Review Modern Physics, Ph 311
It was found that there was no displacement of the
interference fringes, so that the result of the
experiment was negative and would, therefore, show
that there is still a difficulty in the theory itself…
- Albert Michelson, 1907
1/ to 2/ of our
3
3
modern economy !!!
1
Inertial Reference Frame


A reference frame is called an inertial frame
if Newton laws are valid in that frame.
Such a frame is established when a body, not
subjected to net external forces, is observed
to move in rectilinear motion at constant
velocity.
2
Newtonian Principle of Relativity

If Newton’s laws are valid in one reference
frame, then they are also valid in another
reference frame moving at a uniform velocity
relative to the first system.

This is referred to as the Newtonian
principle of relativity or Galilean
invariance/relativity.
3
4
The Galilean Transformation
For a point P


In system K: P = (x, y, z, t)
In system K’: P = (x’, y’, z’, t’)
P
x
K
K’
x’-axis
x-axis
5
Conditions of the Galilean Transformation



Parallel axes
K’ has a constant relative velocity in the x-direction
with respect to K
Time (t) for all observers is a Fundamental invariant,
i.e., the same for all inertial observers
6
The Inverse Relations
Step 1. Replace
with
.
Step 2. Replace “primed” quantities with
“unprimed” and “unprimed” with “primed.”
7
Results of Maxwell's electrodynamics


Visible light covers only a small range of the total
electromagnetic spectrum
All electromagnetic waves travel in a vacuum with a
speed c given by:
(where μ0 and ε0 are the respective permeability and
permittivity of “free” space)
8
Need for Ether

The wave nature of light suggested that there
existed a propagation medium called the
luminiferous ether or just ether.

Ether had to have such a low density that the planets
could move through it without loss of energy

It also had to have an enormous elasticity/toughness to
support the high velocity of light waves
According to classical physics ideas, the ether frame
would be a preferred frame, the only one in which
Maxwell’s equation would be valid as derived

9
An Absolute Reference System


Ether was proposed as an absolute reference
system in which the speed of light was this
constant and in all frames moving with
respect that that frame, there needed to be
modifications of Maxwell’s laws.
The Michelson-Morley experiment was an
attempt to figure out Earth’s relatives
movement through (with respect to) the ether
so that Maxwell’s equations could be
corrected for this effect.
10
The Michelson Interferometer
1. AC is parallel to the motion
of the Earth inducing an “ether
wind”
2. Light from source S is split
by mirror A and travels to
mirrors C and D in mutually
perpendicular directions
3. After reflection the beams
recombine at A slightly out of
phase due to the “ether wind”
as viewed by telescope E.
11
NEVER OBSERVED !!!!
12
The Lorentz-FitzGerald Contraction

Another hypothesis proposed independently by both H. A.
Lorentz and G. F. FitzGerald suggested that the length ℓ1, in
the direction of the motion was contracted by a factor of
…thus making the path lengths equal to account for the zero
phase shift.

This, however, was an ad hoc assumption that could not
be experimentally tested. It turned out to be “less than half
of the story”
13
Length contracted for the
moving muon, it’s own life
time just 2.2 micro seconds
Life time of the muon delayed
for observer on Earth so that it
can travel the whole distance
as observed from Earth
Great thing about special
relativity is that one can
always take two viewpoints,
moving with the experiment,
watching the experiment
move, the observations need
to be consistent in both cases
14
Lorentz Transformation Equations
So there is four-dimensional space time !!!
15
Mary has a light clock. A suitable clock is just any periodic process, the time it takes
for one cycle of the process is the period, its inverse is the frequency.
Tom watching Mary go by figures that her time is delayed due to her moving in a
straight line with a constant high velocity.
16
Atomic Clock Measurement
Figure 2.20: Two airplanes took off (at different times) from Washington, D.C., where the U.S.
Naval Observatory is located. The airplanes traveled east and west around Earth as it rotated.
Atomic clocks on the airplanes were compared with similar clocks kept at the observatory to
show that the moving clocks in the airplanes ran slower.
17
No simultaneity if not also at the same position, just a
consequence of the Lorentz transformaitons
18
The Lorentz Velocity Transformations
In addition to the previous relations, the Lorentz
velocity transformations for u’x, u’y , and u’z can be
obtained by switching primed and unprimed and
changing v to –v:
19
Einstein’s Two Postulates
With the belief that Maxwell’s equations (and with
it all of the known physics of the time) must be
valid in all inertial frames, Einstein proposes the
following postulates:
1) The principle of relativity: The laws of
physics are the same in all inertial systems.
There is no way to detect absolute motion, and
no preferred inertial system exists.
2) The constancy of the speed of light:
Observers in all inertial systems measure the
same value for the speed of light in a vacuum.
20
Relativistic Momentum


