Overview of Modern Physics Quantum Mechanics & Special Relativity

Summary of Important Equations to understand for the HW:

1. c = λ · f

2. E = h · f

3. λ = h/mv

4. Δt' = Δt/√(1 - v2/c

2)

5. E = m · c2

Three ways to transfer energy: With a Particle

With a Wave

With a Field

Summary of Waves Properties:

Wavelength (): distance between two successive crests Frequency (f): number of wave crests passing per second Speed of wave: v=f Amplitude (A): maximum displacement Energy: E A2

Interference

Only waves experience interference this is just adding up the parts of a wave:

Where two crests (or troughs) meet, they add

Where a crest and trough meet, they cancel

Introduction (Reference: The Feynman Lectures, Vol. 1, p. 37-3) Water waves, like all waves, experience interference when

diffracted by two slits As seen here http://www.warren-wilson.edu/~escerbo/Diffraction/2-slit.htm and

here http://www.phy.davidson.edu/introlabs/labs220-230/html/lab10diffract.htm

Imagine a pond in which we start a water wave ripple. The ripple starts out near the center and spreads radially outward (Water Ripples Movie and also video

from The Examined Life) Very far away, the water wave ripple will look like a straight

line. Something interesting happens to these plane waves (in this case, water waves) when they hit two slits.

The water wave ripple will divide up into two new ripples when it hits the slits and, if the slits are spaced just right, the two new water wave ripples will experience interference (Double Slit Movie and Great Java Applet)

We can represent it graphically by plotting the big and small parts as a graph:

Light: Particles vs. Waves Light was originally thought to be a particle

E.g., look at sharp shadows, photoelectric effect, etc. also, Newton endorsed the view that light was made up of particles

Particles behave completely differently when they encounter slits...

E.g., suppose you fired bullets at one slit? What does that look like? Just a simple curve...

What if you fired bullets at two slits? Just two separate curves

So what happens when you fire light at it? Interference like water waves!

Quantum Particle: Wave AND Particle Okay, so maybe light is a wave since it exhibits this wave behaviour.

But we also know light is made up of particles (photons). So okay, light is nutty... light, like all matter, is actually a quantum

particle It exhibits both wave and particle aspects (wave-particle duality)

But that's not all there is to the quantum strangeness... what happens when you fire electron (quantum) particles? Same interference pattern even though you detect individual

electron "particles" at the other end! Well, then electrons are simple waves, right? No!

Reduce flow to a single electron (or photon) at a time and you still get the same pattern!

And we definitely pick up particles at the other end (particle detectors) Each particle somehow knows where it should go...

welcome to the quantum reality, welcome to the real nature of existence itself, which we will now explore in detail...

What Happens at the Slits?

-Top is what we get when slit 1 is covered

-Bottom is what we get when slit 2 is covered

-What do we get when both are open?

What Happens at the Slits?

• The blue curve is what we would get if each electron made a random choice of which slit to pass through

• The orange curve is what we actually get

• Conclusion?

• The electron passes through both slits at the same time

• The electron behaves both as a wave and as a particle

Something is rotten in the state of Denmark

Three problems at the turn of the century were the first clues that all was not how it seemed

Ultra-Violet Catastrophe As the temperature of a BlackBody goes up, more

energy is emitted per second at each wavelength and the peak wavelength shifts to smaller values

Plot of UV catastrophe versus reality (experimental results are shown in Ch. 8 and is also shown in Fig. 10.1 on p. 373 and here)

Why doesn't theory match reality (experiment)?

The Photoelectric Effect• When light hits certain metals, e- are ejected

• Hypothesis: brighter (higher intensity) light should cause more e- to be ejected with higher energies

• But, more light caused more e- to be ejected but it didn't increase their energy

• Only the frequency (colour) of the light affected the energy of the ejected e-

• Higher frequency light ejected e- with higher energy

• But even extremely dim light of the right frequency caused e- to be emitted Why does the energy of the ejected e- depend on the frequency of the light?

Can We Explain This?

Predictions of the Wave Model:

-For high enough intensity, any wavelength of light should produce the photoelectric effect.

-The maximum kinetic energy of the photoelectrons should scale with light intensity.

-The maximum kinetic energy should be independent of the frequency.

-The photoelectrons need a measurable amount of time to gain enough energy to be ejected.

Not what we see.

Not what we see.

Not what we see.

Not what we see.

The Atomic Spectra

Light from bulbs, stars, etc. show a continuous spectrum when seen through a prism

A hot gas, however, has an emission line spectrum made of a few, discrete lines of colour

Why don't hot gases show continuous spectra?

