Download CHAPTER 2: Special Theory of Relativity

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’s 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 inertial reference
frame, then they are also valid in another inertial
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. So the
laws of mechanics are independent on the state
of movement in a straight line at constant
velocity
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 to 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.
As the direction of the ether wind
is unknown, the apparatus has to
be turned around by 90 degrees to
see a shift (starting form many
different initial settings)
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 past, 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 (dilated) due to her moving
in a straight line with a constant high velocity with respect to him.
16
No simultaneity if not also at the same position, just a
consequence of the Lorentz transformations
17
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:
18
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 (special) 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.
19
Relativistic Mechanics
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.
20
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:
21
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.
22
Relativistic and Classical Kinetic Energies
23
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
24
Momentum and Energy
We square this result, multiply by c2, and
rearrange the result.
We replace β2 and find
25
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!
26
Binding Energy, conservation of total energy
The binding energy is the difference between the rest
energy of the individual particles and the rest energy of the
combined bound system. Associated with the potential
energy to keep the system together, we give it a negative
sign as we would need to provide it in order to break the
system up in components (by doing work on it)
Mass-energy conserved, i.e. some mass changed into binding energy in the
formation process
(bound system is lighter)
A couple of eV for chemical reactions.
A couple of MeV for nuclear reactions.
27
A charge moves by us (at rest) and the cations in the wire
B is into the paper because or negative
charge of electrons
We move with that charge (so
that it and the electrons in the
wire are at rest with respect to us)
Description of
the
phenomenon of
repulsion of the
test charge q0
either from the
viewpoint of
magnetism
(Lorentz force)
or electrostatics
(like charges
repel) and
length
contraction of
the distance
between
positive charges
28
Principle of Equivalence

Principle of equivalence (of inertial mass and dynamic
mass): There is no physical 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.
29
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 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 with 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%.
Now time differences due to height differences above the earth of
30 cm can be measures, (Nobel price 2012)

30
Light Retardation
“Shapiro experiment”, 1971




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 of relativity.
c in vacuum is the same
constant in all frame of
references, inertial or
accelerated, no difference, and
with that of the rest of physics
31
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.
32
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?
33
BUT, thank you
very much
indeed Albert !!!
everybody loves
this !!!!
34
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.”
35
Light according to Maxwell
36
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(
x
hf
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
37
Two fitting parameters and no
physical theory behind them !!
38
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.
39
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.
40
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
41
Photoelectric
effect
42
Experimental Results
Only if the energy threshold
to get electrons out of the
metal (work function) is
exceeded.
43
Einstein’s Theory

Einstein suggested that the electromagnetic radiation field is
quantized into particles called (later on) 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
wavelengths it looks more like a wave at short wavelength, high
frequency, high energy it looks more like a particle

The photon travels at the speed of light in a vacuum, and its
wavelength is given by
44
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.
Wave particle duality for light is the much more powerful legacy of the paper
were all of this was reported
45
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.
46
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
47
Bragg’s law
48
No way !!!
Just a relativistic collision
between a mass less particle
and a massive particle.
49
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:
50
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
51
Wave – particle duality, “wavical”
Taoism”
Taijitu (literally
"diagram of
the supreme
ultimate"
No problem, Bohr’s complementarily
52
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.”
53
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.
54
More experiments,
looking at large
angles where one
would not expect any
scattering to show
up, BUT …
55
There is no dough, just
lots and lots of empty
space and a tiny tiny
heavy nucleus where all
of the positive charges
reside.
56
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
57
Fig. 4-21, p. 131
Can all be explained from
Bohr’s model as he puts
physical meaning to the
Rydberg equation.
58
Fig. 4-20, p. 129
Angular
momentum must
be quantized in
nature in units of
h-bar, from that
follows
quantization of
energy levels ….
59
60
Fig. 4-23, p. 133
61
Fig. 4-24, p. 134
The Correspondence Principle
Classical electrodynamics
+
Bohr’s atomic model
with stationary orbits
Determine the nature
of spectral lines
Need a principle to relate the new modern results with classical
ones. Mathematically: h -> 0 or quantum number to infinity
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
62
Bohr’s second paper in
1913. There should be
shells, idea basically
correct, helps explaining
basic chemistry
63
Characteristic X-Ray Spectra and
Moseley’s law (kind of rules only when generalized)

Shells have letter names:
K shell for n = 1
L shell for n = 2

The atom is most stable in its ground state.
after an inner electron has been kicked out by
some process, an electron from higher shells will
fill the inner-shell vacancy at lower energy.


When it occurs in a heavy atom, the radiation emitted is an X ray
photon.
It has the energy E (X ray) = Eu − Eℓ.
64
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 !!!!
65
Moseley’s Results support Bohr’s ideas
for all tested atoms

The x ray photon 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.
66
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. Hg atoms “take in” energy, and radiate it off
again, effect is analogous to spectral lines

67
68
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 !!!
69
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
70
Fig. 5-2, p. 153
5.3: Electron Scattering

Davisson and Germer experimentally observed that low energy electrons were
diffracted much like x rays in nickel crystals. Better to consider this as a case of
Bragg diffraction
Polycrystalline material

George P. Thomson (1892–1975), son of J. J.
Thomson, build a transmission electron
diffraction camera experiments to observe the
diffraction of high energy electrons.
a single
quasi crystal 71
TEM
One operation mode is
transmission diffraction,
there is also electron
energy loss spectroscopy
and X-ray spectroscopy
72
SEM
Short wavelength
and nearly parallel
fine electron beam
results in large
depth of focus, SEM
images “appear”
almost threedimensional
73
One full cycle for envelop wave = 2 pi
74
75
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.
76
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   
dE    d
E    
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
77
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 …
78
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 )
79
L

2
I ( , d ,  )  I (a, d ,  )  I max
a widths of slits,
a < d ≈ λ << L

 Path difference (rad)
a  sin 
2 

sin
( )
2 
2 
cos (  d  sin )  
2
 ( ) 2 
2


80
81
82
83
One cannot determine through which
slit a single electron went and observe
an interference pattern at the same
time: Feynman’s version of the
uncertainty priniple
84
http://www.feynmanlectures.caltech.edu/I_37.html
http://www.informationphilosopher.com/solutions/scientists/feynman/probability
_and_uncertainty.html



Feynman’s version of the uncertaintly principle: It
is impossible to
observe through which slit the quantum mechanical
particle went in a double slit experiment and not to
destroy its contribution to the double slit interference
pattern at the same time.
“The uncertainty principle “protects” quantum mechanics. Heisenberg
recognized that if it were possible to measure the momentum and the position
simultaneously with a greater accuracy, the quantum mechanics would collapse.
So he proposed that it must be impossible. Then people sat down and tried to
figure out ways of doing it, and nobody could figure out a way to measure the
position and the momentum of anything—a screen, an electron, a billiard ball,
anything—with any greater accuracy. Quantum mechanics maintains its perilous
but accurate existence.”
Last lecture in PH 312, experimentally observed violations of so called Bell
inequalities demonstrate that quantum mechanics is complete and Heisenberg’s
uncertainty principle is here to stay
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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 …
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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 of finding the particle to vanish at the walls, we must
have an integral number of half wavelengths in the box.

The energy of the particle is

Putting these two relations together yields quantized energy:

Lowest energy state not zero.
.
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
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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)
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time dependent Helmholtz
due to the
uncertainty principle,
we can only make
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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).
If the particle is not free, it’s wave function need to account 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 (given the uncertainty principle) about the
quantum mechanical system from it.
So the first part of PH 312 is all about how to do these things
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