Rather than abandon the conservation of linear
momentum, let us look for a modification of the
definition of linear momentum that preserves both it
and Newton’s second law.
To do so requires reexamining mass to conclude that:
Relativistic dynamics can be derived by
assuming that mass is increasing with
velocity. The Lorentz factor gets larger when
velocities get larger and so does mass
apparently as we can see from the
relativistic momentum equation. Einstein
derived relativistic dynamics that way. His
derivations are sure correct, but the
foundations are somewhat shaking as there
is no really good definition for mass.
21
Relativistic Energy

Due to the new idea of relativistic mass, we
must now redefine the concepts of work and
energy.

Therefore, we modify Newton’s second law to
include our new definition of linear momentum,
and force becomes:
22
Relativistic Kinetic Energy
Equation (2.58) does not seem to resemble the classical result for kinetic energy, K =
½mu2. However, if it is correct, we expect it to reduce to the classical result for low
speeds. Let’s see if it does. For speeds u << c, we expand in a binomial series as
follows:
where we have neglected all terms of power (u/c)4 and greater, because u << c. This
gives the following approximation for the relativistic kinetic energy at low speeds:
which is the expected classical result. We show both the relativistic and classical kinetic
energies in the next Figure. They diverge considerably above a velocity of 0.1c. Best to
use relativistic dynamics as soon as the speed of something is larger than 1 % of the
speed of light.
23
Relativistic and Classical Kinetic Energies
24
Total Energy and Rest Energy
We rewrite in the form
The term mc2 is called the rest energy and is denoted by E0.
This leaves the sum of the kinetic energy and rest energy to
be interpreted as the total energy of the particle. The total
energy is denoted by E and is given by
25
Momentum and Energy
We square this result, multiply by c2, and
rearrange the result.
We replace β2 and find
26
Momentum and Energy (continued)
The first term on the right-hand side is just E2, and the second term is
E02. The last equation becomes
We rearrange this last equation to find the result we are seeking, a
relation between energy and momentum.
or
is a useful result to relate the total energy of a particle with its
momentum. The quantities (E2 – p2c2) and m are invariant
quantities. Note that when a particle’s velocity is zero and it has no
momentum, “accelerator Equation” correctly gives E0 as the
particle’s total energy.
There can be mass less particles that still have momentum. These can collide with
massive particles. For such a collision one needs to invoke special relativity!
27
Binding Energy
The binding energy is the difference between
the rest energy of the individual particles and
the rest energy of the combined bound system.
A couple of eV for chemical reactions.
A couple of MeV for nuclear reactions.
28
A Conducting Wire
29
Principle of Equivalence


The principle of equivalence
is an experiment in noninertial reference frames.
Consider an astronaut sitting
in a confined space on a
rocket placed on Earth. The
astronaut is strapped into a
chair that is mounted on a
weighing scale that indicates
a mass M. The astronaut
drops a safety manual that
falls to the floor.

Now contrast this situation with the rocket accelerating through space. The gravitational
force of the Earth is now negligible. If the acceleration has exactly the same magnitude g
on Earth, then the weighing scale indicates the same mass M that it did on Earth, and the
safety manual still falls with the same acceleration as measured by the astronaut. The
question is: How can the astronaut tell whether the rocket is on earth or in space?

Principle of equivalence: There is no experiment that can be done in
a small confined space that can detect the difference between a
uniform gravitational field and an equivalent uniform acceleration.
30
Gravitational Time Dilation

Since the frequency of the clock decreases near the Earth, a
clock in a gravitational field runs more slowly (it takes longer
for a hand to move on a clock – so in aggregate the clock gets
slower) according to the gravitational time dilation. This is
because 4D space-time is “bend” – non-Euclidian, so there are
no Euclidian straight lines to follow but Geodesics in a space
whit Riemann’s coordinates

A very accurate experiment was done by comparing the frequency
of an atomic clock flown on a Scout D rocket to an altitude of
10,000 km with the frequency of a similar clock on the ground. The
measurement agreed with Einstein’s general relativity theory to
within 0.02%.
31
Tests of General Relativity
Bending of Light

During a solar eclipse of the sun by the moon,
most of the sun’s light is blocked on Earth,
which afforded the opportunity to view starlight
passing close to the sun in 1919. The starlight
was bent as it passed near the sun which
caused the star to appear displaced.

