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Quantum mechanics
Wave Properties of Matter and Quantum Mechanics I
5.1
5.2
5.3
5.4
5.5
5.6
5.7
X-Ray Scattering
De Broglie Waves
Electron Scattering
Wave Motion
Waves or Particles?
Uncertainty Principle
Probability, Wave Functions, and
the Copenhagen Interpretation
5.8 Particle in a Box
Louis de Broglie
(1892-1987)
I thus arrived at the overall concept which guided my studies: for both matter
and radiations, light in particular, it is necessary to introduce the corpuscle
concept and the wave concept at the same time.
- Louis de Broglie, 1929
5.1: X-Ray Scattering
Max von Laue suggested that if x-rays were a form of electromagnetic
radiation, interference effects should be observed.
Crystals act as three-dimensional gratings, scattering the waves and
producing observable interference effects.
Bragg’s Law
William Lawrence Bragg
interpreted the x-ray
scattering as the reflection of
the incident x-ray beam from
a unique set of planes of
atoms within the crystal.
There are two conditions for
constructive interference of
the scattered x rays:
1) The angle of incidence must equal the
angle of reflection of the outgoing wave.
2) The difference in path lengths must be
an integral number of wavelengths.
Bragg’s Law: nλ = 2d sin θ (n = integer)
The Bragg Spectrometer
A Bragg spectrometer scatters x rays
from crystals. The intensity of the
diffracted beam is determined as a
function of scattering angle by rotating
the crystal and the detector.
When a beam of x rays passes
through a powdered crystal, the dots
become a series of rings.
5.3: Electron Scattering
In 1925, Davisson and Germer
experimentally observed that
electrons were diffracted (much
like x-rays) in nickel crystals.
George P. Thomson (1892–1975),
son of J. J. Thomson, reported seeing
electron diffraction in transmission
experiments on celluloid, gold,
aluminum, and platinum. A randomly
oriented polycrystalline sample of
SnO2 produces rings.
5.2: De Broglie Waves
In his thesis in 1923, Prince Louis V.
de Broglie suggested that mass
particles should have wave properties
similar to electromagnetic radiation.
The energy can be written as:
If a light-wave could also act
like a particle, why shouldn’t
matter-particles also act like
waves?
hn = pc = pln
Thus the wavelength of a matter
wave is called the de Broglie
wavelength:
Louis V. de Broglie
(1892-1987)
Bohr’s Quantization Condition revisited
One of Bohr’s assumptions in his hydrogen atom model was that the
angular momentum of the electron in a stationary state is nħ.
This turns out to be equivalent to
saying that the electron’s orbit
consists of an integral number
of electron de Broglie
electron
wavelengths:
de Broglie
wavelength
Circumference
Multiplying by p/2p, we find the
angular momentum:
5.4: Wave Motion
De Broglie matter waves should be described in the same manner as
light waves. The matter wave should be a solution to a wave equation
like the one for electromagnetic waves:
 2Y 1  2Y
 2
0
2
2
x
v t
It will actually be different,
but, in some cases, the
solutions are the same.
with a solution:
Y(x,t) = A exp[i(kx – wt – q)]
Define the wave number k and the angular frequency w as usual:
and
Electron Double-Slit Experiment
C. Jönsson of Tübingen,
Germany, succeeded in 1961
in showing double-slit
interference effects for
electrons by constructing very
narrow slits and using
relatively large distances
between the slits and the
observation screen.
This experiment demonstrated
that precisely the same
behavior occurs for both light
(waves) and electrons
(particles).
Wave-particle-duality solution
It’s somewhat disturbing that everything is both a particle and a wave.
The wave-particle duality is a little less disturbing if we think in terms
of:
Bohr’s Principle of Complementarity: It’s not possible to describe
physical observables simultaneously in terms of both particles and
waves.
When we’re making a measurement, use the particle description, but
when we’re not, use the wave description.
When we’re looking, fundamental quantities are particles; when
we’re not, they’re waves.
5.5: Waves or Particles?
Dimming the light in
Young’s two-slit experiment
results in single photons at
the screen. Since photons
are particles, each can only
go through one slit, so, at
such low intensities, their
distribution should become
the single-slit pattern.
Each photon
actually goes
through both
slits!
Can you tell which slit the photon went
through in Young’s double-slit exp’t?
When you block one slit, the one-slit pattern returns.
One-slit
pattern
Two-slit
pattern
At low intensities, Young’s two-slit experiment shows that light
propagates as a wave and is detected as a particle.
Which slit does the
electron go through?
Shine light on the double slit and observe with a microscope. After the
electron passes through one of the slits, light bounces off it; observing
the reflected light, we determine which slit the electron went through.
The photon momentum is:
The electron momentum is:
Need lph < d to
distinguish the slits.
Diffraction is significant only
when the aperture is ~ the
wavelength of the wave.
The momentum of the photons used to determine which slit the electron
went through is enough to strongly modify the momentum of the electron
itself—changing the direction of the electron! The attempt to identify
which slit the electron passes through will in itself change the diffraction
pattern! Electrons also propagate as waves and are detected as
particles.
5.6: Uncertainty Principle: Energy
Uncertainty
The energy uncertainty of a wave packet is:
E  h n  h
w
 w
2p
Combined with the angular frequency relation we derived earlier:
Energy-Time Uncertainty Principle:
.
Momentum Uncertainty Principle
The same mathematics relates x and k:
k x ≥ ½
So it’s also impossible to measure simultaneously the precise values
of k and x for a wave.
Now the momentum can be written in terms of k:
h
h
p 
 (h / 2p )k
l 2p / k
So the uncertainty in momentum is:
But multiplying k x ≥ ½ by ħ:

