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Relativity Postulates of Einstein’s theory. There is no preferred frame of reference; the laws of physics are the same in all inertial frames. The speed of light c is the same in all frames. From these Einstein deduced special theory of relativity, which says that space and time are intertwined. If an observer with primed coordinates (x’,y’,z’,t’) moves with speed v relative to an observer with coordinates (x,y,z,t), these coordinates are related by the Lorentz transformation. The same transformation relates the energy-momentum quantities p,E in the two frames. Lorentz contraction. It’s not always necessary to transform all the coordinates. If two observers are measuring two events at the same time, then the effect of relativity reduces to a contraction of the length an object to the observer who sees it moving. The length observed is , where

.

Time dilation. If two observers measure at the same point in space, the effect of relativity simplifies to a difference in the length of the tick of the clocks carried by the two observers. “Moving clocks run slower”, meaning .

Relativistic energy and momentum. Total energy of moving particle is 2

uE mcγ=

where uγ is the relativistic factor γ associated with the particle’s speed u in the lab frame. Since 1uγ = when u = 0, this includes the energy associated with the particle’s mass, the “rest energy” 2mc . In general, I define 2E mc K= + where K is the kinetic energy 2( 1)mcγ − . The momentum of the particle is p mvγ= , and with these definitions one can show that energy and momentum are conserved just as in Newtonian theory. Manipulation of the equations gives another useful equation,

2 2 2( ) ( )E pc mc= + .

For massless particles, E pc= .

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Quantization Several experiments in the early 20th century showed that waves had unexpected properties. Blackbody radiation. The Maxwell theory of light as electromagnetic waves was well established by 1900, so physicists could calculate the spectrum of radiation emitted from a glowing body at a finite temperature T with some confidence. The answer disagreed horribly with experiment, and in particular diverged at high frequencies, the so-called ultraviolet catastrophe. Max Planck found he could fit the experiments using a “guess”:

for the intensity emitted in a given frequency range. He later derived this expression from the statistical physics of gases, but he had to assume that light was emitted by oscillators that had energies that were multiples of E hf= . (This is the beginning of the photon theory of light.) The Planck spectrum peaks at a frequency that is inversely proportional to the temperature, and yields the empirical Wien displacement law , as well as the Stefan-Boltzman law 4I Tσ= , where I is the total intensity and is Stefan’s constant. Photoelectric effect. Another challenge to classical physics of the early 20th century was the photoelectric effect. Classically one expects the emission of electrons bound in a metal to be proportional to the intensity of the light shone on it. What was observed instead was that no electrons were emitted when the light frequency was too low. Only above this frequency increasing the light intensity increased the photoelectron current. Einstein proposed that this could be understood using Planck’s hypothesis, since a photon would only kick out a photoelectron if it had enough energy hf to overcome the metal’s work function (binding energy) φ. The maximum kinetic energy of outgoing electrons is therefore given by .

Compton effect. A photon scatters off a free electron at rest and moves an angle θ relative to its original direction. The wavelength shift of the scattered photon is given by the Compton formula

( )1 cose

hm c

λ λ λ θ′∆ = − = −

The quantity is called the Compton wavelength.

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Nuclear atom, Rutherford scattering, Bohr model Rutherford scattering: Dependence of scattering angle on KE, impact parameter, Z1, Z2. Bohr Model: Based on planetary atomic model with angular momentum quantization condition, . Understand how it explains radii & energy levels & atomic spectra. Also reduced mass and isotope effect, successes and limitations of the Bohr model. Characteristic spectra: Light emission appears at specific wavelengths or photon energies following

where R is the Rydberg, 13.6 eV, and n1 and n2 are integers, n2 > n1.

Particles as waves Louis de Broglie proposed that particles with mass m could behave like waves, just as waves could behave as particles. He assigned them a wavelength of

/h pλ = ,

with p their momentum and h Planck’s const. Because h is very small, only for particles of atomic or nuclear size will the wave behavior be observable. Davisson-Germer experiment. In 1926 Davisson and Germer showed that electrons could diffract from a crystal just like x-rays, proving the wavelike nature of electrons. Double slit experiments with particles. An even more direct test is a Young’s experimental setup with two slits that the electron beam passes though. An interference pattern like light is observed on the screen at large distance D . To a good approximation, the position of an interference maximum on the screen is /ny n D dλ= , where d is the distance between the slits, and n is an integer. Wave packets. The mathematical description of a particle as a wave is a linear superposition of plane waves

( ) ( ) ikxx dk g k eψ∞

−∞

= ∫

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where ( )g k is a function of the wave number k peaked around some 0k . This gives a function which is confined in both position and momentum. Remember momentum p k= .

