular frequency, and n
is the quantum number, b is classical turning point.
Energy Level Diagrams – Simple
Harmonic Oscillator
• The separation between
adjacent levels are equal
and equal to ∆E = ω
• The energy levels are
equally spaced
• The state n = 0
corresponds to the ground
state
– The energy is Eo = ½ ω
• Agrees with Planck’s
original equations!!
Light emission by an oscillating electron
Light emission by an oscillating electron
Molecular Vibrations
Molecular bonds are modeled as quantum harmonic
oscillators energies below the dissociation energy.
Casimir Effect: Model the
Quantum Vacuum as a
harmonic oscillator. The Zero
Point Energy adds up!
Eo = ½ hω
The Correspondence Principle
• Niels Bohr put forward the idea that the average behavior
of a quantum system should begin to look like the
classical solution in the limit that the quantum number
becomes very large—that is, as n → ∞.
• Because the radius of the Bohr hydrogen atom is r =
n2aB, the atom becomes a macroscopic object as n
becomes very large.
• Bohr’s idea, that the quantum world should blend
smoothly into the classical world for high quantum
numbers, is today known as the correspondence
principle.
The Correspondence Principle
As n gets even bigger and the number of oscillations
increases, the probability of finding the particle in an
interval δx will be the same for both the quantum and the
classical particles as long as δx is large enough to include
several oscillations of the wave function. This is in
agreement with Bohr’s correspondence principle.
I think I can safely say
that nobody understands
quantum mechanics.
Richard Feynman
Copenhagen Interpretation of the
Wave Function
•
•
Quantum mechanics is a model of the microscopic world. Like all models, it
is created by people for people. It divides the world into two parts, commonly
called the system and the observer. The system is the part of the world that is
being modeled. The rest of the world is the observer. An interaction between
the observer and the system is called a measurement. Properties of the system
that can be measured are called observables. The initial information the
observer has about the system comes from a set of measurements. The state of
the system represents this information, which can be cast into different
mathematical forms. It is often represented in terms of a wave
function. Quantum mechanics predicts how the state of the system evolves
and therefore how the information the observer has about the system evolves
with time. Some information is retained, and some is lost. The evolution of
the state is deterministic. Measurements at a later time provide new
information, and therefore the state of the system, in general, changes after the
measurements. The wave function of the system, in general, changes after a
measurement.
So quantum mechanics does not really describe the system, but the
information that the rest of the world can possibly have about the system.
Measurement: “Collapsing the Wave
Function” into an eigenstate
•
•
In quantum mechanics, a measurement of an observable yields a value, called an
eigenvalue of the observable. Many observables have quantized eigenvalues, i.e. the
measurement can only yield one of a discrete set of values. Right after the
measurement, the state of the system is an eigenstate of the observable, which means
that the value of the observable is exactly known.
A state can be a simultaneous eigenstate of several observable, which means that the
observer can exactly know the value of several properties of the system at the same time
and make exact predictions about the outcome of measurements of those properties. But
there are also incompatible observables whose exact values cannot be known to the
observer at the same time. A state cannot be a simultaneous eigenstate of incompatible
observables. If it is in an eigenstate of one of the incompatible observables and the
value of this observable is known, then quantum mechanics gives only the probabilities
for measuring each of the different eigenvalues of the other incompatible
observables. The eigenstate of the first observable is a superposition of eigenstates of
any of the other incompatible observables. The outcome of a measurement any of the
other incompatible observables is uncertain. A measurement of one of the other
incompatible observables changes the state of the system to one of its eigenstates and
destroys the information about the value of the first observable.