p  k
k x 
2
And we have Heisenberg’s Uncertainty Principle:
p k
How to think about Uncertainty
The act of making one measurement perturbs the other.
Precisely measuring the time disturbs the energy.
Precisely measuring the position disturbs the momentum.
The Heisenbergmobile. The problem was that when you looked at
the speedometer you got lost.
Kinetic Energy Minimum
Since we’re always uncertain as to the exact position, x  ,
of a particle, for example, an electron somewhere inside an
atom, the particle can’t have zero kinetic energy:
p 
x

The average of a positive quantity must always exceed its
uncertainty:
pave  p 
x

so:
K ave
2
2
pave
(p)2



2m
2m
2m
5.7: Probability, Wave Functions, and
the Copenhagen Interpretation
Okay, if particles are also waves, what’s waving?
Probability
The wave function determines the likelihood (or probability) of
finding a particle at a particular position in space at a given time:
P( x)  Y ( x)
2
The probability of the
particle being between
x1 and x2 is given by:
x2

Y ( x) dx
2
x1



Y ( x) dx  1
2
The total probability of finding the particle
is 1. Forcing this condition on the wave
function is called normalization.
5.8: Particle in a Box
A particle (wave) of mass m is in a one-dimensional
box of width ℓ.
The box puts boundary conditions on the wave. The
wave function must be zero at the walls of the box
and on the outside.
In order for the probability to vanish at the walls, we
must have an integral number of half wavelengths in
the box:
The energy:
2
2
p
h
E  K .E.  12 mv2 

2m 2ml 2
The possible wavelengths
are quantized and hence
so are the energies:
Probability of the particle vs. position
Note that E0 = 0 is not a
possible energy level.
The concept of energy
levels, as first discussed in
the Bohr model, has
surfaced in a natural way
by using waves.
The probability of
observing the particle
between x and x + dx in
each state is
Quantum Mechanics II
6.1
6.2
6.3
6.4
6.5
The Schrödinger Wave Equation
Expectation Values
Infinite Square-Well Potential
Finite Square-Well Potential
Three-Dimensional InfinitePotential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling
Erwin Schrödinger (1887-1961)
A careful analysis of the process of observation in atomic physics has
shown that the subatomic particles have no meaning as isolated
entities, but can only be understood as interconnections between the
preparation of an experiment and the subsequent measurement.
- Erwin Schrödinger
Opinions on quantum mechanics
I think it is safe to say that no
one understands quantum
mechanics. Do not keep saying
to yourself, if you can possibly
avoid it, “But how can it be like
that?” because you will get
“down the drain” into a blind
alley from which nobody has yet
escaped. Nobody knows how it
can be like that.
- Richard Feynman
Those who are not shocked
when they first come across
quantum mechanics cannot
possibly have understood it.
Richard Feynman (1918-1988)
- Niels Bohr
6.1: The Schrödinger Wave Equation
The Schrödinger wave equation in its time-dependent form for a
particle of energy E moving in a potential V in one dimension is:
where V = V(x,t)
where i is the square root of -1.
The Schrodinger Equation is THE fundamental equation of Quantum
Mechanics.
General Solution of the Schrödinger Wave
Equation when V = 0
Try this solution:
Y
 iw Aei ( kx wt )  iwY
t
Y
i
 (i )(iw )Y  w Y
t
This works as long as:
2
k2
w
2m
2Y
2