AbsorptionEmission

Quantum Hypothesis: Energy is quantized What is quantization? It only comes in discrete chunks instead of a

continuous range of energies Transparency #1: Figure 10.3 on p. 373

If you assume the Energy of each atomic oscillator is quantized, you can get the correct BlackBody curve

Planck suggested Energy is quantized in units of h and was proportional to the oscillators frequency: E = hf As is common in physics, he originally just came up with an equation

to fit the curve without knowing anything about the underlying mechanism (he addressed the what but not the how)

This quantization of energy arose as a necessary condition of the equation Planck derived to fit the correct curve

Although he developed a model (energy is quantized) he had no idea why this should be so!

Continuous Discrete

Quantum Hypothesis: Light is quantized Einstein proposed that light is also quantized and its energy

is also determined by its frequency via E = hf Each individual packet of light energy is called a photon and

an EM wave is made of these individual "particles" Brighter light → more photons strike metal each second →

more e- ejected/sec (but it does not increase the energy of each e-)

Higher frequency light ejects e- with more energy because each photon has more energy to give

E hf346.626 10 J sh

maxK hf

Quantum Hypothesis: Orbits are quantized Bohr suggested that the orbits of electrons are also quantized An electron can go from one level to another by absorbing or emitting a

photon of light If light energy is quantized and electron orbits are also quantized, that

would explain why atomic spectra are discrete (since atoms/electrons only absorb or emit a single photon at a time)

The Bohr Model of the Hydrogen Atom

Energy, Light, & Orbits are quantized!

So all three problems at the turn of the century had the same solution: quantization of the fundamental aspects of nature!

This is why it's called quantum mechanics: everything is quantized (comes in discrete chunks instead of a continuous range of values) -- matter (the Bohr "orbitals"), light, and even energy are ALL quantized!

DeBroglie further hypothesized that since electrons also behave as waves, they must also have a wavelength: λ = h/mv This was part of his doctoral thesis which won him a Nobel

prize He also came to physics late in life, contrary to the popular

notion of physics being a young man's game E = mc2 = hf

c=velectron and f = v/λ

→ mv2 = hv/λ → λ = hv/mv2 = h/mv (see http://www.chemistrycoach.com/BohrAssump.htm)

electron wave

In Class Exercise #1: Compare the energies associated with a quantum of:

IR light (f = 3 x 1013Hz), Blue light (f = 6.3 x 1014Hz), and X-Rays (f = 5 x 1018Hz)

Note: h = 6.63 x 10-34J-s = 4.136 x 10-15eV/Hz (see p. 375)

Known Unknown

fIR = 3 x 1013Hz; fBlue = 6.3 x 1014Hz; fX-Rays = 5 x 1018Hz E = ?J

• E = hf

h = 6.63 x 10-34J-s = 4.136 x 10-15eV/Hz

Quantum Mechanics in a nutshell Matter, it seems, is nutty at the sub-microscopic level Quantum particles behave like both waves and particles; even energy

comes in packets or chunks! Look at them one way, they're particles;

another way, they're waves! Transparency #2: Fig. 10.6 on p. 375 (reproduce) If you pass light through slits, it's a wave;

if you aim it at metals, it behaves like a particle "They could but make the best of it, and went around with

woebegone faces sadly complaining that on Mondays, Wednesdays, and Fridays they must look on light as a wave; on Tuesdays, Thursdays, and Saturdays as a particle. On Sundays, they simply prayed." -- Banesh Hoffmann

Electrons are the same way: they behave as both particles and waves All particles have a wave-aspect; higher the momentum, shorter the

wavelength But the incredibly tiny value of h ensures this is only a microscopic

effect

Heisenberg Uncertainty Principle Heisenberg proposed that the wave aspect of an electron

makes it impossible to know both the position and momentum to arbitrary precision

Heisenberg Uncertainty Principle (HUP): Δx • Δ(mv) ≥ h/4π

E.g., if you have a periodic wave (or a standing wave) you can't really tell what its position is (it's spread out over the whole string, e.g.). But you can tell exactly what its wavelength is. Now if you send a wave pulse down the string, you can't tell what its wavelength is (doesn't make sense for a pulse) but you can tell exactly what its position is. (With thanks to Prof. Griffiths)

The Atomic Structure So we can't say where exactly the electron is (it's not like a billiard ball,

or like a wave, or like a puffy cloud, or like anything else we know from ordinary experience)

