PY313 Final Exam

discuss the relationship between mass and energy

(derive from KE) get: E = gamma * mc^2 = K + E0 -particles can contain very large amounts of energy in their mass

describe the Davisson-Germer experiment and discuss the diffraction of electrons by solids

(figure) -electrons are produced by hot filament, accelerated, and focused onto the target (nickel crystal in this case) -electrons are scattered at an angle (phi) into a movable detector -the distribution of electrons is measured as a function of angle phi Results: -electrons were being diffracted like x-rays (applied Bragg's Law -- 2dsin(theta) = n *lambda) lambda = Dsin(phi) / n

describe the scattering of x-rays from crystals and use Bragg's Law

-1912: x-rays are a form of electromagnetic radiation and must behave like a wave -the x-ray wavelength was predicted to be shorter than visible light (~ between 10 ^-10 and 10 ^-11 m) -the distance between atoms in crystals ~ 10 ^-10 m (the diffraction 'grating' should be spaced approx. in the same order of the wavelength) -so, x-rays should scatter from the atoms of crystals (3-D gratings with structure uniformly arranged) n(lambda) = 2dsin(theta) n = integer

describe the J.J. Thomson measurement of e/m, and derive the relevant equations

-Rays from the cathode are attracted to positive potential on A anode, go through B, and travel in a line to strike fluorescent screen that can be visually detected by a flash of light. A voltage across detection plates sets up an electric field that deflects charged particles q/m = (tan(theta) E)/(B^2 l)

derive the radial, azimuthal, and angular equations from the Schrödinger equation given in spherical coordinates

-TISE in spherical coordinates given -V = V(r) -psi(r, theta, phi) = R(r) f(theta) g(phi) -use R f g in the various derivatives of psi and substitute -multiply both sides by (r^2 sin^theta)/(Rfg) -put functions of r and theta on one side, phi on the other - set each side equal to constant -m(l)^2 -get Azimuthal equation: g(phi) = e^(i*m(l)*phi) using phi side -using other side: multiply both sides by 1/(sin^2theta) and rearrange -put function of r on one side, function of theta on other -set equal to constant l(l+1) -using r side: multiply both sides by R/r^2 and set equal to 0 -get radial equation: -use theta side: multiply both sides by f(theta) and set equal to 0 -get angular equation:

explain the thermoelectric effect, thermoelectric power, Seebeck effect, Thompson effect, Peltier effect, and the operation of a thermocouple

-Thermoelectric effect (aka Seebeck Effect): if have a thermal gradient, get an induced electric field E = Q dT/dx, Q = thermoelectric power (constant) -thompson effect: heat is generated at rate I^2*R in normal conductor and temp gradient across conductor causes additional heat to generate -totally reversible: if I point towards higher temp, heat is generated; if I points toward lower temp, heat is absorbed from surroundings -Peltier Effect: occurs when heat is generated at a junction between two conductors as current passes through the junction -thermocouple: operation made possible by difference of thermoelectric power for two different conductors -produces a temp-dependent voltage that can be used to measure temp.

analyze quantum mechanical tunneling and barrier penetration

-V(x) = 0 in regions I and III -V(x) = V0 in region II -psi(I) = Ae^(-ik(I)x) + Be^(-k(I)x) - incident wave + reflected wave -psi(III) = Fe^(ik(II)x) - transmitted wave -psi(II) = Ce^(ik(II)x) + De^(-ik(II)x) -k(I) = k(II) = (2me)^1/2 * (1/h) -k(II) = (2m(E-V0))^1/2 * (1/h) = iK -Reflection coefficient R = reflected^2/incident^2 = B^2/A^2 Transmission coefficient T = transmitted^2/incident^2 = F^2/A^2 -R + T = 1

explain the structural properties of crystals, and know how to estimate the Madelung constant

-crystals form the same way as molecules -V(att) = - a e^2 / 4pi*e0r r = nearest neighbor distance a = madelung constant, a function of the symmetry of the particular crystal ex using NaCl

discuss the concepts of holes, donor levels, acceptor levels, and be able to distinguish n-type and p-type semi-conductors

-e- and h+ are the two types of charge carriers (diagrams) n-type: donor levels more e- than h+ p-type: acceptor levels more h+ than e- distinguish using Hall Effect

describe the criteria for a laser, explain in detail the operation of three and four level lasers, and explain in detail the operation of a He-Ne gas laser