Einstein’s general theory predicted a
deflection of 1.75 seconds of arc, and the two
measurements found 1.98 ± 0.16 and 1.61 ±
0.40 seconds.

Since the eclipse of 1919, many experiments,
using both starlight and radio waves from
quasars, have confirmed Einstein’s predictions
about the bending of light with increasingly
good accuracy.
32
Light Retardation



As light passes by a massive object, the
path taken by the light is longer because
of the spacetime curvature.
The longer path causes a time delay for a
light pulse traveling close to the sun.
This effect was measured by sending a
radar wave to Venus, where it was
reflected back to Earth. The position of
Venus had to be in the “superior
conjunction” position on the other side of
the sun from the Earth. The signal
passed near the sun and experienced a
time delay of about 200 microseconds.
This was in excellent agreement with the
general theory.
33
Spacetime Curvature of Space




Light bending for the Earth observer seems to violate the premise
that the velocity of light is constant from special relativity. Light
traveling at a constant velocity implies that it travels in a straight
line.
Einstein recognized that we need to expand our definition of a
straight line.
The shortest distance between two points on a flat surface appears
different than the same distance between points on a sphere. The
path on the sphere appears curved. We shall expand our definition
of a straight line to include any minimized distance between two
points.
Thus if the spacetime near the Earth is not flat, then the straight line
path of light near the Earth will appear curved.
34
Perihelion Shift of Mercury


The orbits of the planets are ellipses, and the point closest to the
sun in a planetary orbit is called the perihelion. It has been known
for hundreds of years that Mercury’s orbit precesses about the sun.
Accounting for the perturbations of the other planets left 43 seconds
of arc per century that was previously unexplained by classical
physics.
The curvature of spacetime explained by general relativity
accounted for the 43 seconds of arc shift in the orbit of Mercury.
35
Gravitational Wave Experiments


Taylor and Hulse discovered a binary system of two neutron stars that lose
energy due to gravitational waves that agrees with the predictions of
general relativity.
LIGO is a large Michelson interferometer device that uses four test masses
on two arms of the interferometer. The device will detect changes in length
of the arms due to a passing wave.

NASA and the European Space
Agency (ESA) are jointly developing
a space-based probe called the
Laser Interferometer Space Antenna
(LISA) which will measure
fluctuations in its triangular shape.
No success so far, perhaps general
relativity (and special relativity with it) are
not really true, just very very good
approximations to something else?
36
BUT, thank you
very much
indeed Albert !!!
everybody loves
this !!!!
37
Dual nature of light (electromagnetic
radiation) both/neither wave and/nor particle
http://usatoday30.usatoday.com/tech/science/genetics/2008-05-08-platypus-geneticmap_N.htm
“Australia's unique duck-billed platypus is part bird, part reptile and part mammal according
to its gene map.
The platypus is classed as a mammal because it has fur and feeds its young with milk. It
flaps a beaver-like tail. But it also has bird and reptile features — a duck-like bill and
webbed feet, and lives mostly underwater. Males have venom-filled spurs on their heels.”
38
Light according to Maxwell
39
Fig. 3-2, p. 67
Wien’s Displacement Law


The intensity (λ, T) is the total power radiated per unit
area per unit wavelength at a given temperature.
Wien’s displacement law: The maximum of the
distribution shifts to smaller wavelengths as the
temperature is increased.
8f 2
u( f , T )  3 
e
u( f , T )  h(
hf
x
k BT
hf
e
( hf
k BT
)
1
k BT 3
)  u( x )
hc
8  x 3
u( x )  x
e 1
When u(x) is plotted over x, there
is only one peak! One universal
curve for all wavelengths and T
40
Two fitting parameters and no
physical theory behind them !!
41
3.5: Blackbody Radiation


When matter is heated, it
emits radiation.
A blackbody is a cavity in a
material that only emits
thermal radiation. Incoming
radiation is absorbed in the
cavity.
 Blackbody radiation is theoretically interesting
because the radiation properties of the blackbody are
independent of the particular material. Physicists can
study the properties of intensity versus wavelength at
fixed temperatures.
42
Rayleigh-Jeans Formula

