GREview: Recap of Four Past GRE Physics Subject Tests(GR9277, GR8677, GR9677, and GR0177)

Victor V. Albert(Dated: August 26, 2009)

This isn’t about physics, its about getting the right answer.

I CLASSICAL MECHANICS

MATHEMATICAL METHODS & PREREQUISITES

Taylor expansions (around 0 with x << 1). First order is most common in physics:

Order 0th 1st 2nd 3rd

sinx 0 x x x − x3/6cosx 1 1 1 − x2/2 1 − x2/2ex 1 1 + x 1 + x + x2/2 1 + x + x2/2 + x3/6

Taking limits with Taylor expansions: limx→∞

e1/x

x2(e1/x − 1)2≈ e1/x

x2(1/x)2= 1

The formula for a counterclockwise rotational transformation of a coordinate system byangle θ:

( cos θ sin θ− sin θ cos θ

)

Calculating eigenvectors and eigenvalues of a 2 × 2 matrix.

Normalizing functions.

A form of the wave equation, with amplitude A, wavenumber k, radial frequency ω, andposition x: A sin(kx ± ωt). If the sign is a plus (minus), then the wave is in thepositive (negative) x-direction. The sign in front of the kx term does not matter.Other equations to remember (with T = period, λ =wavelength, v = speed, and f =frequency): fT = 1, ω = 2πf , k = 2π/λ, and λf = ω/k = v.

Vector identities for any fieldÐF :

Ð∇ ⋅ (Ð∇ ×ÐF ) = 0

I. CLASSICAL MECHANICS

When in doubt, try units! For some seemingly difficult problems, simple dimensionalanalysis will work: working out the units of the answer in terms of the quantitiesgiven may often significantly narrow down the possible choices or even single out thecorrect answer. Common units include: [Force] = kg ⋅m/s2, [Acceleration] = m/s2,[Energy] = [Force] ⋅m, [Density] = kg/m3.

In the kinematic approximation, when a projectile is launched at an angle 0 < θ < π fromthe earth, its horizontal velocity vx is constant while its vertical velocity vy decreaseslinearly due to gravitational acceleration.

Some useful relations involving force (in the 1D case) are:

F = dpdt

= −dUdx

=md2x

dt2(1)

As stated above, force is the time derivative of momentum p, the negative spatialderivative of potential energy U , and, as Newton’s Second Law states, is also propor-tional to acceleration. In the first relation, multiplying both sides by dt yields theequation for dp⇒∆p, or impulse: ∆p = ∫ Fdt.

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I CLASSICAL MECHANICS

In many-body force diagrams, each object in the system will have its own set of forcesacting on it and its own acceleration. Thus, for each object in the system, and foreach dimension of movement of the object, ∑F = ma. At times, one will be asked tobalance out (equate) several forces and solve for a variable in one or both of them.Some important types of forces are as follows:

• Weight of a mass mg;

• Normal force η: when an object is being pushed by a force, it exerts an equaland opposite force (usually denoted as η) on whatever is pushing it, accordingto Newton’s 3rd Law. Also, if a mass is resting against any surface, that surfacepushes against the mass with the normal force, which cancels out any force push-ing perpendicularly on that surface. One force which always has a componentpushing on the surface is weight, so often η =mg;

• Centripetal force Fcent =mv2/r, used in the case of a body in circular orbit;

• Static frictional force, Ffric = µη, with normal force η and coefficient of staticfriction µ. This can be used in the case of someone standing on a rotatingsurface;

• Hooke’s law, i.e., force on a spring with equilibrium position x0 and spring con-stant k: Fspr = −k(x − x0). This force, when balanced with the usual F = mx(and x0 set to 0), gives the differential equation for a harmonic oscillator:

mx + kx = 0 with frequency ω =√

k

m(2)

Using the frequency-period relation fτ = 1 and f = 2πω, the period and frequencycan also be worked out from the above. When identical springs are lined up inparallel (series), the overall k constant is doubled (halved).

• Air resistance, Fair = −kv, with k being a proportionality constant. This forceslows down any object flying through air in any direction. For example, whenan object is tossed up into the air with velocity v0, its velocity going down willalways be less than v0. Also, the acceleration of the object will only be equal tog when v = 0, i.e., at the top of its flight.

• Tension of a string T . A string can be thought of as a very strong spring, andwhen something is hanging on and stretching such a spring, the spring tries itsbest to condense back to its equilibrium shape. Thus, like the spring force, Twill always pull something toward itself. When a string is cut, as often happens,letting T →ma, i.e., substituting the tension with motion, may solve the problemmore quickly.

• Gravitational force between two masses, Fgrav = GMm/r2, where M and m aremasses and G is the gravitational constant. This can be used whenever there isa satellite in circular orbit around a planet.

• Buoyancy force: Fbouy = −ρgV . According to Archimedes, “any object, wholly orpartly immersed in a fluid, is buoyed up by a force equal to the weight of thefluid (usually air or water) displaced by the object.” This is a glorified version ofweight, with ρ and V being the density and volume of the displaced fluid.

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I CLASSICAL MECHANICS

Just as tangential velocity always points tangent to the motion of a rotating body, cen-tripetal acceleration (acent) always points into the center of curvature. However, if thebody also has tangential acceleration atan, i.e., its tangential velocity is not constant,then the total acceleration (vector added centripetal and tangential) does not neces-sarily point to the center. An example would be a pendulum under gravity at thepoint of highest potential energy (maximum height); the pendulums total acceleration(centripetal plus gravitational) vector is in the direction of the point of highest kineticenergy (minimum height).

When dealing with velocity, several equations are important, including: v = ωr (angularvelocity and radius), v = λf (wavelength and frequency), v = d/τ (total distance overtotal time, or the period in the case of a periodic motion). In the case of a circularorbit, v = 2πR/τ .

When dealing with elliptical planetary motion of a satellite (mass m) around a planet (massM), several equations are worth noting:

• With rmin, rmax being the minimum and maximum radii between the satelliteand planet, the semi-major axis a = 1/2(rmin + rmax).

• τ 2 = (2π)2

GMa3 (Kepler’s Third Law): “The square of the period of orbit is propor-

tional to the cube of the semi-major axis.”

• E = T +U = −GMm

2a: total energy for an elliptical orbit.

Circular orbit is a special case of elliptical orbit and any perturbation (quick push orpull) will make a circular orbit turn into an elliptical orbit.

