Modern Physics I - Thomaz Notes

MODERN PHYSICS

Time Dilation-

Length Contraction-

Relativistic Doppler Shift-

Lorentz Transformation-

Velocity Transformation-

Relativistic Momentum-

Relativistic Energy-

Blackbody Radiation-

Photoelectric Effect-

X-Ray Production-

Compton Effect-

Rutherford Scattering-

Bohr Atom-

Rydberg Constant-

De Broglie-

Particle/Group Waves-

Heisenberg Uncertainty-

By: Thomaz L. Santana http://www.thomazsantana.com

Fall 2010, PHY3106 - Modern Physics I

Born Interpretation-

Schrödinger Wave Equation-

Schrödinger Time-Independent Eq.-

Energy Quantization (Schrödinger)-

Wavefunction Quantization (Schrödinger)-

Modern Physics I - Thomaz Notes

http://www.thomazsantana.com

d

O'

d = t'c

To simplify we will consider just one side of the triangle. Let (t) equal time for the rest observer frame and (t') equal time in moving reference frame.

tct'c

t

Consider a fast moving train with a laser aiming vertically and mirrors perpendicular to the laser on the roof and floor. Take [S'] to be the observer in the train & [S] to be the observer next to the train track. They see the laser covering different distances at different times.

O

S' S

Time DilationSunday, September 12, 20107:42 PM


Solving for time (t)

now we create a Scalarand call it gamma

The formula is only capable of giving us the difference in time rate of one inertial frame relative to another inertial frame.

is the time rate of an inertial frame relative to a observed inertial frame with a velocity and time rate .

Time DilationSunday, September 12, 20107:52 PM


What speed would a clock have to be moving in order to run at one-half the rate of a clock at rest?

An atomic clock is placed in a jet airplane. The clock measures a time interval of 3,600s when the jet moves with a speed of 400m/s. How much longer or shorter a time interval does an identical clock held by an observer on the ground measure?

If: Then:

The clock on the ground measured 3.2 nano seconds more then the clock on the Jet.

Time Dilation ExamplesSunday, September 12, 20106:29 PM


Recall from Time Dilation section:

(c) is a constant velocity & velocity * time gives a length (L)

Recall:

: Is the length measured by an infinite amount of observers spanning the path of travel of the object as the object passes the observers with a velocity. Alternately the length can also be measured by an observer using a stopwatch to measure the time [ ] it took the object to pass & multiplying the objects velocity.

L : Is the length measured by an infinite amount of observers spanning & at rest with respect to the object.

Length ContractionSunday, September 12, 20106:18 PM


The spaceship observer sees Earth 106 m away.

Distance example: An observer on earth sees a spaceship at an altitude of 435m moving downward toward Earth at 0.970c. What is the altitude of the spaceship as measured by an observer in the spaceship?

Length example: A spaceship at rest is measured to be 100m long. Now this spaceship flies by an observer with a speed of 0.99c, what length will the observer measure?

Length Contraction ExamplesSunday, September 12, 20106:50 PM


Muon reference frame

Earth reference frame

Time for muon to travel 3230m in earths reference frame is coincidently 11 µs, which is the half-life of the muon.

# of half-lifes =

(both have to be from the same reference frame)

If done without special relativity, we would not know the muons half-life relative to earth. We would have divided time of travel (11us) by the half-life (2.2us) and have concluded that 5 half-lifes have gone by. Giving us...

Example: Muons are unstable elementary particles that have charge equal to an electron and mass

207 times that of an electron. Muons are naturally produced by the collisions of cosmic radiation with atoms high in earths atmosphere. Muons have a half-life of 2.2 µs measured in a reference frame at rest with respect to them. Muons travel with a speed of 0.98c relative to earth observer. Muons travel 643m in their reference frame.

If detector (D1) measured [N] muons what would detector (D2) measure? Detector (D2) is 3230m directly below detector (D1).

Muons

Length Contraction & Time Dilation ExampleSaturday, September 18, 20109:42 AM


Observer Source

Lambda is a length; velocity * period [T] =

Within the time period [T], the light pulse moved away from the source at the speed of light [c] & the observer moved towards the pulse with a velocity [v].