Lord Rayleigh used the classical theories of electromagnetism and
thermodynamics to show that the blackbody spectral distribution
should be
k: Boltzmann’s
constant 8.614
10-5 eV/K

It approaches the data at longer wavelengths, but it deviates badly at
short wavelengths. This problem for small wavelengths became
known as “the ultraviolet catastrophe” and was one of the outstanding
exceptions that classical physics could not explain.
43
Planck’s Radiation Law

Planck assumed that the radiation in the cavity was emitted (and
absorbed) by some sort of “resonators” that were contained in the
walls. These resonators were modeled as harmonic oscillators. He
effectively invented new physics in the process. His result cannot
be explained with classical Boltzmann-Maxwell statistics.
Planck’s radiation law, only one
fundamental constant h left that
can explain Wien’s and Stephan’s
constants …, significant progress

Planck made two modifications to the classical theory:
1)
2)
The oscillators (of electromagnetic origin) can only have certain discrete
energies determined by En = nhf, where n is an integer, f is the frequency,
and h is called Planck’s constant.
h = 6.6261 × 10−34 J·s.
The oscillators can absorb or emit energy in discrete multiples of the
fundamental quantum of energy given by
44
Photoelectric
effect
45
Experimental Results
Only if the energy threshold
to get electrons out of the
metal (work function) is
exceeded.
46
Einstein’s Theory

Einstein suggested that the electromagnetic radiation field is
quantized into particles called photons. Each photon has the
energy quantum:
where f is the frequency of the light and h is Planck’s constant. Also
he came up with the wave particle duality of light, at long
wavelengts it looks more like a wave at short wavelenght, high
frequency, high enerly it looks more like a particle

The photon travels at the speed of light in a vacuum, and its
wavelength is given by
47
Einstein’s Theory

Conservation of energy yields:
where
is the work function of the metal.
Explicitly the energy is

The retarding potentials measured in the photoelectric effect are
the opposing potentials needed to stop the most energetic
electrons.
48
X-Ray Production

An energetic electron passing through matter will radiate photons and lose kinetic
energy which is called bremsstrahlung, from the German word for “braking
radiation.” Since linear momentum must be conserved, the nucleus absorbs very little
energy, and it is ignored. The final energy of the electron is determined from the
conservation of energy to be

An electron that loses a large amount of energy will produce an X-ray photon.
Current passing through a filament produces copious numbers of electrons by
thermionic emission. These electrons are focused by the cathode structure into a
beam and are accelerated by potential differences of thousands of volts until they
impinge on a metal anode surface, producing x rays by bremsstrahlung as they stop
in the anode material.
49
Inverse Photoelectric Effect.

Conservation of energy requires that the
electron kinetic energy equal the
maximum photon energy where we
neglect the work function because it is
normally so small compared to the
potential energy of the electron. This
yields the Duane-Hunt limit which was
first found experimentally. The photon
wavelength depends only on the
accelerating voltage and is the same for
all targets.
Let’s have 10 – 50 keV, very short wavelengths, very energetic photons
50
Bragg’s law
51
No way !!!
Just a relativistic collision
between a mass less particle
and a massive particle.
52
Compton Effect

When a photon enters matter, it is likely to interact with one of the atomic
electrons. The photon is scattered from only one electron, rather than from
all the electrons in the material, and the laws of conservation of energy and
momentum apply as in any elastic collision between two particles. The
momentum of a particle moving at the speed of light is

The electron energy can be written as

This yields the change in wavelength of the scattered photon which is
known as the Compton effect:
53
X-Ray Scattering, modern crystallography


Max von Laue suggested that if x rays were a form of
electromagnetic radiation with wavelengths on the 0.1 nm scale,
interference effects should be observed for a crystal, which can be
thought of as a 3D diffraction grating.
Friedrich and Knipping did the experiments and modern
crystallography was born !!! Almost all of of our knowledge of atomic
structures comes from such (and electron and neutron) diffraction
experiments
54
Wave – particle duality, “wavical”
Taoism”
Taijitu (literally
"diagram of
the supreme
ultimate"
No problem, Bohr’s complementarily
55
Dual nature of quantum mechanical objects
both/neither particle and/nor wave
http://usatoday30.usatoday.com/tech/science/genetics/2008-05-08-platypus-geneticmap_N.htm
“Australia's unique duck-billed platypus is part bird, part reptile and part mammal according
to its gene map.
The platypus is classed as a mammal because it has fur and feeds its young with milk. It
flaps a beaver-like tail. But it also has bird and reptile features — a duck-like bill and
webbed feet, and lives mostly underwater. Males have venom-filled spurs on their heels.”
56
Thomson’s Atomic Model