Sometimes, instead of equating forces, one has to use conservation of energy. For example,this can be effectively used whenever something is falling or rolling down a hill, orduring an elastic (pool-ball-like) collision. Different types include:

• Translational kinetic energy: Ttrans = 12mv

2 = p2/(2m), or the energy of motion.

• Rotational kinetic energy: Trot = 12Iω

2. This occurs whenever something is rotat-ing, with moment of inertia I and rotational (angular) velocity ω.

• Work: W = Fd with F = average force and d = distance an object traveled. Thiscan occurs when some energy is applied to move something of a distance. Theinitial energy applied is equal to the work done.

• Gravitational potential energy: Ugrav =mgh. This is the gravitational “potential”to roll a ball down a hill of height h or force an object to fall once let go from aheight h. This energy is relative (like voltage) due to the height h being relative,and thus h should equal the difference between the object’s highest and lowestallowed heights.

• Spring potential energy: Uspr = 12k(x − x0)2, with spring constant k and equilib-

rium spring position x0.

The Hamiltonian of a system H = T + U is the sum of the total kinetic energy T andtotal potential energy U .

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II ELECTROMAGNETISM

When dealing with “sticky” (inelastic) collisions, energy is NOT conserved. Thus, in aninelastic collision, Einit = Efinal +U , where U is the non-conservable energy, which canbe positive of negative. Momentum p is conserved in both types, elastic and inelastic:an important formula is ∑imivi = ∑f mfvf for all moving constituents. Sticky colli-sions will be specified, but when in doubt, it is a good idea to start with momentumand move on to energy conservation if the momentum conservation equation is notexactly solvable and if no other equation is given. Sometimes a problem may consistof both (8677:5), in which case energy conservation works for only parts of the systemand momentum conservation provides the missing link.

The Parallel Axis Theorem states that the moment of inertia I ′ around an axis for a bodyof mass m and moment of inertia I around a parallel axis at the body’s center of mass(with distance d to the axis of I ′) are related as such:

I ′ = I +md2 (3)

Accordingly, the kinetic energy of something rotating can either be expressed as1/2Iω2 + 1/2mv2

CM with (vCM = velocity of the center of mass, or the “translationalspeed”) or 1/2I ′ω2 (with I ′ related by Parallel Axis Theorem). Common moments ofinertia at center of mass and around symmetrical axes:

Rod w/length L 112mL

2

Solid disk/cylinder 12mR

2

When dealing with things that rotate, the magnitude of the angular momentum L isconserved (for closed systems). Angular momentum has two different formulations,listed below, where I is the moment of inertia of an object, ω is the angular velocity,Ðp = mÐv is the momentum, and Ðr is the radius of rotation. Note that angularmomenta can only be added if they are for objects rotating around the same point.Also, if v is the tangential velocity, then L =mvr.

L = ∣Ðr ×Ðp ∣ = Iω (4)

The Lagrangian of a system L = T − U is the difference between kinetic energy T andpotential energy U . For each coordinate q in a system, the equation of motion, i.e.,the equation relating q and q, can be obtained from the Euler-Lagrange equation:

∂

∂t(∂L∂q

) = ∂L∂q

(5)

II. ELECTROMAGNETISM

In electro- (magneto-) statics, electric (magnetic) field lines originate at the positive charge(north pole) and travel toward the negative charge (south pole). Thus, the lines oftwo positive charges (two north poles) do not connect.

The equation for the magnitude of the electric field of a point charge Q at a distance d is:

E = 1

4πε0

Q

d2(6)

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II ELECTROMAGNETISM

The field at the center of a configuration of equally-spaced like charges is zero, i.e., thefield in the center of a circle of n equally spaced charges (positive or negative) is zero.

The field inside any ideal conductor is zero. Moreover, the field inside any sphericalshell, conducting or nonconducting, is also zero. One can use the above equationfor any spherical object for points outside of the object’s radius because spherical ob-jects act as point charges when d is greater than their radius. If the charge distributionis linear inside the sphere, the field inside increases linearly with the radius. Similaranalogies can be made for planets with evenly distributed mass.

Inside an insulator, the electric field is non necessarily zero. In insulators, the envi-ronment (permittivity) is different due to charges floating around, so ε0 → ε, wherethe new permittivity constant ε is represented as a multiple of ε0: ε = εrε0. This εr iscalled the material’s relative permittivity, or its dielectric constant. In vacuum, thefield E ∝ 1/ε0. Thus, in a dielectric, E ∝ 1/ε and E → E/εr.

The equation for the magnetic field magnitude from a wire with current I at perpendiculardistance r is below.

B = µ0I

2πr(7)

Moreover, one needs to multiply the left side by θ/2 to obtain the magnetic field atthe focal point of a wire spanning an angle θ.

Similar to the permittivity changing in insulators, the permeability also changes whennot in vacuum: µ0 → µ where µ is also expressed as a factor of µ0: µ = µrµ0, where µris the material’s relative permeability.

The speed of light in a medium v can be expressed in terms of the permittivity andpermeability of the medium. Rearranging the terms to include the relative permittivityand permeability, εr, µr, we obtain (with c = speed of light in vacuum):

v = 1√εµ

=(1/√ε0µ0)√

εrµr= c

√εrµr

(8)

The quantity√εrµr is the index of refraction n of the material, to be used in the optics

section.

The dielectric constant εr can be between 1 and infinity. As εr → 1 (εr → ∞), itapproaches the behavior of a vacuum (a conductor). This can be explained in termsof the equation above: when εr = 1, then light speed is c, and when εr =∞, no light ismoving as no light moves through metals.

Electric fields act on charges like gravity acts on masses, i.e., charges move parallel (orantiparallel) to the electric field. Stationary charges will get accelerated by an electricfield. Magnetic fields will not affect stationary charges, but will deflect moving chargesas long as the charges are not moving parallel to the magnetic field. The Lorentz forceequation sums it together and involves both the forces of electric and magnetic fields,ÐE and

ÐB respectively, on a charge q with velocity Ðv and mass m:

ÐF Lor = q(

ÐE +Ðv ×ÐB ) (9)

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II ELECTROMAGNETISM

If the charge is an electron, q < 0. When a moving charge is in the presence of both

fields, it’s possible that it will not be deflected ifÐE = −Ðv ×ÐB . From balancing out the

kinetic energy of the charge with the electric potential energy, we can also determinewhat electric potential V that can be used to accelerate the charge from rest to velocity

v: v =√

2qV /m. This force can be balanced with the forces listed in Section ??. Forexample, one can obtain the frequency of orbit (cyclotron frequency) of a charge in aB field by setting E = 0 and equating with the centripetal force.