-emitted photons in lasers must have coherent radiation (same energy, direction, and phase) 3-level system: 1. pump e- from E1 to E3 2. choose atom such that prob. of transition from E3 to E2 is HIGH & prob. of transition from E2 to E1 is LOW 3. e- transitions from 3 to 2 4. e- is in level 2 for long time 5. send in trigger photon * have population inversion between level 2 and 1 by pumping e- to level 3 and having it decay quickly to level 2 * fails due to self absorption: tasing photons encounter an atom that has spontaneously decayed to GS; this atom then absorbs a photon removing it from the beam 4-level system: 1. pump e- from E1 to E4 2. e- falls to E3 quickly 3. e- waits for trigger in E3 4. e- makes transition emitting hf = E3 - E2 5. e- falls quickly to GS NO self-absorption - trigger between 2 excited states, not GS He-Ne Laser: lasing transition from 5s to 3p excited He is metastable - 2s to 1s transition is forbidden

discuss the bonding of atoms to form molecules, and justify the potential energy expression used to describe bonding

-for a molecule to form, there must be a long range attractive force (potential) -for a molecule to be stable, have to have short range repulsive force F = -dV/dr F = q1q2/4*pi*e0*r^2 F is -VE dV/dr is +VE in general, V(r) = (A / r^n) - (B / r^m)

discuss wave-particle duality

-matter can behave like a wave and a particle -wave: can be diffracted -- constructive and destructive interference -energy is localized in a particle but continuous in a wave * It is NOT possible to describe physical observables simultaneously in terms of both particles and waves

describe in detail the quantum theory of electrical conduction, derive expressions for the density of states, the Fermi energy, the mean energy, and be able to show how the quantum theory succeeds where the classical theory fails

-not all electrons can accept thermal energy -- e- can only go to unoccupied states -n(E) = g(E) FDE(E) -Nr = 2 * 1/8 * volume -solve for EF -solve for g(E) = dNr/dE -solve for mean electron energy by (1/N) * integral 0 to EF of Eg(E) dE -get mean energy = 3/5 EF Breakdowns of classical theory 1. conductivity - replace mean v by vF 2. proportionalities 3. only fraction of electrons that can absorb thermal energy: alpha*kT / EF

describe in detail the Stern-Gerlach experiment, and explain the introduction of spin angular momentum

-send bean of ground state H atoms into an uneven B field where S is stronger than N -only explanation for only 2 points on the detector (not including at z=0), is that there's an intrinsic magnetic moment for the electron, in addition to the orbital magnetic moment -spin angular momentum s -uS = -2uB / h *s -ul = -uB / h * L

discuss classical wave motion and be able to define wavelength, period, wave number, and angular frequency

-sine/cosine curve: Asin((2*pi/lambda)(x-vt)) instantaneous displacement -wavelength (lambda): the distance between points in the wave with the same phase -period (T): the time required for a wave to travel a distance of one wavelength lambda = vT -wave number (k): k = (2*pi) / lambda -angular frequency (w): w = (2*pi) / T

explain the thermal expansion of solids

-solids expand when heated -equilibrium separation of atoms in crystals must get bigger ex: get V(x) = ax^2 - bx^3 <x> = 3bkT / 4a^2

discuss the failures of the Bohr model

1. It could only successfully be applied to single-electron atoms (H, He+, Li++) 2. It was not able to account for the various intensities of the fine structure of the spectral lines 3. discovery of doublets: pairing of lines very close together (improved technology)

state and understand the boundary conditions for wave functions

1. Psi(x,t) must be finite everywhere (if was infinity anywhere, probability would also be infinite - NOT PHYSICAL) 2. Psi must be single values (or else, you'd get 2 probabilities for 1 value of x) 3. Psi must approach zero at x= +/-infinity (has to or can't normalize) 4. Psi and dPsi/dx must be continuous

state Einstein's postulates of special relativity and discuss their consequences

1. principle of relativity: -laws of physics are the same in all internal frames -no way to detect absolute motion -no preferred frame 2. constant c of speed of light in a vacuum consequences: 1. the distance between 2 points and the time interval between the 2 events depend of the frame which they are measured (NO absolute time or length) 2. events that are simultaneous in one frame will not be in another

analyze in detail the quantum wave behavior of a particle in three dimensional infinite potential wells