Uncertainty principle. Analysis of wave packet solutions to the Schrödinger equation led Heisenberg to his “uncertainty principle”

.

The Uncertainty principle says that the momentum and position of a particle cannot be measured infinitely accurately at the same time as in classical physics. This has immediate consequences for the double-slit experiment. If I try to determine which slit the particle passes through, I confine it in position space (x). Doing so leads to complete uncertainty in its momentum p, or wavelength via the de Broglie formula. Thus because λ is not well-defined, the interference pattern is smeared out.

Schrödinger equation Wavefunction ψ and probability: All information about a particle is in the wavefunction ψ but I cannot measure ψ directly. The probability for a particle for being in a small region of size dV at position x is

P = .

ψ must be normalizable so that the total probability is 1. Operators: Momentum, energy, angular momentum are operators:

, , ,

etc. are differential operators. Expectation values: Expectation value for function or operator is

.

I calculated the expectation values of x, x2, p, p2 for various solutions of the SE.

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Schrödinger equation: The SE is merely the equation for energy conservation for a particle in a potential written in operator form and applied to a wavefunction ψ.

(A second-order differential equation) Solving gives ψ and thus all information about a particle. Stationary states: Time independent potential U(x) use separation of variables & find

,

where satisfies time-independent SE.

Solutions to SE: Important: The requirement of a normalizable wavefunction and smooth boundary conditions gives quantized energies for bound states. Free particle: For

U(x) = 0 ,

where and

.

The solution describes simple wave motion for a particle with momentum px, wavenumber and wavelength (de Broglie wavelength).

Particle in a box (1D), infinite potential walls: Solutions inside box are

=

with energies

(n = 1, 2, 3…). Energy quantization comes from the boundary conditions.

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Particle in a box (1D), finite potential walls: Bound state solutions are sinusoidal within box and exponentially falling outside it in classically forbidden region. Since ψ does not have to go to 0 at the boundaries, the wavelength for each energy level is larger than the corresponding wavelength for the infinite box and thus energy is lower. Barrier transmission: A quantum particle can tunnel through a barrier. The wave function is exponentially falling in the barrier. It emerges with probability set by barrier height and width. But it as the same kinetic energy it approached with Harmonic oscillator (1D): Solutions are of the form

ψ(x) =

where H is a Hermite polynomial. The energy is

with ,

i.e. equally spaced energies. Energy quantization comes from requirement that ψ(x) is normalizable. HO is very important in physics because all potentials with a stable minimum can be expanded as a quadratic in x near the minimum, so first few energies obtained from the harmonic oscillator are reasonable approximations to true energies.

Uncertainty principle: , (precise definition requires these be the standard deviations). One can apply the uncertainty principle to estimate the ground state energy (and size) of a particle in a box, harmonic oscillator, atoms, etc. Particle in a box (3D) with infinite potential walls: Solve using separation of variables.

with .

This leads to energy degeneracy. One ground state (E = 3E0). (111). threefold degenerate 1st excited state (E = 6E0), (112, 121, 211).

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Hydrogen atom Radial potential U(r): The SE for any radial potential can be separated into (1) radial equation depending on U(r), energy E and angular momentum quantum number l and (2) an angular part independent of U(r). Angular solution: The angular wavefunctions for any radial potential are spherical harmonics

,

a product of a polynomial in and a phase factor in φ. It is normalized as

.

The solution depends only on l and ml, where (2l + 1 wavefunctions). Increasing l introduces additional nodes in the wavefunction. Radial part: The solution

,

which depends on 3 quantum numbers: n, l, ml, satisfying , and .

The energies are

(independent of l), same as Bohr model.