Magnetic and electric field interactions with electrons has nothing to do with spin inthe realm of electromagnetism (8677:27).

The method of images allows us to replace “infinite grounded conducting planes” by chang-ing the sign and then mirroring whatever is to one side of the plane. For example, fora charge q a distance a above such a plane, we know that we can replace the planeby placing a charge −q a distance −a from the plane (mathematically, this does notviolate the boundary conditions because the potential at the plane is still zero). Thus,the charge induced in the plane is that of the mirror charge that we had to put in tosubstitute for the plane, i.e., −q.

Only accelerating (including spinning, deflecting, and turning) charged particles emit ra-diation. Bremsstrahlung (braking radiation in German) is the radiation emitted fromcharged particles when they are accelerating due to forces from other particles.

Whenever a quantity such as power, work or electric field is oscillating in time at a constantfrequency, the total power, work (or electric field) is zero. This has to do with the factthat the time average of a sinusoid is zero. In an analogy in classical mechanics, if onewalks up and down the stairs for a week at the same rate, then one has done no workas the work cancels out. This is the reason “root mean square“ voltage and currentare implemented in alternating circuits.

While fields propagate at the speed of light, the actual electrons in a circuit move at a speedon the order of 10−4 m. The classical formula for electron drift velocity is J = nev,where J = current density, n = number of electrons N per volume V , e = the charge ofan electron, and v = the drift velocity. Remembering other definitions of J , we obtainthe following set of useful formulas (with I = current, A = cross-sectional area of awire, σ = conductivity, and E = electric field):

J = I

A= σE = nev (10)

Based on the above relation, the equation that relates voltage V , current I, and resistanceR is called Ohm’s Law:

V = IR (11)

Resistance in a wire with cross-sectional area A, length L, and resistivity ρ is R = ρL/A.

The energy stored in a capacitor is 12CV

2, where C = the capacitance and V = thepotential. Capacitors in parallel are added like resistors in series (they can be thought

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II ELECTROMAGNETISM

of as metal plates, and thus two parallel metal plates would just form a bigger one).Capacitors in series are added together using the parallel formula:

CT = 11

C1

+ 1

C2

= C1C2

C1 +C2

. (12)

In a circuit with one battery, current always flows from the “+” end to the “-” end.Kirchoff’s Law of Voltage allows one to go along a wire loop and determine an equationfor the voltages of the various components of the loop. The voltage of a componentwill either be negative or positive, depending on the direction of the loop. A good wayis to start with a device that creates voltage, such as a battery or generator and goin the direction of the current. Thus the first term will be positive. Then the voltageacross any resistor will be negative as there is a voltage drop. In addition, the voltageof voltage-generating device lined up opposite the direction of the current will alsobe negative. Some loops will be connected to other loops and thus the current mightbe opposite the initial current in a segment, thereby reversing the signs of all of thecomponents in that segment of the loop.

In a direct current RL (RC) circuit, the current in the circuit is proportional to 1 − e−t/τ(e−t/τ ), where τ is the time constant L/R (RC). As a mnemonic, a capacitor “shutsdown” the current from a battery while an inductor only “fights” the current. Con-sequently, the voltage across an inductor (capacitor) in an RL (RC) circuit is propor-tional to −e−t/τ (1− e−t/τ ). The inductor starts with nonzero voltage across it and actsas a regular wire after enough time has passed.

In an alternating current (AC) circuit, the impedance of a capacitor (inductor) is1

ωC(ωL).

Thus, as ω →∞, a capacitor (inductor) will act like an open (closed) circuit. A high(low) pass filter is a circuit that allows high (low) frequencies to pass. A capacitor(inductor) is a high (low) pass filter.

In an AC RLC circuit, the formula for the resonant frequency, i.e., the driving frequencyω at which the current is maximized, is given by the formula ω = 1/

√LC.

The boundary conditions for a given fieldÐF ∈ ÐE,ÐB can be illustrated in the Figure

below:

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II ELECTROMAGNETISM

ld

ad ⊥tF

||tF

⊥iF ||

iF

tF

iF

FIG. 1: Boundary conditions of a surface (grey) with initial fieldÐ

Fi and final fieldÐ

Ft

ÐFi is the (incident) field coming into the grey surface while

ÐFt is the (transmitted) field

coming out. Both of those field vectors have components parallel (∥) and perpendicular() to the surface. The parallel (perpendicular) vector components are dotted with the

infinitesimal length (area) element dÐl (dÐa ). The area elements point away from the

surface while the length element is clockwise around the light blue path. Thus, twotypes of integrals arise: surface integrals (flux, Φ) and line integrals (curl, Γ). Thereare four criteria that need to be satisfied that are made up of those integrals: the fourMaxwell equations in integral form, with S being the entire red matchbox and L beingthe light blue path and Qenc, Ienc being the respective charge and current inside thematchbox:

ΦÐE

= ∮S

ÐE ⋅ dÐa = 1

ε0Qenc ΦÐ

B= ∮

S

ÐB ⋅ dÐa = 0

ΓÐE

= ∮L

ÐE ⋅ dÐl = 0 ΓÐ

B= ∮

L

ÐB ⋅ dÐl = µ0Ienc

(13)

Given the above rules, four boundary conditions arise which relate the magnitudes

of the four vectors ÐFi∥,ÐFi,

ÐFt∥,

ÐFt for both fields (where σ,K are the respective

surface charge and current magnitudes):

Et −Ei = 1

ε0σ Bt

= Bi

Et∥ = Ei

∥ Bt∥ −Bi

∥ = µ0K(14)

E&M waves are simply two oscillating E and B fields traveling in a direction perpendicular

to both of them - the Poynting VectorÐS = ÐE ×ÐB . When an E&M wave strikes a

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III OPTICS AND WAVE PHENOMENA

surface perpendicularly, there are no components of the wave perpendicular to thatsurface (Et

= Bt = 0), so there is no surface charge. There is also a reflected vector

ÐFr, not shown in Fig. ??, which is simply vector traveling in a direction opposite theincident vector. Thus, in the boundary condition equations in (??), Ei ⇒ Ei +Er forboth E and B.