"Now we know how the electrons and light behave. But what can I call it? If I say they behave like particles I give the wrong impression; also if I say they behave like waves. They behave in their own inimitable way, which technically could be called a quantum mechanical way. They behave in a way that is like nothing that you have ever seen before. Your experience with things that you have seen before is incomplete. The behavior of things on a very tiny scale is simply different. An atom does not behave like a weight hanging on a spring and oscillating. Nor does it behave like a miniature representation of the solar system with little planets going around in orbits. Nor does it appear to be somewhat like a cloud or fog of some sort surrounding the nucleus. It behaves like nothing you have ever seen before." -- Richard P. Feynman, The Character of Physical Law

Since we can't talk about its exact location, it's more useful to concentrate on the electron's energy

Predicted Energy Levels Instead of looking at orbits, we now look at energy

levels, which are the certain, allowed energy states Lowest energy level (corresponding to innermost

orbit in Bohr theory) is called the ground state and higher energy states are excited states

The structure of the atom is shown schematically on an energy-level diagram labeled with a quantum number n Transparency #3: Fig. 10.31 on p. 390

As quantum number ↑, Energy associated with that state ↑

Transition of the electron from one orbit to another is now represented as the atom going from one energy level to another

Transition achieved by absorption or emission of a photon with an energy corresponding to the difference in energy between the two levels, or states When white light hits an atom, only photons with the

right energy are absorbed!

In Class Exercise #2: What is the frequency and wavelength of the photon absorbed by a

hydrogen atom that takes it from the n=1 state to the n=2 state (see Fig. 10.30 on p. 390)? Note: c = λ f (see p. 391)

Known Unknown

ni = 1

nf = 2

f = ?Hz

• E = E2 – E1 = hf f = E/h

• c = f = c/f

• = hc/E

Ei = E1 = eV

Ef = E2 = eV

h = 6.63 x 10-34J-s = 4.136 x 10-15eV/Hz

c = 3 x 108m/s

λ = ?m

Some Atomic Physics Atom can gain or lose energy by absorption or

emission of photons or by collisions Pauli Exclusion Principle: two electrons cannot occupy

the same quantum state at the same time Number of quantum states in a given energy level

given by 2n2 If even one electron is in a higher energy level, the atom

is said to be in an excited state Properties of each element determined by the ground-

state configuration of its atoms (e.g., valence electrons, etc.)

What about relativity?

Everything is relative…

In conversation, this implies there is no correct answer, no preferred point of view, nothing is fixed or constant.

In physics, the meaning is quite different: if I know how your motion and mine differ, I can predict exactly what you will see when you observe me.

Some things look the same to all viewers everywhere at any time (invariants). Others do not. Thus, very important to find these invariants.

One essence of science: PREDICTIONS

Inertial Reference Frames

Describing a Physical Phenomenon requires:

1. Event

2. Observer

3. Frame of reference (the Point of View!)

A reference frame is a coordinate system and a clock.

An inertial reference frame is one that is not accelerated.

Newton’s Laws are valid in inertial reference frames only.

An underlying assumption in Newtonian mechanics: The laws of physics are the same in all inertial reference frames (Relativity).

Galilean vs. Special Relativity Galilean relativity: throw ball from a speeding truck and the

speeds add. In the late 19th century, Michelson and Morley designed an

experiment that replaced the football with light, and the truck was replaced by the entire Earth. 

But what they found was not the obvious solution they found that light travelled at a constant speed. If the quarterback shines a flashlight from a standing position, it

goes at the same speed as if the quarterback shines the flashlight from the back of a moving truck.

Invariance of Maxwell’s Equations (Skip) Einstein noticed a contradiction between

classical mechanics and electromagnetism Consequence of reconciling them (i.e.,

invariance of Maxwell's equations) leads to the constancy of the speed of light

Maxwell’s equations showed that light is: An electromagnetic wave EM waves don’t need a medium EM waves propagate at speed c

They also said that EM waves ALWAYS propagate at the same speed, c

Newtonian mechanics predicts that any motion relative to empty space will reveal a different value for the speed of electromagnetic waves. THIS HAS NEVER BEEN OBSERVED

Maxwell’s equations say: changing E changing B changing E

They look like:

Laws of Mechanics must be the same in all Inertial Frames of References (Skip) No Experiment involving laws of mechanics can differentiate between any two inertial frames of reference Only the relative motion of one frame of ref. w.r.t other can be detected Notion of ABSOLUTE motion is meaningless There is no such thing as a preferred frame of reference Some good links:

"Unlike Newton's equation, however, Maxwell's equations were not invariant under the Galilei transformation. This is evident from the fact that Maxwell's equations predicted the speed of light. But the speed of something depends on which inertial frame you were observing it from. So Maxwell's equations could only be correct in one particular inertial frame!This did not go well with the belief that the laws of physics should be the same regardless of which inertial frame you were making the observation from." (See http://www.phys.vt.edu/~hcp/special_relativity/notes/section7.html)