3-D: -V(x) = infinity everywhere outside box -V(x) = 0 inside box (0<x<L1, 0<y<L2, 0<z<L3) -use V(x) = 0 in TISE -get psi(x,y,z) = Asin(n1*pi*x/L1)sin(n2*pi*y/L2)sin(n3*pi*z/L3) -get E = (pi^2*h^2)/(2m) * (n1^2/L1 + n2^2/L2 + n3^2/L3)

compare the full quantum mechanical (Schrödinger) description of the hydrogen atom and the Bohr model of the hydrogen atom

BOHR: 1. semi-classical, classical in orbit, still a particle 2. 1 quantum number, L=nh E is proportional to n^2 3. only works for atoms with a single e- SCHRODINGER: 1. quantum, non-classical, describes particle as a wave function 2. 4 quantum numbers: m, l, ml, ms E is proportional to -1/n^2 3. no restriction on the number of electrons IN COMMON: energy is quantized L is quantized -use radial equation and assume l=0, ml=0 -V(r) = -e^2/(4*pi*e0*r) -substitute and rearrange -R(r) = Ae^(-r/a) -rearrange and set equal to 0, both quantities in brackets must be 0 -solve for a in one bracket --- get Bohr radius!! -solve for E in other bracket -- get Bohr value for ground state

describe in detail the classical theory of electrical conduction, and identify areas where this theory fails

Classical theoryL 1. e- exist as a gas of free particles 2. the metal itself is a lattice of positive ions 3. electrons have random speed 4. no net flow of e-, unless potential in applied 5. when E field is applied, e- move but there is resistance called draft velocity vd derive conductivity Failures: 1. n can be measured, t can be estimated -- value of conductivity is 1/3 of measured value 2. temperature dependence of conductivity predicted to be proportional to (T)^-1/2 but its measured proportional to (T)-1/2 3. Cv is approx 3R but we expected Cv = 4.5 R

show the Newtonian expression for KE is a limiting approx. of the relativistic expression

K = mc^2 (gamma - 1) expand neglect terms u^4 / c^4 or higher since c^4 >> u^4 get: K = 1/2 mu^2

identify Maxwell-Boltzman and Fermi-Dirac energy distribution functions, and understand the circumstance where they apply

Maxwell-Botlzman F(E) classical theory, identical parts are distinguishable Fermi-Dirac F(FD) quantum theory, identical parts are indistinguishable

discuss the relativistic expressions for momentum, energy, and KE

Momentum: P = mu * gamma Force: F = dP/dt F = d/dt (mu * gamma) KE: (derivation) W12 = KE2 - KE1, KE1 = 0, initially at rest ultimately get K = mc^2 (gamma - 1)

discuss in detail the origin of x-rays and the Compton effect

Röntgen (1897) discovered x-rays accidentally -bremmstrahlung: process by which protons are emitted by an electron slowing down E(f) = E(i) - hf -electrons are produced by thermionic emission (hot filament) -electrons are accelerated by huge potential difference and impinge on metal anode -KE lost by heating anode -x-rays are produced -each element used in target displayed different characteristic x-ray wavelengths, but the same minimum wavelength for a fixed voltage Duane-Hunt Rule min. wavelength = (hc)/(eVo) Compton effect -sends x-rays into solid -most emerge at same wavelength, same direction -some emerge at longer wavelengths (lower energy) and at different directions -Compton postulated that photon had momentum p = E/c = hf/c = h/lambda change in wavelength = (h (1-cos(theta)) / mc

discuss characteristic x-ray spectra and the excitation of atoms by electrons (Franck-Hertz experiment)

characteristic x-ray: -photons emitted from a solid at quantized energies, specific to the element -occurs when an external electron hits on of the closest electrons to the nucleus of an atom, ejects it, external electron continues on, and there is an empty level -another electron in a higher level on the atom falls down, leaving an ionized atom Franck-Hertz Experiment: -run a current of electrons through a gas of heavy atoms and measure current as the voltage is varied (with retarding voltage**) -quantized atomic energy levels (graph) Results: -for mercury (Hg) atoms: electron at 4.9 eV is able to excite the least highly bound electron of Hg into the first excited state -at 9.8 eV, the electron has enough energy to excited two Hg atoms in inelastic collisions PROVED: quantization of atomic electron energy levels -- the bombarding of electron's KE can change only by certain discrete amounts determined by the atomic energy levels on the Hg atom