Furthermore, when an E&M wave strikes a conducting surface perpendicularly, no

fields are transmitted (E∥

t = B∥

t = 0). Thus, according to the boundary conditions inEq. (??), everything is reflected (Ei +Er = Et = 0) and, by the Poynting vector, themagnetic field for both the incident and reflected waves points in the same direction.

When dealing with moving loops of current, Faraday’s Law can help. It states that theinduced electric potential E (which one can think of as voltage) is related to themagnetic flux through one loop of wire S by:

E = −∂ΦÐ

B

∂t= − ∂

∂t ∫SÐB ⋅ dÐa (15)

One can then substitute this E for V in Ohm’s Law to determine current I or resistanceR in the loop. With N loops, the mnemonic NBA comes to mind.

If magnetic monopoles exist, then two of Maxwell’s equations will change:Ð∇ ⋅ÐB = 0 ⇒

Ð∇ ⋅ÐB = µ0ρm andÐ∇ ×ÐE = −∂

ÐB

∂t⇒ Ð∇ ×ÐE = −∂

ÐB

∂t− µ0

ÐJm where ρm and

ÐJm are the

magnetic charge and current densities, respectively.

Lenz’s law states that fields induced by other changing fields try to keep the changingfields the same. In other words, if an external increasing (decreasing) magnetic fieldgoes though the center of a coil, then the field induced in the coil will point in theopposite (same) direction of the external field. Taking the external field to be out ofthe page and looking down at the coil, the induced current in the coil for the increasing(decreasing) external field will be clockwise (counterclockwise).

III. OPTICS AND WAVE PHENOMENA

The two wave-like phenomena that humans can perceive in this world are light and sound.Since both are waves, they follow the Doppler effect. In the classical Doppler effect, asound source traveling at a speed of vs with frequency f0 reaches an receiver travelingat a speed of vr (with all velocities fractions of the speed of sound). Since the source’sand receiver’s velocities have two directions (forward and back), there are four possiblescenarios. If the receiver and source are moving toward each other, then the frequencyshift at the receiver will be highest and the observed frequency f will be:

f = (1 + vs1 − vr

) (16)

The other extreme case is when both are moving away, in which case the signs switch.The remaining two cases are when both the source and receiver are moving in thesame direction. Also, it is possible that one of the velocities is 0, which significantlysimplifies the reasoning. The main point is that if the distance between the receiverand source is decreasing (increasing), f > f0 (f < f0).

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III OPTICS AND WAVE PHENOMENA

The human eye can perceive oscillations of only up to 25 Hz. Anything faster than thatwill seem like its no moving (8677:13).

Lenses take in objects and create “images” of the objects. The thin-lens formula relatesdo, di, and f (distance between the lens and the object, distance between the lens andthe image it created, and the lens’s focal length):

1

do+ 1

di= 1

f(17)

One can (relatively) easily determine the image’s position for both mirrors and lenses byusing “ray-tracing.” Basically, drawing the object as an arrow, draw two “rays” fromthe top of the arrow, one parallel to the principal axis (to point A in the example witha convex mirror in Fig. ??) and the other through the center of the lens/mirror (pointB). Then reflect and distort the rays according to the structure of the lens/mirror,remembering that the focal point is where incident light is usually focused. Afterthat, try to make them connect, a process which may involve extending the lines inthe other direction if they do not converge in the reflected direction. In the end, theline parallel to the focal axis will wind up crossing a focal point.

2F'2F F'F

Object

Image

Principal axis

(center of curvature)

(Focus)

A

B

FIG. 2: Example of ray-tracing: concave mirror.

Since an aperture (hole which light comes through) is similar to a two-dimensional single-slit experiment, light going through the aperture to the lens interferes with itself,creating a blur in the image. The formula that calculates the angular resolution (θ,radians) for an aperture (in a telescope, camera, or microscope, for example) is theRayleigh criterion (with D being the diameter of the lens’ aperture and λ the incidentwavelength):

sin(θ) = 1.22λ

D(18)

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III OPTICS AND WAVE PHENOMENA

θ

d

1n

2n

3n

FIG. 3: Light incident on two media.

The standard approximation for double-slit interference for bright fringes ismλ

d= Yl

where

m is an integer multiple of wavelength λ, Y is the position of the bright fringe, and l isthe length of the screen. For destructive interference, i.e., dark bands, let m→m+1/2.

In a Michelson interferometer, a beam of light is split into two rays which then meet backup at a detector. One of rays bounces off of an extra mirror that can be adjusteda distance d. Every time the total distance traveled (2d) is a multiple of λ (λ/2), abright (dark) fringe is observed. Thus, the number of fringes m which occurs whenthe mirror is moved a distance d is: m = 2d/λ.

When light/sound waves travel through different media and/or change speeds, their wave-length remains the same and only the frequency changes. Based on that concept,the new speed of light v through a medium i with index of refraction ni = c/vi isvi = λif = λi(c/λ) (where λ is the wavelength in vacuum). Thus, λi = λ/ni.When light traveling in a medium with index n1 strikes a medium with index n2,the reflected wavelength λ1 will change phase if n2 > n1 (e.g. a pulse sent through ajump-rope tied to a wall will change phase). Fig. ?? depicts what happens when lighttraveling in n1 strikes n2 and gets partially transmitted to n3.

In Fig. ??, if θ << 1, which is a condition of near-incident light, then the distancetraveled in medium two is ≈ 2d. Usually n2 > n1, so the phase change in medium 1,∆P1, is λ1/2. Table ?? depicts the possible conditions on n2 and n3, the phase changein medium 2, and the formula for thickness d in order to obtain full constructive or

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IV THERMODYNAMICS AND STATISTICAL MECHANICS

destructive interference of the reflected waves (with integer factor of wavelength m).