Because the observer is measuring, we can't use the time period [T] from the source. We have to use the observers time period [T'].

Substituting using these relations

Relativistic Doppler ShiftSunday, September 12, 20108:33 PM


Or algebraically manipulated...

Note: [+ velocity]: means our assumption when we derived this formula (source at rest and observer moving towards source) is correctly applied to the problem.[- velocity]: means the problem is not similar to the one the formula was derived from, thus the observer is moving opposite to the source.

Example: Calculate, for the judge, how fast were you going when you ran the red light because it appeared Doppler-shifted green to you. (Wavelengths: Red: 650nm, Green: 550nm)

Relativistic Doppler ShiftSunday, September 12, 201010:00 PM


S S'

r'r

O O'

At time [t = t' = 0], a light pulse is emitted and arrives at [P] at time [t] according to [S] and at [t'] according to [S']P

r = ct r' = ct'

Assume:

6-

7- 8-

Apply 7 at object [P] at [O']

7 becomes: 9-

Apply 8 and 9 to light pulse described by 6 Then:

In 3-D & only considering velocity in [x]-axis:

Lorentz TransformationSaturday, September 18, 201011:38 AM


10-

11-

12-

13-

Inverse

14-

15-

Simultaneity Example: Suppose for each observer, lightning strikes occurred at [t = 0] and at the locations [X1 = 0] and [X2 = L]. Observer [S] sees both lightning strikes at the same time. What will observer [S'] see?

[S] & [S'] would see event1 at the same time

[S'] would see event2 happen before [S]

Event1 Event2

Observer [S]

Observer [S']

[S'] observed event2 [ ] seconds ago.

x = 0 x = L

Lorentz TransformationSaturday, September 18, 201012:27 PM


Taking derivative of equations 10 & 13

16-

17-18-

From differential equations we get [y] & [z] components...

[ ]: is [x] component of velocity of object according to observers in [S]

[ ]: is [x'] component velocity of object according to observers in [S']

[ ]: is velocity of [S'] relative to [S]

Velocity TransformationsSaturday, September 18, 20101:18 PM


S (earth) S' (A)

B A

Rockets [A] & [B] move towards each other at 0.9c relative to earth.What is the velocity of [B] relative to [A]?

Note: The given speeds are both relative to earth. We can fix ship [A] to the [S'] frame and assign earth [S] as a rest frame. Now we can say ship [B] has speed 0.9c relative to earth and [S'] frame has speed 0.9c relative to [S]

A

B

S (earth) S'As seen from earth, two spaceships are approaching along perpendicular directions. If [B] is observed by earth to have a speed of 0.8c along the positive x-axis and [A] observed to have a speed 0.9c traveling in the negative y-axis, find the speed of [A] as measured by [B].

A

Important Concept: Because [A] is measuring the speed of [B], [A] must be at rest with the [S'] frame to obtain [

] (check definition of

Velocity Transformations ExamplesSaturday, September 18, 20101:59 PM


S' S'

Before After

S

Before

S

After

Proving momentum is not conserved in the [S'] frame. Consider the exact event that occurred in the [S] frame from the [S'] frame. Recall definitions of [ ], [

] & [ ] to use in [S'] frame.

Because [S'] is unaware of its [ ], it only recognizes [ ]

Relativistic MomentumSaturday, September 18, 201010:46 PM


We must create a new expression for relativistic momentum such that momentum is conserved for all observers.

RelativisticMomentum

: Mass of the object measured by an observer at rest with respect to the object.

: Velocity in the direction of interest of object measured by observer in [S]

: Magnitude of the velocity of an object measured by observer in [S]

Relativistic MomentumSaturday, October 09, 20104:23 PM


A collision of two balls of equal mass, in a reference frame [S].

Find: &

First we will solve for the initial linear momentum in the direction. Then we equate it to the final momentum and solve for .

Important Concept: Recall the definition of and , velocity in a direction and magnitude of velocity.

Because the final component is only in the x direction the magnitude velocity is equal to velocity in .

Comparing initial velocity in direction to final vel. in direction, , we see that the final vel. has increased which is absolutely not what we would expect from classical momentum.