explain the thermal conductivity of solids, derive the Weider-Franz Law, and derive a classical and Quantum expression for the Lorenz number

dQ/dt = -KA dT/dx K = (1/2) nvlk Lorentz number = 4k^2 / pi* e^2 K/o = LT corrected value = K/o = (pi^2 k^2 / 3 e^2) T do derivations

define the density of states and state both the definition and the physical significance of the Fermi energy, Fermi temperature, and Fermi velocity

density of states: g(E) number of states available per unit energy Fermi energy: EF the energy of the highest occupied state in the system at T = 0 Fermi temperature: Tf the temperature at which thermal effects are comparable to quantum effects Fermi velocity: Vf the velocity that corresponds to the Fermi energy of a particle

understand the significance of the Schrödinger equation

describes the wave function of a quantum-mechanical system; it cannot be derived, and its validity comes from verification and ability to predict. The equation can be used to find the allowed energy levels of quantum-mechanical systems. The TDSE describes how the wave function of a particle evolves over time while the TISE describes the allowed energies of the particle

describe the Millikan Oil Drop measurement of e, and derive the relevant equations

drops of oil fall to the bottom, some will acquire a chard due to friction (Ff = -bv) q = (mgd)/v m = (4/3) pi r^3 p

discuss the shell structure of atoms and describe the structure of the periodic table of the elements

electron shells: label/letter codes for different principle quantum numbers/binding energies n = 1 2 3 4 = K L M N sub shells: descriptions of nl -- add superscript H: (1, 0, 0, +/- 0.5) He: (1 0 0 +0.5) and (1 0 0 -.5) Li: (1 0 0 +0.5) and (1 0 0 -.5) and (2 0 0 +/- 0.5) -- [1s2 2s1- Be: [1s2 2s2] Periodic Groups: Inert Gases (He Ne Ar Kr Xe): full outer shell Alkali Metals (Li Na K Rb Cs): 1e- in outer shell Alkaline Earths (Be Mg Ca Sr): 2e- in outer shell Halogens (F Cl Br I): outer shell filled but missing 1e-

discuss the band theory of solids, use it to explain the differences between electrical conductors, insulators, and semiconductors

energy gap: range of E which e- cannot have -when we have multiple atoms close together, discrete energy levels broaden into energy bands (show diagrams for conductors, insulators, and semiconductors)

describe the relationship between these three quantum numbers and the quantization of energy and angular momentum in the hydrogen atom

energy: En = -E0/n^2 angular momentum: L^2 = Lx^2 + Ly^2 + Lz^2 let l = 2, m(l) = -2 -1 0 1 2 L = (6)^1/2 * h Lz = 2h h 0 -h -2h

discuss ionic and covalent bonds

ionic: an e- is transferred from one atom to another & electrostatic BOND formed (Ex: NaCl) covalent: atom one are NOT easily ionized -they share their outer electrons most large molecules have this bond -all diatomic molecules of identical atoms (H2, O2, N2)

discuss the internal rotational and vibrational energy of molecules, and discuss molecular spectroscopy

molecular spectroscopy: study of internal bonding in molecules - examine the emission and absorption of radiation by molecules Consider a diatomic molecule: internal energy of molecule has 3 components: 1. e- can rearrange (small effect) 2. molecule can rotate 3. atoms in molecule can vibrate E = E(rot) + E(vib) + E(el) can ignore E(el) E(rot) = h^2 l(l+1) / 2I E(vib) = (n + 0.5) hw

describe the origins of the 3 quantum numbers n, l, m(l), and know the possible values of these numbers

n principle quantum number: -results from the solution of the radial wave equation R(r) -En = -E0 / n^2 -n = 0, 1, 2, ... l orbital angular momentul quantum number: -associated with the R(r) and f(theta) parts of the wave equation -L = (l (l+1))^1/2 * h -l = 0, 1, 2, 3, 4, 5, ... n-1 = s p d f g h m(l) magnetic quantum number -solution for g(phi) specifies ml is an integer and related to the z-component of the angular momentum -Lz - m(l) * h

understand and apply the Lorenz Transformations

on eq. sheet

explain in detail the operation of p-n junction diodes, zener diodes, light emitting diodes, and photovoltaic cells