Condition ∆P2 Constructive Destructive

n3 > n2 2d + λ2/2 d = m2

(λ1

n2

) d = m + 1/22

(λ1

n2

)

n3 < n2 2d d = m + 1/22

(λ1

n2

) d = m2

(λ1

n2

)

Bragg Reflection is the interference of waves reflected from two layers of a crystal.It is similar to Fig. ?? except that it occurs deep inside the crystal so all indexesof refraction are assumed to be the same and the incident angle can be any angleas opposed to only being near-incident. Since the near-incident approximation is out,sin(θ) is no longer about 1 and the resulting formula for constructive interference (mthorder reflection) is:

d sin(θ) = m2λ (19)

Snell’s Law, a consequence of boundary conditions described in Section ??, governs theangle of refraction θ2 of a beam of light incident at θ1 on the boundary betweenmaterials with indices n1 and n2:

n1 sin(θ1) = n2 sin(θ2) (20)

Light, an inter-twining of E and B fields, is randomly polarized with intensity I0 ∝ E20 .

When it goes through a polarizing filter, each polarization can be divided into a vectorcomponents parallel and perpendicular to the slits. Since only the parallel componentsgo thorough, the new intensity I1 = I0/2. After that, the light is polarized and onlythe parallel components go through the ith filter: Ii = Ii−1 cos2(θi).

IV. THERMODYNAMICS AND STATISTICAL MECHANICS

Thermodynamics

The basic energy equation for gases is dU = dQ − dW (energy equals heat minus work). Inthis, work is defined to be as the area under a P − V diagram (with pressure P beinga function of V ):

W = ∫Vb

Va

PdV (21)

Thus, when a gas expands (Vb > Va), it does work on its surroundings (W > 0), andit loses energy as a result. There are four variables (T,Q,V, and P ) with two relatedvariables (U and W ), so we have four processes: isothermal (dT = dU = 0), adiabatic(dQ = 0), isochoric (dV = dW = 0), and isobaric (dP = 0). A P − V diagram for anideal gas with all four processes is shown in Fig. ??.

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IV THERMODYNAMICS AND STATISTICAL MECHANICS

P1

P2

V1 V2 V3

AdiabaticIsobaricIsochoricIsothermal

III

IV

III

V

FIG. 4: Iso-processes on a P − V diagram.

Process I is isochoric, II is adiabatic, III is isothermal, and IV and V are isobaric. Themagnitude of the work done during each process is the are under the curve for eachprocess. The sign of the work depends on the initial and final volumes.

Using Eq. ??, the work done during the cyclical process II → IV → I is (approximatelyequal to but strictly less than) the area of the triangle that the process lines make.Clearly, if V3 > V2, the work done by III → V → IV → I is more than the work doneby II → IV → I. Thus, the work an ideal gas does during an adiabatic process is lessthan the work it would do during an isothermal process.

For an ideal gas, two arbitrary points i and j in the figure above are related byPiVi = PjVj.

A Carnot cycle is a type of thermodynamic process which, if perfectly efficient, does notincrease entropy. As any heat engine, it takes heat from a warmer region and transfersit to a colder region while doing work. It consists of two adiabatic steps and twoisothermal steps. There are a couple of equations that need to be remember for it thatrelate the total work W , the heat absorbed/expelled Q1/Q2 with Q1 < 0 and Q2 > 0,the two temperatures T1/T2, and its efficiency n.

W = Q1 +Q2

n = W

Q2

= 1 − T1

T2

(22)

For one mole of an ideal gas with pressure P , temperature T , gas constant R, and volumeV , PV = RT . The specific heat at constant pressure, dQ

dT Pis greater than than that at

constant volume, dQdT V

, specifically because the definition of work being PdV . An idealgas can do work at a constant pressure but cannot do work at a constant volume.

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IV THERMODYNAMICS AND STATISTICAL MECHANICS

For isothermal and adiabatic processes, change in entropy dS = dQ/T . Thus, in an adiabaticprocess, entropy is constant. In a Carnot cycle, a cycle with two adiabats and twoisotherms, the total entropy is constant because the respective entropy changes for thetwo isotherms are equal and opposite.

A useful series of ratios between micro- and macroscopic variables is:

NA = R

kB= Nn

= Mm

m≈ 6.02 ⋅ 1023 (23)

In the above equation, Avogadro’s constant is the ratio of gas constant R to Boltzmannconstant kB, total number of constituents N to number of moles of constituents n,and molar mass Mm to mass of each constituent m.

Using Eq. (??), entropy can be defined from a statistical perspective to be S =R/NA ln(Ω), with Ω = total number of states. This can be applied to Joule expansion inwhich a gas expands in vacuum to occupy, for example, twice its initial volume. Thus,the number of possible states for each particle has now increased to two. Thereby, forall N particles, there are a total of 2N states and the change in entropy is S = NR ln(2).RMS (Root Mean Square) velocity can be defined as vRMS =

√3RT /Mm, meaning

that vRMS ∝√T and vRMS ∝ 1/

√Mm.

Statistical Mechanics

According to the Equipartition Theorem, the average energy of a system (⟨H⟩) is a sumof the energy contributions of the various components of the system and that anyquadratic relation in the Hamiltonian contributes 1

2kBT to the total energy. Theenergy of an ideal gas is 3

2kBT because each particle in the gas has three degreesof freedom of translational motion and each contributes 1

2kBT to the total energy.Translational kinetic energy and rotational kinetic energy are all quadratic and alsocontribute 1

2kBT . Whenever an oscillator is involved, it adds a harmonic potentialenergy (quadratic 1

2kx2) and it also has its own “vibrational” kinatic energy. Thus an

oscillator will add a total of kbT to the total energy.

For example, in a 3D harmonic oscillator, there are three vibrational degrees of freedomand three harmonic potentials, so ⟨H⟩ = ⟨KE⟩ + ⟨PE⟩ = 2

3kBT + 23kBT = 3kBT .

Another example, a springy dumbell (used as a model for a diatomic molecule) freeto rotate about axes perpendicular to the spring. It can rotate about 2 axes (since 2axes are perpendicular), so the rotational contribution is kBT . It can translate in allthree directions, so the translational contribution is 3

2kBT . Finally, there is a springin involved, so the spring contribution is kBT . In total, 7

2kBT .

Similar to total energy, the specific heat at constant volume cv can also be determinedfrom this analysis.