J. J. Thomson’s “plum-pudding” model of the atom had the positive
charges spread uniformly throughout a sphere the size of the atom,
with electrons embedded in the uniform background.
Not quite, electrons
repulse each other
as much as possible
but what is the
dough?

In J. J. Thomson’s view, when the atom was heated, the electrons
could vibrate about their equilibrium positions, thus producing
electromagnetic radiation.
57
More experiments,
looking at large
angles where one
would not expect any
scattering to show
up, BUT …
58
There is no dough, just
lots and lots of empty
space and a tiny tiny
heavy nucleus where all
of the positive charges
reside.
59
Planetary model of the atom
would not work on the basis of
classical physics, would not
explain why atoms are forever,
when a molecule breaks up,
the atoms are just as before
60
Fig. 4-21, p. 131
Angular
momentum must
be quantized in
nature in units of
h-bar, from that
follows
quantization of
energy levels ….
61
62
Fig. 4-23, p. 133
63
Fig. 4-24, p. 134
Can all be explained from
Bohr’s model as he puts
physical meaning to the
Rydberg equation.
64
Fig. 4-20, p. 129
Bohr’s second paper in
1913. There should be
shells, idea basically
correct, helps explaining
basic chemistry
65
The Correspondence Principle
Classical electrodynamics
+
Bohr’s atomic model
Determine the properties
of radiation
Need a principle to relate the new modern results with classical
ones. Mathematically: h -> 0
Bohr’s correspondence
principle
In the limits where classical and quantum
theories should agree, the quantum
theory must reduce the classical result.
Bohr’s third paper in 1913
66
4.6: Characteristic X-Ray Spectra and
Atomic Number

Shells have letter names:
K shell for n = 1
L shell for n = 2

The atom is most stable in its ground state.


An electron from higher shells will fill the innershell vacancy at lower energy.
When it occurs in a heavy atom, the radiation emitted is an x ray.
It has the energy E (x ray) = Eu − Eℓ.
67
X-ray spectroscopy on the basis of the
characteristic X-rays
Atomic Number Z,
L shell to K shell
M shell to K shell


Kα x ray
Kβ x ray
Atomic number Z = number of protons in the nucleus.
Moseley found a relationship between the frequencies of the
characteristic x ray and Z.
This holds for the Kα x ray.
Explanation on the basis of Bohr’s model for H and shielding for all other
atoms !!!!
68
Moseley’s Results support Bohr’s ideas
for all tested atoms

The x ray is produced from n = 2 to n = 1 transition.

In general, the K series of x ray wavelengths are
Moseley’s research clarified the importance of the electron shells
for all the elements, not just for hydrogen.
69
Frank-Hertz experiment
Accelerating voltage is below 5 V.
electrons did not lose energy as they are scattered
elastically at the much heavier Hg atoms.
Accelerating voltage is above 5 V.
sudden drop in the current because there is now inelastic
scattering instead.

70
71
There are also matter waves, not only
classical and electromagnetic waves !!!
Wave particle duality for matter leads us into
quantum mechanics, condensed matter
physics ….
1/ to 2/ of our
3
3
modern economy !!!
72
Instantaneous (linear)
momentum is quantized
as well in a bound system
= 32 ao
momentum =
h
/ wavelength
for particles
with mass as
well, not only
photons
73
Fig. 5-2, p. 153
5.3: Electron Scattering

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 the effects of electron
diffraction in transmission experiments. The first
target was celluloid, and soon after that gold,
aluminum, and platinum were used. The randomly
oriented polycrystalline sample of SnO2 produces
rings as shown in the figure at right.
74
TEM
One operation mode is
transmission diffraction,
there is also electron
energy loss spectroscopy
and X-ray spectroscopy
75
SEM
Short wavelength
and nearly parallel
fine electron beam
results in large
depth of focus, SEM
images “appear”
almost threedimensional
76
One full cycle for envelop wave = 2 pi
77
78
Wave Packet Envelope

The superposition of two waves yields a wave number and angular
frequency of the wave packet envelope.

