1. 1/3 to 2/3 of our modern economy !!!
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
1/3 to 2/3 of our modern economy !!!

2. 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.

3. 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.

5. 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

6. 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

7. The Inverse Relations Step 1. Replace with .
Step 2. Replace “primed” quantities with
“unprimed” and “unprimed” with “primed.”

8. 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)

9. 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

10. 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.

11. 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.

12. NEVER OBSERVED !!!!

13. 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”

14. Length contracted for the moving muon, it’s own life time just 2
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
Figure 1.11 (a) Muons traveling with a speed of 0.99c travel only about 650 m as measured in the muons’ reference frame, where their lifetime is about 2.2 s. (b) The muons travel about 4700 m as measured by an observer on Earth. Because of time dilation, the muons’ lifetime is longer as measured by the Earth observer.

15. Lorentz Transformation Equations
So there is four-dimensional space time !!!

16. 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.
Figure 1.10 (a) A mirror is fixed to a moving vehicle, and a light pulse leaves O’ at rest in the vehicle. (b) Relative to a stationary observer on Earth, the mirror and O’ move with a speed v. Note that the distance the pulse travels measured by the stationary observer on Earth is greater than 2d. (c) The right triangle for calculating the relationship between t and t’.

17. 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.

18. Figure 1. 9 Two lightning bolts strike the ends of a moving boxcar
Figure 1.9 Two lightning bolts strike the ends of a moving boxcar. (a) The events appear to be simultaneous to the stationary observer at O, who is midway between A and B. (b) The events do not appear to be simultaneous to the observer at O’, who claims that the front of the train is struck before the rear.
No simultaneity if not also at the same position, just a consequence of the Lorentz transformaitons

19. 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:

20. 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:
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.
The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

21. 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.

22. 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:

23. 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.

24. Relativistic and Classical Kinetic Energies

25. 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

26. Momentum and Energy
We square this result, multiply by c2, and rearrange the result.
We replace β2 and find

27. 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!

28. 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.

29. A Conducting Wire

30. Principle of Equivalence
The principle of equivalence is an experiment in non-inertial 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.

31. 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%.

32. 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 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.

33. 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.

34. 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.

35. 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.

36. 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?

37. BUT, thank you very much indeed Albert !!! everybody loves this !!!!

38. Dual nature of light (electromagnetic radiation) both/neither wave and/nor particle
“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.”

39. Light according to Maxwell
Figure 3.2 A light or radio wave far from the source according to Maxwell and Hertz.
Fig. 3-2, p. 67

40. 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.
When u(x) is plotted over x, there is only one peak! One universal curve for all wavelengths and T

41. Two fitting parameters and no physical theory behind them !!

42. 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.

43. Rayleigh-Jeans Formula
Lord Rayleigh used the classical theories of electromagnetism and thermodynamics to show that the blackbody spectral distribution should be
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.
k: Boltzmann’s constant eV/K

44. 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 made two modifications to the classical theory:
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 = × 10−34 J·s.
The oscillators can absorb or emit energy in discrete multiples of the fundamental quantum of energy given by
Planck’s radiation law, only one fundamental constant h left that can explain Wien’s and Stephan’s constants …, significant progress

45. Photoelectric effect

46. Experimental Results
Only if the energy threshold to get electrons out of the metal (work function) is exceeded.

47. 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

48. 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.

49. 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.

50. 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

51. Bragg’s law
(a) Schematic diagram of Compton’s apparatus. The wavelength was measured with a rotating crystal spectrometer using graphite (carbon) as the target. The intensity was determined by a movable ionization chamber that generated a current proportional to the x-ray intensity.

52. No way !!!
Just a relativistic collision between a mass less particle and a massive particle.
Figure 3.22 X-ray scattering from an electron: (a) the classical model, (b) the quantum model.

53. 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:

54. 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

55. No problem, Bohr’s complementarily
Wave – particle duality, “wavical”
Taoism” Taijitu (literally "diagram of the supreme ultimate"
Figure 3.16 (a) A classical view of a traveling light wave. (b) Einstein’s photon picture of “a traveling light wave.”
No problem, Bohr’s complementarily

56. Dual nature of quantum mechanical objects both/neither particle and/nor wave
“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.”

57. 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.
In J. J. Thomson’s view, when the atom was heated, the electrons could vibrate about their equilibrium positions, thus producing electromagnetic radiation.
Not quite, electrons repulse each other as much as possible but what is the dough?

58. More experiments, looking at large angles where one would not expect any scattering to show up, BUT …
Figure 4.10 A schematic view of Rutherford’s  scattering apparatus.

59. There is no dough, just lots and lots of empty space and a tiny tiny heavy nucleus where all of the positive charges reside.
Figure 4.11 Scattering of  particles by a dense, positively charged nucleus.

60. 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
Figure 4.21 The classical model of the nuclear atom.
Fig. 4-21, p. 131

61. Angular momentum must be quantized in nature in units of h-bar, from that follows quantization of energy levels ….
Figure 4.22 Diagram representing Bohr’s model of the hydrogen atom.