Relativistic Momentum ExampleSaturday, October 09, 20104:52 PM


Consider work done by a force.

Substituting & changing limits of integration for W

Recall work-energy theorem states that work done by all forces acting on a particle equals the change in kinetic energy of a particle. We integrated from 0 to , forming an equation in which kinetic energy is initially at zero.

Relativistic Kinetic Energy

If:

Simplified Kinetic Energy

Recall is velocity in [S]

Relativistic EnergySaturday, October 09, 20106:01 PM


Total Energy:

Recall is velocity in [S]

When velocity, , of a given particle is zero we observe that the particle still contains energy.

Rest Energy

We can deduce that mass is an incredibly dense form of energy.

If we relate total energy and momentum by squaring both sides we obtain the following expression.

For the case of particles that have zero mass, photons, we set = 0.

Getting familiar with eV as a unit of energy.

Rest energy of electron:

Relativistic EnergySaturday, October 09, 20107:02 PM


We can no longer think of separate laws of conservation of mass and the conservation of energy. We unify them and call it Conservation of Mass-Energy.

Change in mass:Note: M is always greater then the sum of individual masses thus, is always positive.

Example: How much energy is released during the formation of 1g of 4He ?

Known: Tritium molecule 3H fuses with a proton 1H to form a helium nucleus 4He

4He = 3727.4 3H = 2808.9 1H = 938.3

2808.9 + 938.3 - 3727.4 = 19.8

19.8 of energy is released during the formation of one nucleus of 4He

1 g of 4He contains 1.51 X 1023 nuclei

Conservation of Mass-EnergySaturday, October 09, 201011:38 PM


A blackbody [BB] is one that absorbs all electromagnetic radiation falling on it.

is the power emitted per unit area per unit frequency.

[ ] power per unit area emitted at the

surface of a BB at all frequencies.

[ ] coefficient if not an ideal BB

[ ] Stefan-Boltzmann constant

[ ] 5.67

[ ] Temperature in unit Kelvin

Wien's displacement law:

Example: estimate the surface temperature of the sun. Sun radius , Earth-Sun distance m, the power per unit area measured on earth is 1,400 W/m2

We can also use Wien's displacement law, measure on earth to be m

BlackBody RadiationSunday, October 10, 201012:58 AM


Spectral energy density, energy per unit volume per unit frequency of the radiation within the blackbody cavity.

[ ] Energy per unit volume per unit frequency

[ ] Planck's constant

[ ] [ ] Boltzmann's constant

[ ]

[ ] Temperature in unit Kelvin

[ ] Frequency

Discrepancy in Wien's exponential law for longer wavelength's led Planck to his formula (which perfectly agrees with experimental data)

Planck's formula

Planck's Contribution:Oscillation energy is an integer multiple of [ ], emitting or absorbing discrete amounts of energy n*[ ]

BlackbodySunday, October 10, 20103:34 PM


Light knocks electrons off metal plate, then they get repealed by like charges and shoot to other side.

When a potential difference is applied, only electrons with enough kinetic energy will make it to the cathode. [Stopping Potential] is when we increase the voltage to the point where no electrons reach the cathode.

The kinetic energy of the most energetic electron is equal to the Stopping Potential times the electron charge.

[ ] Stopping Potential

[ ] Kinetic energy of most energetic electron

[1eV] is the kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt, thus the Voltage needed to stop an electron is equal to the electron's kinetic energy

Photoelectric EffectSunday, October 10, 20104:33 PM


(expected)1)For given , is independent of (unexpected)2) varies linearly with (unexpected)3)Delay in current is zero (unexpected)4)

Results:

Note: [ ] Intensity of light (number of electrons per second)

With the results we must change our view of light, consider light as a wave and particles.

Light behaves like particles (Photons) in its interaction with other matter. Each photon has a discrete energy

PhotoelectricSunday, October 10, 20105:25 PM


[ ] Discrete Energy value per photon

[ ] Work function

[ ] Kinetic energy of most

energetic electron

Note: [ ] is the energy with which an electron is bound in the metal.