p-n junction diode: -focus on majority carrier and first focus on e- in n-type -as soon as junction is formed, the e- in n-side CB diffuse to p-type -get diffusion current aka recombination current -e- flow from n to p, leaving an excess +VE charge on n-side -likewise, h+ flow from p to n, leaving excess -VE on p-side -flow of majority carriers induces electric field at junction that acts against I(R) -examine minority carrier: get a second current due to minority carriers called thermal current I(T) I(R) + I(T) = 0 reverse-bias: E(ext) is parallel to E(int) forward-bias: E(ext) points opposite of E(int) reverse-bias: I = I0(e^eV kT -1) (derive) solar cells: solid state population inversion between e- on n-side and h+ on p-side zener diodes: made to operate under reverse-bias once a sufficiently high voltage has been reached LED: whenever an e- transitions from CB to VB, photon is emitted

define proper time and proper length, and be able to discuss their relation to time and length intervals measured in other reference frames

proper time: the time difference between 2 events occurring at the same position in a system as measured by a clock at rest in the system T0 Time dilation: train example derive: gamma = 1 / (1 - v^2/v^2)^-1/2 dT = gamma * dT' dT > dT' always where dT is the time interval between events measured by stationary frame, dT' moving frame proper length: length measured when object is at rest length contraction: spaceship ex: L = L0 / gamma -mover observes shorter length

discuss the concept of simultaneity, and the twin paradox

simultaneity: the relation between 2 events assumed to be happening at the same time in a given frame of reference - simultaneity is not absolute (Einstein) Twin paradox: -Mary and Frank are twins -Mary gets on spaceship (8ly from Earth and travels close to 0.9c) at age 30 -Frank remains on Earth - Mary is younger when she returns According to Frank: -mary's time to star: 10 years there and to return (8ly / 0.8 c) = 10 y -frank will be 50 (30+10+10) when Mary returns According to Mary: (slower clock) -time to star: 6 years there and to return 10 (1-0.8^2 y)^1/2 = 6 y -Mary will return at 42 with respect to frank's rest clock

state the Heisenberg Uncertainty Principle, show how it can arise by scattering a single photon from a stationary electron, and discuss its implications

statement: if a measurement of position is made with a precision (delta x) and a simultaneous measurement of momentum is made with precision (delta p), then the product of the two uncertainties can never be smaller than h(bar) / 2 (delta x)(delta p) >- h(bar) / 2 scattering of single photon from stationary electron: (figure) alpha: scattering angle of photon theta: max angle -if alpha > theta, then the photon misses the lens and is NOT detected -from conservation of momentum, the momentum of the electron in the x-direction can vary from -hsin(theta) / lambda to +hsin(theta) / lambda --uncertainty in electron momentum is (delta p) = 2hsin(theta) / lambda from optics, the smaller distance between two points that we can distinguish is by (delta x) = lambda / (2sin(theta)) (given) derive by substituting (delta x) and (delta p) to get = h >- h(bar) / 2

discuss the concepts of absorption, stimulated emission, and spontaneous emission, and derive relationships between the Einstein A and B coefficients

stimulated emission: triggered spontaneous: random in time absorption: opposite of emission U(f) = energy density per unit freq. in the incident beam [J/m^3/Hz] B12 * U(f) = prob. of absorption per unit time, per atom B12 = Einstein coefficient of absorption B21 * U(f) = prob. of stimulated emission B21 = Einstein coefficient of stimulated emission *spontaneous emission does not dement on U(f) A21 = prob. of spontaneous emission N1 = # atoms in state E1 N2 = # atoms in state E2 Use Maxwell-Boltzman stats N1 / N2 = e^ hf / kT (derive) Assume U(f) >> 0, B12 = B21 U(f) = A21 / Be^hf/kT - B get A21 / B = 8 * pi * hf^3 / c^3 B is proportional to A21

state the physical significance of the Maxwell velocity, speed, and energy distributions

velocity f(v)d^3v : prob of finding a particle with velocity between v and v+d^3v speed F(v) dv: prob of finding a particle with speed between v and v+dv energy F(E): prob of finding a particle with energy between E and E+dE

describe the concept of the wave function and define normalization and probability density

wave function: a mathematical function that describes matter; it contains all the information that can be known about a particle: energy, momentum, position... normalization: scaling of the wave functions so that all the probabilities add to 1 probability density: (psi squared) probability per unit volume of finding the particle at time t, in a small volume, around x,y,z

state the De Broglie hypothesis

wave-particle duality applies to matter lambda = h/p (momentum)