62. Figure 4.23 The first three Bohr orbits for hydrogen.
Fig. 4-23, p. 133

63. Figure 4. 24 An energy-level diagram for hydrogen
Figure 4.24 An energy-level diagram for hydrogen. In such diagrams the allowed energies are plotted on the vertical axis. Nothing is plotted on the horizontal axis, but the horizontal extent of the diagram is made large enough to show allowed transitions. Note that the quantum numbers are given on the left.
Fig. 4-24, p. 134

64. Figure 4.20 The Balmer series of spectral lines for hydrogen (emission spectrum).
Can all be explained from Bohr’s model as he puts physical meaning to the Rydberg equation.
Fig. 4-20, p. 129

65. Figure 4.25 Bohr’s sketches of electronic orbits.
Bohr’s second paper in There should be shells, idea basically correct, helps explaining basic chemistry

66. 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

67. 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.
When it occurs in a heavy atom, the radiation emitted is an X ray photon.
It has the energy E (X ray) = Eu − Eℓ.
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.

68. L shell to K shell Kα x ray M shell to K shell Kβ x ray
Atomic Number Z, X-ray spectroscopy on the basis of the characteristic X-rays
L shell to K shell Kα x ray
M shell to K shell 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 !!!!

69. 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.

70. 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

71. Figure 4.28 Current as a function of voltage in the Franck–Hertz experiment. To obtain these data, the filament voltage was set at 6.0 V and the tube heated to 185C. (Data taken by Bob Rodick, Utica College, class of 1992)

72. 1/3 to 2/3 of our modern economy !!!
There are also matter waves, not only classical and electromagnetic waves !!! Wave particle duality for matter leads us into quantum mechanics, condensed matter physics ….
1/3 to 2/3 of our modern economy !!!
Figure 4.27 Franck–Hertz apparatus. A drop of pure mercury is sealed into an evacuated tube. The tube is heated to 185°C during measurements to provide a highenough density of mercury to ensure many electron–atom collisions.

73. momentum = h / wavelength
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
Figure 5.2 Standing waves fit to a circular Bohr orbit. In this particular diagram, three wavelengths are fit to the orbit, corresponding to the n = 3 energy state of the Bohr theory.
Fig. 5-2, p. 153

74. Polycrystalline material
5.3: Electron Scattering
Davisson and Germer experimentally observed that low energy electrons were diffracted much like x rays in nickel crystals.
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. 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.
a single quasi crystal

75. TEM
One operation mode is transmission diffraction, there is also electron energy loss spectroscopy and X-ray spectroscopy
Figure 5.11 (a) Schematic drawing of a transmission electron microscope with magnetic lenses. (b) Schematic of a light-projection microscope.

76. SEM
Short wavelength and nearly parallel fine electron beam results in large depth of focus, SEM images “appear” almost three-dimensional
Figure 5.15 The working parts of a scanning electron microscope.

77. One full cycle for envelop wave = 2 pi
Figure 5.18 Beats are formed by the combination of two waves of slightly different frequency traveling in the same direction. (a) The individual waves. (b) The combined wave has an amplitude (broken line) that oscillates in time.
One full cycle for envelop wave = 2 pi

78. Figure 5.19 Superposition of two waves of slightly different wavelengths resulting in primitive wave groups; t has been set equal to zero in Equation 5.14.

79. 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.

80. Modern physics backed up by experiments Mathematical uncertainties
Heisenberg's uncertainties
Figure 5.17 Representing a particle with matter waves: (a) particle of mass m and speed v0; (b) superposition of many matter waves with a spread of wavelengths centered on 0 / h/mv0 correctly represents a particle.

81. 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 …
Figure 5.27 The Heisenberg microscope.

82. 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 !!
dy for no particular reason, its just 1D dx

83. The Copenhagen Interpretation
Copenhagen’s interpretation of the wave function (quantum mechanics in its final and current form) consisted of 3 (to 4) principles:
The complementarity principle of Bohr
The uncertainty principle of Heisenberg
The statistical interpretation of Born, based on probabilities determined by the wave function
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 …

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 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

85. 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)

86.  Path difference (rad)L
 Path difference (rad)
Figure 5.28 Electron diffraction. D is much greater than the individual slit widths and much less than the distance between the slits and the detector.
a widths of slits, a < d ≈ λ << L

87. Figure 5.29 Gradual accumulation of interference fringes after the diffraction of an increasing number of electrons. (From P. G. Medi, G. F. Missiroli, and G. Pozzi, Am. J. of Phys. 44:306, 1976).

88. Figure 5.30 The probability of finding electrons at the screen with either the lower or upper slit closed.

89. Figure 5.31 Accumulated results from the two-slit electron diffraction experiment with each slit closed half the time. For comparison, the results with both slits open are shown in color.

90. 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
Figure 5.33 A thought experiment to determine through which slit the electron passes.

91. time dependent Helmholtz
due to the uncertainty principle, we can only make statistical inferences

92. So the first part of PH 312 is all about how to do these things
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