At the point where kinetic energy is zero:

Example: Cesium illuminated with a 300nm light source gives off electrons with a maximum kinetic energy of 2.23eV Find: a) work function b) stopping potential for a 400nm light source c) threshold frequency

PhotoelectricSunday, October 10, 20105:48 PM


A heated filament boils electrons off. Then a huge potential difference accelerates the electrons to collide into a metal target, X-Rays are given off from the collision.

If electron loses all its energy in a single collision...

When an electron collides with an electron in orbit, an electron from a higher orbit will drop in energy level to fill in the gap releasing a photon. The spikes in the graph is when a low orbit electron gets knocked off.

X-Ray ProductionSunday, October 10, 20106:47 PM


Atoms in plane [A] will scatter constructively if the angle of incidence equals the angle of reflection.

Atoms in plane [A] & [B] will scatter constructively if the path difference for rays (1) & (2) is an integer multiple of the wavelength. Constructive interference will occur when:

Condition for constructive interference(Bragg equation)[ ] Distance from plane [A] to [B][ ] Angle the ray makes with the plane (assuming a 2D diagram)[ ] Integer multiple[ ] Wavelength

Example: X-radiation ( is incident on a crystal with adjacent atomic planes spaced ( apart. Find the three smallest angles for which constructive interference will occur.

X-Ray DifractionSunday, October 10, 20107:34 PM


Initially there was no [y] component, thus for there to be a [y] component a force must be applied. Every action has an equal and opposite reaction, so the [y] component of momentum of the photon must equal the [y] component of momentum of the electron.

Photon

Electron

Compton Shift

Compton EffectSunday, October 10, 20108:05 PM


Photons of 1.02 MeV are scattered from electrons in a metal at rest. The scattering is symmetric, find the angle and energy of the scattered photons.

Solve for E'

Proof a photon cant transfer all its energy to a free electron.

Initial = Final

assuming: In the final condition all kinetic energy from photon is transferred to electron.

[ ] momentum of photon[ '] momentum of electron

Compton ExampleSunday, October 17, 201012:53 AM


Effective inertial mass determines how the photon responds to an applied force. Gravitational mass determines the force of gravitational attraction.

Gravity and LightSunday, October 10, 20108:59 PM


We can deduce that the alpha radiation experienced a force for its momentum to change. Assuming the force acting on the particle is columbic & all charges are concentrated in a small radius at the nucleus, we derive a formula...

b

Au

(impact parameter)

Integrate and solve for [b]

Rutherford ScatteringSunday, October 10, 20109:20 PM


Au

b

Cross Section ( )

A

microscope

(A') gold foil R

Predict amount of radiation to detect in any given location

Then substitute

[ ] /sec from source

[ ] /sec scattered at specific area

[ ] # nuclei per unit area of foil

(proportional to thickness)

[ ] area of detector

[ ] coulomb constant

[ ] nuclear charge, # of electrons

[ ] distance from foil to detector

[ ] angle of detector measured from

axis parallel to source

[ ] elementary charge

[ ] kinetic energy of (2 ) radiation

[ ] mass 1 atom

[ ] density

[ ] thickness

[ ] mass of foil

[ ] # atoms, # nuclei

Rutherford Scattering

Skipped steps...

Rutherford ScatteringSunday, October 17, 20101:00 AM


Electrons are assumed to orbit without radiating "stationary state".

Angular momentum is quantized.

Substituting velocity solved from equating Coulombic force to centripetal force into quantized angular momentum equation, we have an expression for

allowed orbit radius where [ ] is any real integer.

Allowed Orbital Radius

[ ] Integers. 1, 2, 3…[ ] Coulomb constant[ ] Radius of [ ] Bohr Radius

[ ] Bohr Radius ( = 1)

Analyzing total system energy (negative because potential energy is zero at infinity) and substituting the radius for quantized radius expression.

Allowed Energy

or

or

Bohr AtomFriday, October 22, 201012:20 AM


The change in energy when the electron changes orbit, can be interpreted as the energy released when the electron goes to a lower state (smaller ) or the energy required to be in an excited state (larger ).

Rydberg Constant

[ ] Nuclear mass[ ] "Reduced mass" [ ] Nuclear charge [ ] Rydberg Constant

Generalized Equation

Important concept: The equations can only be used on Hydrogen or "Hydrongenic" one-electron atom.