The range of wave numbers and angular frequencies that produce the
wave packet have the following relations:

A Gaussian wave packet has similar relations:

The localization of the wave packet over a small region to describe a
particle requires a large range of wave numbers. Conversely, a small
range of wave numbers cannot produce a wave packet localized
within a small distance.
79
k x  x  1
k y  y  1
k z  z  1
Modern physics backed up by experiments
Mathematical
uncertainties
  t  1
px  x  
p y  y  
pz  z  
E  t  
E   
E    
dE    d
p x  k x
Heisenberg's
uncertainties
dpx  dk x
px  k x
p y  k y
dp y  dk y
p y  k y
pz  k z
dpz  dk z
pz  k z
80
Since the uncertainty
principle is really a
statement about accuracy
rather than precision, there
is a kind of systematic rest
error that cannot be
corrected for
In classical physics this is
simply ignored as things are
large in comparison to
electrons, atoms, molecules,
nano-crystals …
81
Probability, Wave Functions, and the
Copenhagen Interpretation

The square of the wave function determines the likelihood (or
probability) of finding a particle at a particular position in space at
a given time.

The total probability of finding the electron is 1. Forcing this
condition on the wave function is called normalization.
If wavefunction is normalized !!

  * ( y, t )   ( y, t )  dy  something
normalized( y, t )  1

dy for no particular reason, its just 1D dx
something
 ( y, t )
82
The Copenhagen Interpretation

Copenhagen’s interpretation of the wave function (quantum
mechanics in its final and current form) consisted of 3 (to 4)
principles:
1)
The complementarity principle of Bohr
2)
The uncertainty principle of Heisenberg
3)
The statistical interpretation of Born, based on probabilities
determined by the wave function
4)


Bohr’s correspondence principle (for quantum mechanics being
reasonable
Together these concepts form a logical interpretation of the physical
meaning of quantum theory. According to the Copenhagen
interpretation, physics needs to make predictions on the outcomes
of future experiments (measurement) on the basis of the theoretical
analysis of previous experiments (measurements)
Physics is not about “the truth”, questions that cannot be answered
by experiments (measurements) are meaningless to the modern
physicist. Philosophers, priests, gurus, … can be asked these
questions and often answer them. Problem: they tend to disagree …
83
Probability of finding the Particle in a certain region of
space

The probability of observing the
particle between x and x + dx in each
state is
Since there is dx, we need to integrate over
the region we are interested in
All other observable quantities will be
obtained by integrations as well.

Note that E0 = 0 is not a possible
energy level, there is no quantum
number n = 0, so E1 is ground state
also called zero point energy if in a
quantum oscillator

The concept of energy levels, as first
discussed in the Bohr model, has
surfaced in a natural way by using
matter waves.
We analyze the same model in the next chapter with operators on wave
functions and expectation value integrals (that tell us all there is to know)
84
Particle in an infinitely deep Box, no potential
energy to be considered




A particle of mass m is trapped in a one-dimensional box of width L.
The particle is treated as a standing wave. It persist to exist just like a standing
wave.
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 of the particle is
.

The possible wavelengths are quantized which yields the energy:

The possible energies of the particle are quantized.
There is a ground state energy, zero point energy, particles that are confined
can never stand still, always move, no way to utilize this energy for mankind
85
L

2
I ( , d ,  )  I (a, d ,  )  I max
a widths of slits,
a < d ≈ λ << L

a  sin 
2 

sin
( )
2 
2 
cos (  d  sin )  
2
 ( ) 2 
2


 Path difference (rad)
86
87
88
89
90
time dependent Helmholtz
due to the
uncertainty principle,
we can only make
91
statistical inferences
Given the wave particle duality, we need a new way of
thinking.
The whole physical situation is described by a wave function
(which is complex for a traveling matter wave).
That wave function accounts for the physical boundary
conditions, which encode the nature of the physical problem.
To check if the wave function we came up with makes physical
sense, we put it to the “Schrödinger equation test”. (It’s a test
if our wave function obeys the conservation of total energy
(while ignoring rest energy and with that special relativity – if
we need to include that, i.e. v > 0.01 c, we need to make the
Dirac equation test)
If our wave function is OK, we can calculate anything we are
allowed to know about the quantum mechanical system from it. 92