In Maxwell-Boltzmann statistics, the partition function of a system with n possible discretestates (microstates) is ∑nj=1 gje−Ej/(kBT ) where gj is the degeneracy of each microstate.If there are a total of N molecules in the system, the fraction of molecules that are in

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V QUANTUM MECHANICS

an arbitrary state Ni is given by:

Ni

N= gie−Ei/(kBT )

n

∑j=1

gje−Ej/(kBT )

(24)

Instead of using the total number of molecules, defined by the partition function, onecan just as easily find ratios between two arbitrary states. Also, following the samepattern of ratios, the average, Q, of a quantity Q is given by:

Q =

n

∑j=1

Qie−Ei/(kBT )

n

∑j=1

e−Ej/(kBT )

(25)

V. QUANTUM MECHANICS

All physical solutions to the Schrodinger equation must be square-integrable, i.e., theirprobability P over all space, in the 1D case: ∫

+∞

−∞∣ψ(x)∣2dx, must be finite. Thus,

1/r is not a physical wavefunction form while e−ar is. In fact, the integral for P isnormalized such that, for the radial distribution in the 3D case (without forgettingthe radial element):

∫+∞

−∞

∣ψ(r)∣24πr2dr = 1 (26)

In order to find the most probable value of ψ(r), one needs to take the derivative ofthe stuff inside the integral and set it to 0.

Arbitrary discrete eigenstates ∣1⟩ and ∣2⟩ are orthonormal if ⟨1∣2⟩ = δij.Symmetric and Hermitian matrices have real eigenvalues and orthogonal eigenvectors.

If an operator Ω has eigenvector ∣λ⟩ and eigenvalue λ, then Ω∣λ⟩ = λ∣λ⟩. The possible out-comes of a measurement of a given state ∣ψ⟩ are the eigenvalues λ of the correspondingobservable operator Ω. The expectation value of the operator ⟨ψ∣Ω∣ψ⟩ for a given state∣ψ⟩ is the average of many measurements taken over a long period of time. Any state∣ψ⟩ can be expressed as a linear combination of the N normalized eigenvectors ∣λ⟩:∣ψ⟩ = ∑Nλ=1 aλ∣λ⟩.

The Hamiltonian H is the sum of kinetic and potential energies, just like in classicalmechanics, except x, p now become operators. If there is a discrete number of eigen-states ∣n⟩, H ∣n⟩ = En∣n⟩ for all integers n. The first-order energy correction ∆E to aHamiltonian H perturbed by ∆H is the expectation value of the perturbation withnon-perturbed eigenstates:

∆E = ⟨n∣∆H ∣n⟩ (27)

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V QUANTUM MECHANICS

The wave function for two particles x1 and x2 at respective states ψα and ψβ is:

ψα(x1)ψβ(x2) if the particles are distinguishable

1

2(ψα(x1)ψβ(x2) + ψα(x2)ψβ(x1)) if the particles are bosons

1

2(ψα(x1)ψβ(x2) − ψα(x2)ψβ(x1)) if the particles are fermions (28)

Bosons (deuterons, photons) have symmetric wavefunctions (you can switch ψa ↔ ψβand still have the same function) and integer spin. Fermions (protons, neutrons,electrons, positrons) have anti-symmetric wavefunctons and half-integer spin.

Equations for eigenvalues l,ml of the angular momentum operators L2, Lz, respectively,are:

L2∣l,ml⟩ = l(l + 1)h2∣l,ml⟩ (29)

Lz ∣l,ml⟩ = mh∣l,ml⟩ (30)

Since the spin of an electron is any value in the vector space of spin-up and spin-downvectors (∣+⟩ and ∣−⟩, respectively), adding the spins of two electrons means that weneed to make sure that both spins can still have all of their possible values. Thus, wepick an orthonormal basis for the vector space of a two electron system. For a twoelectron system, the space consists of a singlet and triplet. Each vector ∣s,ms⟩ in thespace represents a state of total spin s and z-component ms, which follow the samerules as l,ml, the angular momentum eigenvalues:

Triplet

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∣1,+1⟩ = ∣ + +⟩∣1,0⟩ = 1

√

2(∣ + −⟩ + ∣ − +⟩)

∣1,−1⟩ = ∣ − −⟩(31)

Singlet ∣0,0⟩ = 1√

2(∣ + −⟩ − ∣ − +⟩)

The operator Sz, for which ms is an eigenvalue, can be expressed as h2σz, with σz being

a Pauli matrix. Thus ms is an eigenvalue of σz. Similarly, spin projections in to the xand y directions (Sx and Sy, respectively) can also be expressed in terms of matrices.

Several important commutator identities are worth noting:

[A,B] = AB −BA (32)

[A,B] = − [B,A][A,BC] = [A,B]C +B [A,C] similar to product rule d(fg) = (df)g + f(dg)

A free particle has energy E = p2/(2m) with wavenumber k, momentum p, and de Brogliewavelength λ. When such particle enters a region of potential (similar to a ball beingrolled up a hill), it loses an amount of its kinetic energy equal to V , so its new kineticenergy becomes E−V . Using various de Broglie-related relations E = hω, p = hk = h/λ,ω = 2πf , k = 2π/λ, and h = h/(2π), one can relate the de Broglie wavelength of a freeparticle to that of a particle confined to a potential.

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V QUANTUM MECHANICS

The symbolic Mendeleev’s Table of Elements is shown below, with emphasis on the quan-tum numbers of each electron in the table.

n 1234567

llmsm

0 1230 3,2,1,0,1,2,3 −−− 2,1,0,1,2 −− 1,0,1−

21

±21,

21,

21,

21,

21,

21,

21

±±±±±±±21,

21,

21,

21,

21

±±±±±21,

21,

21

±±±

2 14 10 6

FIG. 5: Periodic Table with Quantum Numbers.

All in all, each electron in every atom has the following numbers describing it: n =energy level, l = orbital angular momentum, ml = projection of orbital angular mo-mentum on z-axis, s = spin, and ms = projection of spin. Since s = 1/2 for an electron,this is usually not explicitly stated, so an electron is described by n, l,ml, and ms. Thelevel n goes up to 7, but l = 1,2, ..., n−1 only to 3 because the most important occupiedorbitals have l = 0,1,2,3 (s, p, d, and f , respectively). The above figure describes howthe periodic table is arranged to take those numbers into account. This is useful inremembering what the angular momenta of electrons are. For example, Sodium hasa 3s1 configuration, with the angular momentum of its outermost electron being 0.Also, the outermost electron (l = 0,ml = 0) has a spherically symmetric state, i.e., it’swavefunction doesn’t depend on the spherical angles θ, φ because the spherical har-monic with l = 0 and ml = 0 is simply a constant. Also, filled energy sub-shells (s, p, d,and f) have a spherically symmetric wavefunction because all spherical harmonics areincluded (???).