The Generalized Equation takes the effect of nuclear mass and the different nuclear charge of "Hydrongenic" atoms into account.

Also...Example: A hydrogen atom at n=3 decays to the ground state with emission of a photon. Find: wavelength, recoil energy & momentum of atom.

Photon momentum = Atom momentum

Rydberg ConstantSunday, November 07, 20105:57 PM


"because photons have wave and particle characteristics, perhaps all forms of matter have wave as well as particle properties"

De Broglie wavelength

Recall...

Electron matter waves bent into a circle around the nucleus can only support integral number of wavelengths that fits exactly into the circumference of the orbit, otherwise destructive interference.

This is precisely the Bohr condition for quantization of angular momentum.

Example: What is the Broglie wavelength of a 140g baseball traveling at a speed of 27 m/s ?

Note: Must use relativistic momentum when velocity is fast (also when there is a big ratio of momentum to mass).

De BroglieSunday, November 07, 20106:39 PM


Particle/Group WavesSunday, November 07, 20108:03 PM


It is impossible to determine simultaneously with unlimited precision the position and momentum of a particle.

Momentum-PositionUncertainty

Energy-TimeUncertainty

Example: Estimate the kinetic energy of an electron confined within a nucleus of size 100pm.

Heisenberg Uncertainty PrincipalSunday, November 07, 20108:38 PM


[ ] Is the "complex conjugate" of

Complex number contains two bits of information, real & imaginary.

(all real)

Single valued-

Continuous-

Remain finite-

Over an infinite interval, probability must equal 1

-

Must be:

Greek letter "Psi"

Sunday, November 07, 20104:33 PM


The probability that a particle will be found in the infinitesimal interval dx about point x, denoted by P(x), is:

"Normalized" when we equate it to 1. (means there is a 100% probability that the particle is in the integrated range)

The fundamental problem in quantum mechanics is this: Given the wavefunction at some initial instant, say t=0, find the wavefunction at any subsequent time t.

Example: Normalize the function and find the probability of the particle being within ( a/2 ).

a-a

Given

Normalized

Evaluate integral, equate to one & solve for A.

Or 87.5%

Born InterpretationSunday, November 07, 20109:44 PM


A differential equation whose solutions give

for a particle as its acted by a force.

Connecting Equations "C.E."

Instead of force, consider the potential energy UE.

Starting point: Total = Kinetic + Potential

Substituting with "C.E."

[1] [2]

Substitute [ ] & [ ] into equation [2](note: * = -1)

Schrödinger's Wave Equation

Schrödinger WaveSunday, November 07, 201011:09 PM


The Schrödinger equation propagates the initial wave forward in time. at , the initial rate of change of the wavefunction. From this we compute the wavefunction a short time, , later as . This allows the LHS to be reevaluated, now at . With each such repetition, is advanced another step into the future. Continuing the process generates at any later time .

Assume solutions exist of form:

Starting point: Substitute into Schrödinger wave equation

[1]

[2] Notice [2] the LHS is a function of only, and the RHS is a function of only. Since we can assign any value

of independently of , the two sides can only be equal if each is equal to the same constant, denoted

[E]. This leads to equation [3].

[3]

Substitute [3] into [2]

Time-Independent Schrödinger equation

Schrödinger Time-IndependentMonday, November 08, 20101:25 AM


For a free particle with no forces acting on it confined in a "box" of length . Because it is confined to the box, the particle can never be found outside, which requires to be zero

in the exterior regions and . Inside the box, .

Rearranging the Time-Independent Schrödinger equation:

With

The solution to is:

Allowed Energies for a Particle in a Box

The lowest energy state allowed is , which is called the ground state or zero-point energy.

Energy QuantizationSunday, November 14, 20104:53 PM


For each value of the quantum number there is a specific wavefunction describing the state of the particle with energy .

Recall conditions:

Stationary States for a Particle in a Box

Probability density:

Normalizing:

Wavefunction QuantizationSunday, November 14, 20107:11 PM


Modern Physics I - Thomaz Notes

Documents

Transcript of Modern Physics I - Thomaz Notes