"Another great problem was that Maxwell's equations did not appear to obey the principle of Galelian Relativity i.e. they were not invariant under the Galelian transformations. This means that in a moving space ship the electric and optical phenomena should be different from those in a stationary ship!!! Thus one could use for example optical phenomena to determine the speed of the ship. One of the consequences of Maxwell's equations is that if there is a disturbance in the field such that light is generated, these electromagnetic waves go out in all directions equally and at the same speed c. Another consequence of the equations is that if the source of the disturbance is moving, the light emitted goes through space at the same speed c. This is analogous to the case of sound being likewise independent of the motion of the source. This independence of the motion of the source in the case of light brings up an interesting problem." (See http://vishnu.mth.uct.ac.za/omei/gr/chap1/node2.html)

"Einstein, in 1905, explained this observation in the basis of the assertion that the laws of nature including Maxwell's equations with the same velocity of light are the same to you whether you are moving or not.Fitzgerald and Lorentz had shown how to modify the equations of ordinary mechanics to give them the same invariance properties as Maxwell's equations." (see http://www-math.mit.edu/18.013A/chapter29/section06.html) "The theory of Electromagnetism, summarized and completed by J. Maxwell (1831-1879), appeared to alter the situation. Maxwell's equations allow a wave solution which represents electromagnetic waves propagating with the same speed in all directions. The Galilean transformation cannot retain the constancy of the speed of light and there could be only one inertial frame in which the light travels with the same speed in all directions. One could designate this frame the absolute rest frame. However, all experimental searches for this frame have failed. Einstein accepted the constancy of the speed of light in all inertial frames as a postulate and showed that the coordinate transformation between inertial frames has to be the Lorentz transformation to retain the invariance of the speed of light. All inertial frames are on the same footing again and the concept of relativity can remain. But the problem is that Newton's equation of motion, being invariant under the Galilean transformation, is not invariant under the Lorentz transformation. In the process of rescuing the concept of relativity in Electromagnetism, one in turn faces the possibility of ruining the concept of relativity in Mechanics. Einstein, postulating that all laws of physics are the same in all inertial reference frames, chose to modify Newton's equation of motion and invented the Relativistic Dynamics which is invariant under the Lorentz transformation." (See http://hepth.hanyang.ac.kr/~kst/lect/relativity/c13.htm)

"Maxwell's equations are not invariant with respect to the Galilean transformation. That transformation introduced extra terms into the form of the equations, and this meant that different inertial observers would observe different electromagnetic effects and therefore, by performing a suitable experiment you would be able to determine your speed with respect to the ether. Several experiments were done in order to observe the effects that these extra terms introduced, in an attempt to measure the speed of the ether wind. Naturally, they all failed to discover those effects, thus people began to believe that, somehow, Maxwell's equations were wrong. Interestingly enough, Maxwell's equations turn out to be invariant with respect to the Lorentz transformation. This was all very confusing. Apparently, either Maxwell's equations or the Galilean transformation had to be wrong. They couldn't possibly both be correct!" (See http://mathforum.org/library/drmath/view/56248.html)

See also http://www.geocities.com/autotheist/Physics/

Foundation of Relativity: The laws of physics are the same for all observers

moving uniformly in an inertial reference frame I.e., no special places in the universe (continuation of

Copernican revolution) Galileo, Newton, and Einstein all agree that mechanics looks

the same in all inertial frames Einstein added that electromagnetism looks identical in all

inertial frames (therefore, c is invariant) Galileo & Newton thought time and spatial length are invariant Einstein, noticing the invariance of c, suggested what’s invariant

is the “interval”: 2 = c2t2 – x2

In addition, time becomes just another coordinate (albeit imaginary)

• Perceived order in which events occur depends on the observer’s frame of reference

Special Theory of Relativity

Consequences of the invariance of the speed of light, c: Two observers, in uniform relative motion, find space and time

behave differently for them These effects are:

Time dilation Length contraction Mass increase

Totally real effects! Not an illusion of perspective

or apparent effect… these are VERY real consequences!

So how does this happen?

Light Clocks & Time Dilation Imagine a clock that uses light pulses When it’s motionless, the light beam goes up and down and

clicks off one second (tick-tock) Now, imagine the clock in motion… it still ticks off one

second just like the ball on the truck still goes up and down BUT, the distance travelled, for both the ball and the

clock, is much greater What does this mean?