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The radial wavefunctions are normalized as

and have the form

where K is a polynomial and a0 = Bohr radius. Energies and radii change if electron mass replaced by heavier mass or nucleus charge is Ze. Quantization: Energy and angular momentum quantization (n, l) come from the requirement that the wavefunctions be normalizable. ml quantization comes from periodicity in φ,. Angular momentum quantization: The square of the total angular momentum is

L2 = ,

where l ≥ 0. The z component of angular momentum is quantized with values .

(2l + 1 values).

Magnetic moment and energy splitting: Orbital magnetic moment of an electron in a hydrogen atom is

and is quantized as

.

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The energy of particle in a magnetic field B is

,

thus an orbital with angular momentum specified by l is split into 2l + 1 states each separated by energy .

Intrinsic spin: For spin ½ particles magnetic moment ,

where S = ½ and gs ~ 2 for electrons, protons, neutrons. States are split into two (spin up/down) by . Each spin shift is the same as the split for l = 1 atomic transitions, but there are only two states, not three. Selection rules for atomic transitions: These are

1. ∆n = anything 2. ∆l = ±1 3. ∆ml = 0, ±1 4. ∆ms = 0

The second and third rules result from the fact that the photon has intrinsic spin 1. You should be able to draw photon transitions from an energy level diagram.

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Atomic Phsics Pauli exclusion principle: This principle explains why electrons fill shells of increasing energy and not just all electrons in the ground state. Two identical fermions (electrons, neutrons, protons…) cannot be in the same quantum state. It requires two-fermion wave functions be antisymmetric under exchange of particle labels, so for particles with quantum numbers {A} and {B}, respectively, in single-particle eigenstates i and j,

{ } { } { } { }( , ) ( ) ( ) ( ) ( )i j j iA B A B A Bψ ψ ψ ψΨ = −

so that if {A} and {B} are identical the wave function vanishes. For bosons ( , )A BΨ looks the same except there is a + sign between the two terms. Filling atomic shells: I fill subshells and shells based on nuclear screening by the inner electrons, which affects the energy of the new electron to be added. Order of filling, within a given n, is s, p, d, f, …. Periodic table:

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Molecular physics Bonds: Ionic, covalent, van der Waals H2: This is an example of a covalent bond. The 1s spatial wave functions can form symmetric or antisymmetric combinations. The symmetric combination builds up electron charge between the protons and the antisymmetric combination reduces it. The symmetric in space combination is the bonding orbital and is lower in energy than the antibonding orbital. Rotational energies: H2 is a symmetric dumbbell. It has rotational freedom about an axis through the middle of the bond. The rotational energies are

with L the angular momentum, I the moment of inertia, and l the quantum number, 0, 1, 2 … The energies are in the meV range. Vibrational energies: The masses and bond in H2 are a harmonic oscillator. The vibrational energies are

Solid State physics Crystal Structures. Atoms arrange themselves most of the time in periodic structures (“crystal lattices”) in a solid, where the atoms are held in place by the bonds to their nearest neighbors. Each atom has 3 degrees of freedom corresponding to translation in 3 directions, and 3 degrees of freedom corresponding to compressing the “spring” between them. Chemical bonding in solids: ionic, covalent, and metallic. In ionic bonds electrons spend most of their time localized on a single atom, whereas in covalent bonds electrons are shared between the two. In metals, some of the “conduction electrons” become delocalized from particular atoms altogether, so that they can conduct electricity. Metals: Fermi-Dirac Distribution. A good model of a metal is a set of free electrons (one from each atom) subject to the Pauli principle. You add 2 electrons to each energy level until you have used up all the electrons, let’s say 1023 of them. The highest filled level is called the Fermi energy,

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typically of order a 2-10 eV, and is typically of order of 20,000-100,000 K. The typical speed of such electrons is vF, the Fermi velocity, much higher than the average thermal velocity expected for classical particles because the Pauli principle forces only two states per energy level. The scale of the Fermi velocity is

vF = c/200.

Conductivity of metals. The conductivity of metals is given by the Drude result 2 /ne mσ τ= , where n is the number density of conduction electrons, τ is the mean time between collisions, and m is the electron mass. While this form is a classical result, a quantum mechanical calculation of τ is required to understand the experimental results on metals. Electrons do not collide with ions in the crystal lattice, but with ionic vibrations, as well as impurities and other electrons. Superconductivity. Superconductors are metals that are perfect conductors of electricity below a critical temperature cT . They also expel magnetic fields from their interior, a phenomenon known as the Meissner effect. The electrons in a superconductor exist as Cooper pairs, a coherent superposition of two elecrons with k, spin up and –k, spin down:

Superconductors have a gap for excitations. To break a Cooper pair requires (at T = 0)

yielding two unpaired electrons. D is then the binding energy per electron.