k
Y
2
t
2 2
 2 2Y
k

Y
2
2m x
2m
which says that the total
energy is the kinetic energy.
General Solution of the Schrödinger
Wave Equation when V = 0
In free space (with V = 0), the general form of the wave function is
which also describes a wave moving in the x direction. In general the
amplitude may also be complex.
The wave function is also not restricted to being real. Notice that this
function is complex.
Only the physically measurable quantities must be real. These
include the probability, momentum and energy.
Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is
given in the equation
The probability of the particle being between x1 and x2 is given by
The wave function must also be normalized so that the probability
of the particle being somewhere on the x axis is 1.
Properties of Valid Wave Functions
Conditions on the wave function:
1. In order to avoid infinite probabilities, the wave function must be
finite everywhere.
2. The wave function must be single valued.
3. The wave function must be twice differentiable. This means that it
and its derivative must be continuous. (An exception to this rule
occurs when V is infinite.)
4. In order to normalize a wave function, it must approach zero as x
approaches infinity.
Solutions that do not satisfy these properties do not generally
correspond to physically realizable circumstances.
Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time.
The dependence on time and position can then be separated in the
Schrödinger wave equation. Let:
which yields:
Now divide by the wave function y(x) f(t):
The left side depends only on t, and the right side
depends only on x. So each side must be equal to
a constant. The time dependent side is:
Time-Independent Schrödinger Wave Equation
We integrate both sides and find:
where C is an integration constant that we may choose to be 0.
Therefore:
But recall our solution for the free particle:
In which f(t) = e -iw t, so: w = B / ħ or B = ħw, which means that: B = E !
So multiplying by y(x), the spatial Schrödinger equation becomes:
Time-Independent Schrödinger Wave Equation
This equation is known as the time-independent Schrödinger
wave equation, and it is as fundamental an equation in quantum
mechanics as the time-dependent Schrodinger equation.
So often physicists write simply:
Ĥy  Ey
where:
2

Hˆ  
V
2
2m x
2
Ĥ is an operator.
Stationary States
The wave function can be written as:
The probability density becomes:
The probability distribution is constant in time.
This is a standing wave phenomenon and is called a stationary state.
6.2: Expectation Values
In quantum mechanics, we’ll compute expectation values.
The expectation value, x , is the weighted average of a
given quantity. In general, the expected value of x is:
x  P1 x1  P2 x2 
 PN xN 
P x
i
i
i
If there are an infinite number of possibilities, and x is continuous:

x  P( x) x dx
Quantum-mechanically:

x  Y ( x)Y * ( x) x dx 

Y * ( x) x Y ( x) dx
And the expectation of some function of x, g(x):
g ( x) 

Y * ( x) g ( x) Y ( x) dx
Momentum Operator
To find the expectation value of p, we first need to represent p in terms
of x and t. Consider the derivative of the wave function of a free particle
with respect to x:
With k = p / ħ we have
This yields
This suggests we define the momentum operator as
The expectation value of the momentum is
.
Position and Energy Operators
The position x is its own operator. Done.
Energy operator: The time derivative of the free-particle wave
function is:
Substituting w  E / ħ yields
The energy operator is:
The expectation value of the energy is:
Deriving the Schrodinger Equation
using operators
The energy is:
p2
 EY 
Y V Y
2m
p2
E  K V 
V
2m
Substituting operators:
E:
EY  i
Y
t
p2
1 
 
Y V Y 

i

 Y V Y
2m
2m 
x 
2
K+V:
2Y

V Y
2
2m x
2
Substituting:
2
Y
 2Y
i

V Y
2
t
2m x
6.3: Infinite Square-Well Potential
The simplest such system is that of a particle
trapped in a box with infinitely hard walls that
the particle cannot penetrate. This potential is
called an infinite square well and is given by:
0
L
x
Clearly the wave function must be zero where the potential is infinite.
Where the potential is zero (inside the box), the time-independent
Schrödinger wave equation becomes:
where
The general solution is:
Quantizatio
n
Boundary conditions of the potential dictate
that the wave function must be zero at x = 0
and x = L. This yields valid solutions for
integer values of n such that kL = np.
0
The wave function is:
We normalize the wave function:

L
x
½  ½ cos(2npx/L)

A  2/ L
The normalized wave
function becomes:
These functions are identical to those obtained for a vibrating string
with fixed ends.
Quantized Energy
The quantized wave number now becomes:
Solving for the
energy yields:
Note that the energy depends on integer values of n. Hence the energy
is quantized and nonzero.
The special case of
n = 1 is called the
ground state.
6.4: Finite SquareWell Potential
The finite square-well potential is
The Schrödinger equation
outside the finite well in
regions I and III is:
Letting:
Considering that the wave function
must be zero at infinity, the solutions
for this equation are
yields
Finite Square-Well Solution
Inside the square well, where the potential V is zero, the wave equation
becomes
The solution here is:
The boundary
conditions require that:
so the wave function
is smooth where
the regions meet.
Note that the
wave function is
nonzero outside
of the box.
where
Penetration Depth
The penetration depth is
the distance outside the
potential well where the
probability significantly
decreases. It is given by
The penetration distance
that violates classical
physics is proportional to
Planck’s constant.
6.6: Simple Harmonic Oscillator
Simple harmonic
oscillators describe
many physical
situations: springs,
diatomic molecules
and atomic lattices.
Consider the Taylor expansion of a
potential function:
Simple Harmonic
Oscillator
Consider the second-order term
of the Taylor expansion of a
potential function:
Substituting this into Schrödinger’s equation:
Let
and
which yields:
The Parabolic
Potential Well
The wave function solutions
are
where Hn(x) are Hermite
polynomials of order n.