The total angular momentum j = l + s, where the two vectors can be added likescalars, is the eigenvalue of the total angular momentum operator J . The z-projectionof j, mj, and is found by adding two quantized angular momentum vectors usingClebsch-Gordan coefficients outside the scope of this work. The standard spectroscopicnotation for an atom is 2s+1lj, where s is the sum of the spins of each electron in theatom.

During excitations or relaxations in which a photon is absorbed or emitted by an electron,the initial and final quantum numbers of the electron (n, j,mj, l,ml, s,ms) follow cer-tain selection rules. The transitions that are dealt with are the easiest - electric dipole

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VI ATOMIC PHYSICS

transitions. Basically, to radiate electromagnetic waves (emit a photon), an oscillatingelectric dipole has to be created. By conservation of angular momentum and since aphoton has been observed to have angular momentum ±1, then:

∆l = ±1

∆j = 0,±1 (33)

Since angular momentum does not depend on n and there is no conservation law forn:

∆n = anything (34)

Since the electric dipole operator does not act on the spin of the particles:

∆s = 0

(35)

Also, with no intuitive explanation:

∆mj = 0,±1 (36)

The numbers ml,ms do not follow rigorous rules like the ones above and are determinedby factors in the system. For example, for a system with zero net spin (s = 0), therules for j,mj will apply to l,ml because j = l + s = l.

When one examines electromagnetic fields in terms of special relativity, the Lorentz trans-formation of a solely electric field allows for existence of both electric and magneticfields in another reference frame. This is because in the Lorentz transformation, amagnetic field component in the new frame is a function of an electric field componentin the old frame.

VI. ATOMIC PHYSICS

Using the Bohr Model of an atom, which correctly predicts the energy of electrons in H,we know that energy (eV) of a level n of hydrogenic (one-electron) atoms with nucleusof charge Z is:

En = −13.6Z2

n2(37)

Thus, for He+, for example, the energy to remove the last s electron is −13.622

12= −54.4

eV. Also, the change in energy from n = 2 to n = 1 for H (called Lyman-α radiation) is

∆E = 13.6( 1

n2f

− 1

n2i

) = 13.6( 1

12− 1

22). When modeling things other than Hydrogen,

there are several considerations:

• For Ps, or positronium (an electron and positron), the mass approximationdoesn’t work as the masses are the same for both, so the energy is half of the Hatom energy: EPs = 1/2EH .

• The same model can approximate larger atoms, but if ni is the innermost shell(called the K-series), then the effective charge is now Z − 1 due to screening.

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VII SPECIAL RELATIVITY

Formulas that relate the frequency of the photon to the change in energy are ∆E = hfand c = λf (h = Planck’s constant, f = frequency of photon, λ = wavelength, c = speedof light). Using these, we can determine all kinds of proportionalities between n, f, λ,and ∆E.

Given a complicated molecule, there are several ways for that molecule to absorb energyfrom photons (E = hf).

First, its atoms may vibrate or rotate with respect to each other and thus absorb energyquanta equal to the energies of their respective vibrational and rotational frequencies.A vibrational/rotation spectrum is the spectrum of frequencies/wavelengths/energies(they’re all related) which are absorbed by the nuclear vibrations/rotations. Thesetransitions occur in the infrared section of the E&M spectrum. The specific frequenciesof vibration can reveal information about the nuclear structure of the sample.

Second, the electrons around the atoms can become excited (by a photon) and jumpto a higher energy state (excited state). The absorption spectrum of a molecule is thespectrum of frequencies/wavelengths/energies that the molecule’s electrons absorb toexcite themselves. Conversely, the emission spectrum of a molecule is the spectrum offrequencies/wavelengths/energies that the molecule’s electrons emit when they relaxto a lower-energy state. These transitions occur in the visible section of the E&Mspectrum. The specific frequencies of excitation can reveal information about theelectronic and nuclear structure of the sample.

In blackbody (e.g. a hot glowing piece of metal) radiation, the energy pressure rate (energyradiated per second per area, or “intensity”) u ∝ T 4, with T = temperature. The onewavelength which is radiated the most for a given temperature, λmax = b/T with b =Wien’s displacement constant.

VII. SPECIAL RELATIVITY

For ease of use, let the speed of light c = 1. Some useful equations for relating energy E,momentum p, and rest mass m include:

E = γm =√p2 +m2 (38)

γ = (1 − v2)−1/2 (39)

p = γmv (except for photon) (40)

Photons have momentum/energy but do not have rest mass. Using the unit convention,the energy of a photon is equal to its momentum.

Similar to collisions described in Section ??, momentum is always conserved whileenergy is conserved only in non-sticky (elastic, “pool-ball”) collisions. Various atomicreactions, such as electron-positron annihilation, can be considered as inelastic unlessspecified otherwise.

When dealing with time dilation and length contraction, as with classical mechanics, thereare two frames: the rest frame (in which the object in question is stationary) and thelaboratory frame (with respect to which the object is moving). Rest frame variables

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VIII LABORATORY METHODS

(change in distance d, time t) in the equations below will have a “p” subscript, meaning“proper.” As usual, everything is divided by c:

d = 1

γdp (41)

t = γtp (42)

v = (1

c) dt

(43)

Since γ ≥ 1 and therefore 1/γ ≤ 1, in the real-world laboratory frame, distance contracts(“moving objects are shortened”) and time dilates (“moving clocks run slow”).

In the Galilean system, velocities are added directly, such as determining the velocity ofobject A with respect to C given an intermediate object B: vAC = vAB + vBC . In SR,velocity additions are contracted such that one can never surpass c:

vAC = vAB + vBC1 + vABvBC

(44)

There is a conserved quantity s that is true for all reference frames, including the usualrest and lab frames: s2 = d2 − (ct)2 = d2

p − (ctp)2. The following chart shows differentcharacteristics for s for d and t in an arbitrary frame:

Criteria Consequence Label Possible frame exists in which...

s < 0 0 ≤ d < t Timelike ...2 events occur at the same places > 0 0 ≤ t < d Spacelike ...2 events occur at the same time

When a light source traveling at a relativistic velocity v passes an observer, the initial lightemitted from the source will be observed having a frequency higher than the frequencyin the source’s frame. On the approach of the source, a “blue-shift” is observed therelation between the frequency in the observer’s frame (f) and the frequency of the

source (fs) is f =√

1 + v1 − vfs. After the source passes the observer, the light emitted

from the source will be observed having a frequency lower than the frequency in thesource’s frame. When the source is traveling away from the observer, a “red-shift” isobserved at v → −v in the above equation.