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Energy bands and semiconductors: When the periodic potential of the atoms in the crystal is included in the Schrödinger equation, the electron has a (periodic in lattice constant a) potential energy V(r) as well as a kinetic energy. The wave functions are still indexed by a momentum, k = p/ħ. Start with a 1-dimensional chain of atoms, lattice constant a. The states that are most affected by the periodic potential are those which interfere constructively on being reflected (twice so k goes to –k and then back to +k). The wavelength is 2a, or k = 2π/λ = ± π/a. I get bonding (lower energy) and antibonding (higher energy) linear combinations of energies for the wave functions with this value of k, as well as nk, where n is an integer. There is a gap in the energy spectrum at these k values. Each quantized energy level Ek can hold two electrons, so if there are 2 (or 4, 6, …) electrons per atom, the band is “full” and the solid is a semiconductor. One, or 3, and it a metal. (In three dimensions and in compounds it is more complicated, but the basic idea holds.) Silicon has 4 electron in its n = 2 level, so it is a semiconductor, with a gap between its full valence band and empty conduction band. Silicon may be doped with atoms from neighboring columns in the periodic table. These act as donors (P) or acceptors (B) and change the conductivity. The donors and acceptors have hydrogen-like levels of the extra electron or acquired hole at low temperatures but at room temperature they are ionized and control the carrier density. If there are acceptors, the charge carriers are holes in the valence band and the crystal is “p-type.” If donors, it is electrons in the conduction band, “n-type.” One may make p-n junctions to make a diode. More complex structures give transistors, integrated circuits, and computer cpus.

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Particle physics Nuclear structure basics. The nucleus is made up of protons and neutrons, each of size ~1 fm, or 10-15 m. Nuclei are charcterisded by A, the atomic mass, and Z, the atomic number. Nuclei with the element (same Z) but different A are called isotopes. They correspond to the same chemical element but have different masses (numbers of neutrons). Each nucleon has a mc2 of about 1 GeV. Nucleons are held together in the nucleus by the attractive nuclear strong force, much stronger than the electromagnetic force between protons that tends to force them apart. Radioactive decay. A nucleus typically decays by emitting an alpha particle (He nucleus), beta particle (electron or positron plus accompanying neutrino), or gamma particle (photon). Fundamental forces. There are 4 fundamental forces in nature: strong, weak, electromagnetic, and gravitational. Electricity and gravity are “infinite-ranged” in the sense that they fall off as 1/r2 away from the source of the force. Strong and weak nuclear forces act on very short length scales of the size of the nucleus or smaller. The characteristic times for each of the forces can be used to identify which force is involved in the reaction. Different kinds of elementary particles. The various elementary particles have been classified according to their mass and a set of intrinsic quantum numbers characteristic of each particle: electric charge, spin, lepton number, baryon number, strangeness, charm, etc. Antiparticles. Each particle has its own antiparticle, with opposite quantum numbers. Deacays and reactions. Elementary particle reactions and decay must obey the known conservation principles. Electric charge, baryon and lepton number, strangeness, charm, and angular momentum must be consistent on both the left and right of a reaction equation. Note that lepton number is conserved within lepton generations. The electron lepton number must be conserved simultaneously with the muon lepton number, etc. Of course energy and momentum must be conserved also. Quark model. In 1964 Gell-Mann proposed the quark model to explain the clustering of observed particles in mass, charge, and strangeness space. The stable common baryons like proton and neutron were made up of u and d quarks. u has charge 2/3 and d is −1/3. Thus p = (uud) and n = (udd). Quarks also come in 3 generations with the charm, strange, top and bottom quarks more massive than the u and d quarks. All known baryons and mesons can be classified in terms of their quark content. Quarks have their own quantum numbers (“flavor” such as strangeness, charm, bottom, top) that must be conserved in strong and electromagnetic reactions. Weak decays can change flavor.

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