The Parabolic
Potential Well
Classically, the probability
of finding the mass is
greatest at the ends of
motion and smallest at
the center.
Contrary to the classical
one, the largest
probability for this lowest
energy state is for the
particle to be at the
center.
Analysis of the Parabolic Potential
Well
As the quantum number increases, however, the solution
approaches the classical result.
The Parabolic Potential Well
The energy levels are given by:
The zero point
energy is
called the
Heisenberg
limit:
6.7: Barriers and Tunneling
Consider a particle of energy E approaching a potential barrier of
height V0, and the potential everywhere else is zero.
First consider the case of the energy greater than the potential barrier.
In regions I and III the wave numbers are:
In the barrier region we have
Reflection and Transmission
The wave function will consist of an incident wave, a reflected wave, and
a transmitted wave.
The potentials and the Schrödinger wave equation for the three regions
are as follows:
The corresponding solutions are:
As the wave moves from left to right, we can simplify the wave functions
to:
Probability of Reflection and
Transmission
The probability of the particles being reflected R or transmitted T is:
Because the particles must be either reflected or transmitted we
have: R + T = 1.
By applying the boundary conditions
x → ±∞, x = 0, and x = L, we arrive at
the transmission probability:
Note that the transmission probability can be 1.
Tunneling
Now we consider the situation
where classically the particle
doesn’t have enough energy
to surmount the potential
barrier, E < V0.
The quantum mechanical result is one of the most remarkable
features of modern physics. There is a finite probability that the
particle can penetrate the barrier and even emerge on the other
side!
The wave function
in region II becomes:
The transmission probability that
describes the phenomenon of tunneling is:
Tunneling wave function
This violation of classical physics is allowed by the uncertainty
principle. The particle can violate classical physics by E for a
short time, t ~ ħ / E.
Analogy with Wave Optics
If light passing through a glass prism reflects
from an internal surface with an angle greater
than the critical angle, total internal reflection
occurs. However, the electromagnetic field is
not exactly zero just outside the prism. If we
bring another prism very close to the first one,
experiments show that the electromagnetic
wave (light) appears in the second prism The
situation is analogous to the tunneling
described here. This effect was observed by
Newton and can be demonstrated with two
prisms and a laser. The intensity of the second
light beam decreases exponentially as the
distance between the two prisms increases.
Alpha-Particle Decay
The phenomenon of tunneling explains alpha-particle decay of heavy,
radioactive nuclei.
Inside the nucleus, an alpha particle feels the strong, short-range
attractive nuclear force as well as the repulsive Coulomb force.
The nuclear force dominates inside the nuclear radius where the
potential is ~ a square well.
The Coulomb force dominates
outside the nuclear radius.
The potential barrier at the nuclear
radius is several times greater than
the energy of an alpha particle.
In quantum mechanics, however,
the alpha particle can tunnel
through the barrier. This is
observed as radioactive decay.
6.5: Three-Dimensional Infinite-Potential Well
The wave function must be a function of all three spatial coordinates.
Now consider momentum as an operator acting on the wave function.
In this case, the operator must act twice on each dimension. Given:
So the three-dimensional Schrödinger wave equation is
The 3D infinite potential well
It’s easy to show that:
y ( x, y, z )  A sin(k x x) sin(k y y ) sin(k z z )
where:
k x  p nx / Lx
kz  p nz / Lz
2
2
2 

n
nx
n
p
y
E
 2  2  z2 
2m  Lx Ly Lz 
2
and:
k y  p n y / Ly
2
When the box is a cube:
E
p2
2
2
2
2
n

n

n

x
y
z 
2
2mL
Try 10, 4, 3
and 8, 6, 5
Note that more than one wave function can have the same energy.
Degeneracy
The Schrödinger wave equation in three dimensions introduces
three quantum numbers that quantize the energy. And the same
energy can be obtained by different sets of quantum numbers.
A quantum state is called degenerate when there is more than
one wave function for a given energy.
Degeneracy results from particular properties of the potential energy
function that describes the system. A perturbation of the potential
energy can remove the degeneracy.