VIII. LABORATORY METHODS

Accuracy means how close ones measurements are to the true value, while precision meanshow close ones measurements are to each other.

Gaussian distributions (bell curves) are often used to describe distributions of measure-ments in which unquantifiable (random) factors affect those measurements. However,when describing the probability of events occurring in a fixed amount of time, theslightly skinnier Poisson distribution is used. The uncertainty of a symmetric distri-bution of measurements is the standard deviation of the distribution. The standard

deviation of a Poisson distribution with mean N is√N . The percent uncertainty,

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IX SPECIALIZED TOPICS

σ%, is the uncertainty scaled by the mean, i.e.,

√N

N= 1√

N. Given an average num-

ber of events per second, N0, the average number of events per measurement periodN = τN0, where τ is the duration of the measurement period. This, the relationshipbetween the uncertainty and the measurement period assuming a Poisson distribution

is σ%(τ) = 1√τN0

. Qualitatively, the longer the measurement period, the less the

uncertainty in the number of events, similar to the Law of Large Numbers.

IX. SPECIALIZED TOPICS

Radioactive Decay One type of decay of radioactive sources is emitting gamma radiation,or high-frequency invisible light. A source emits gamma rays evenly in a sphere. Whenplaced next to a circular detector, the percent of gamma rays detected is the solid angleof the cone of gamma rays that make it to the circle divided by the total solid angleof the sphere, 4π. The formula for a solid angle is 4πsin2(θ/2) where θ is the angleof the cone. However, when θ << 1, i.e., when the circular detector is small and faraway from the point of radiation, we can approximate the percent of gamma radiationdetected as the ratio of the area of the detector to the surface area of the sphere.

Half-lives of various decays such as β or γ of an element are added “in parallel” (similarto capacitances) in order to obtain the total half life.

Particle Physics Various particles decay into various other particles. An important con-version to remember is that of Beta decay (with n = neutron, p = proton, e− = electron,and ν = antineutrino):

n → p + e + ν +∆E (45)

Using the above conversion and the principle of particle/antiparticle annihilation, wecan add a positron p+ and a neutrino ν to both sides of the equation to obtain theformula for β+ decay: ∆E + p→ n+ e+ + ν. Similarly, the formula for electron capture:∆E + p + e− → n + ν.

Nuclear transmutation (conversions from one element to another) occurs via the abovethree decay processes along with some others. Remembering that the superscript of anelement is its mass number (protons + neutrons) and a subscript is its atomic number(protons), an element E either turns into element B via β− decay or becomes elementC via β+ decay or electron capture. Also, it can turn into element D via alpha decay,or emission of a He nucleus:

AZE → A

Z+1B + e− + ν (46)AZE → A

Z−1C + e+ + νAZE + e− → A

Z−1C + νAZE → A−4

Z−2D + 42He

An important conservation rule to remember is the Conservation of Lepton Number.It is consistent for the entire Standard Model, as opposed to Baryon Conservation.

A muon is a MUch heavier version of the electrON.

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IX SPECIALIZED TOPICS

Electron-Photon Interaction When a material is bombarded by photons, three effectsoccur depending on the energy. Light or an X-ray will cause the electron to leavethe material, i.e., the Photoelectric Effect. At a higher energy (a gamma ray), thephoton will give off some of its energy to an electron in the material, which is calledthe Compton effect or Compton scattering. At extremely large energies, the photonwill perform something not readily intuitive: it will create an electron and positronpair, a phenomenon called Pair Production and is the opposite of annihilation.

Binding Energy Binding energy is the difference in mass between an element and itsnuclear constituents. This difference in mass is due to E = mc2 and can be thoughtof as the potential energy of the element. The process of fission can be thought ofas the opposite of an inelastic collision where Einit = Efinal + U , with Einit and Efinalbeing the binding energies of the initial and final constituents and U being the kineticenergy of the final constituents that have been ejected as a result of the fission. Thus,the kinetic energy of the particles if the difference between the binding energies.

In addition, the binding energy can be divided by the amount of nucleons (protons andneutrons) in a given element to obtain the binding energy per nucleon. This number isimportant as it is used to judge the stability of an element. Lead (Pb) has the highestbinding energy; elements with an atomic number higher than lead thus tend to fallapart (undergo fission) to attain greater stability and elements with atomic numbersbelow that of lead tend to combine (fuse) to attain greater stability.

Fusion in the Sun (and other large planets) is governed by 4H + 2e− → He + 2ν + 6γ (withν = neutron and γ = photon). We know that energy and mass are interconvertibleduring fusion/fission, according to ∆E = ∆mc2, with the mass difference ∆m for thisreaction being the difference between the fused He atom mass and the four H atommasses.

Solid State A Bravais lattice is a lattice which is made up of only one type of atom. A unitcell of that lattice is the shape made by arbitrary basis vectors. The density of a unitcell is measured in atoms/volume. Atoms are counted in the cell by being depictedas spheres; only the fractions of the spheres that are inside the cell are counted. Aprimitive cell is one with a density of 1 atom/volume. For example, if one knew ofa unit cell of volume V that contains 2 atoms, then the volume of the primitive cellwould be V /2.

Metals contain positive ions in a “sea” of nearly-free electrons. Metals have conductionelectrons at 0K and only a slight change in potential is needed to perturb the electronsand create current. The electrons form a Fermi gas, which the analog of an ideal gas inquantum mechanics. Metals conduct worse as temperature increases becausemore it is harder to push faster-moving electrons in the direction of thecurrent. Adding impurities to metals does not necessarily decrease their conductivity:if a metal impurity is added, the conductivity will remain the same.

Undoped semiconductors, unlike metals, do not have any conduction electrons at 0K.They need some sort of excitation, either thermal or photo-excitation, to conduct elec-trons. Thus, the resistance, and therefore the resistivity, of undoped semiconductors isinfinite at 0K and goes down ∝ 1/T . Semiconductors conduct better as temper-ature increases because more electrons are excited to the conduction band.

23 of ??

IX SPECIALIZED TOPICS

Doping the semiconductors with atoms that can donate electrons to the conductionband decreases their resistivity.

24 of ??