Time Dilation: slowing down time (OLD) Synchronize two light clocks on two ships One spaceship moves with a speed, v, relative to the

observer on the stationary ship on the Earth Light path on the spaceship, according to the observer on Earth,

follows a longer zig-zag path Speed = distance/time = c (speed of light)

c is invariant

Therefore, longer distance means longer time (since t = d/c) – As measured by a stationary observer on EARTh

time dilation (as measured by the stationary observer on Earth!)

But to the people on the moving ship, the Earth observer's clock runs slower! Both observers are correct: "Time, then, is not an absolute, innate quality of nature"

Δt' = Δt/√(1 - v2

/c2

)

Original distance (tick)

Distance light travels

Distance ship travels

Time, as measured by the STATIONARY OBSERVER

Trucks to spaceships… Synchronize two light clocks on two ships One spaceship moves with a speed, v, relative to the observer on the

stationary ship on the Earth Light path on the spaceship, according to the observer on Earth, follows a

longer zig-zag path Just like the guy throwing the ball on the truck He saw the ball go straight up and down But stationary observer saw it follow a longer, parabolic path

Stationary vs. Moving So the distance of the light beam, as measured by the stationary observer on Earth, is the longer zig-zag path

But Speed = distance/time = c (speed of light)

And c is invariant and constant

Therefore, the light, as measured by the stationary observer on Earth, has to travel a longer distance

Since time = distance/(speed of light), that longer distance takes a longer time (as measured by a stationary observer on Earth)

This is time dilation (as measured by the stationary observer on Earth!)

Different observers, different strokes observations

Although the guy on the moving ship still measures the tick-tock as taking one second, the stationary observer on Earth measures that same tick-tock as taking much longer than a second.

This is just like the guy on the truck: he sees the ball go straight up an down whereas the observer on the sidewalk sees it as taking a longer, parabolic path.

So who’s right?

Both observers are correct:

"Time, then, is not an absolute, innate quality of nature"

Time Dilation: the gruesome derivation… We can derive this relationship quite easily… Distance light travels, as seen by the moving observer

(straight up and down): d = ct/2 Distance light travels, as seen by the stationary observer (the

longer zig-zag path): ct/2 Distance the ship travels, as seen by the stationary observer

(assume speed = v): vt/2 Apply Pythagorean theorem: c2 = a2 + b2 (ct/2)2 = (vt/2)2 + (ct/2)2 Rearrange: Δt' = Δt/√(1 - v

2/c

2)

Original distance (tick)

Distance light travels

Distance ship travels

Time, as measured by the STATIONARY OBSERVER

Some more consequences… Time dilation only significant at high speeds:

Transparency #5: Fig. 12.5 on p. 451 Another consequence: length contraction

ΔL' = ΔL • √(1 - v2/c2) Yet another consequence: rest energy is Eo = mo c2

In inelastic collisions, mass is transformed into energy and energy may be converted into mass because mass and energy are equivalent!

This means, according to Einstein himself, "[Mass and energy] are only different expressions for the same thing."

Erel = KErel + mc2 = mc2/√(1 - v2/c

2)

Correspondence Principle: Special Relativity reduces to Galilean Relativity at low speeds

KErel reduces to ½ mv2 for low speeds (for higher speeds, higher the speed, higher the energy/mass)

General Relativity Einstein further realized that mechanics and light

should look the same to an observer falling freely in a gravitational field as to an observer in an inertial frame! This is a more general statement of relativity. It makes natural the “principle of equivalence”

equating inertial and gravitational mass. Truly explains gravity

Matter tells space how to “bend” and space tells matter how to “move”

Space is “bent” by matter:

Newton vs. Einstein

Newton

Einstein

Why Do We Think Einstein’s Theories are Correct?

General relativity predicts exactly the observed orbits of the planets. Newtonian mechanics come close but do not exactly match observation.

We have actually observed time dilation with highly accurate synchronized clocks.

His theories account for experimental observations for which we have no other explanation (i.e. the detection of muons on the surface of the Earth).

Four Known Forces Two familiar kinds of interactions:

gravity (masses attract one another) and electromagnetism (same-sign charges repel, opposite-sign charges attract)

What causes radioactive decays of nuclei ? Must be a force weak enough to allow most atoms to

be stable. What binds protons together into nuclei ?

Must be a force strong enough to overcome repulsion due to protons’ electric charge

Previously, we peered inside the atom

We recalled that electrons orbit the atom’s massive nucleus and determine an element’s chemical behavior.

We explored the proton and neutron content of nuclei and the phenomena of radioactivity, fission, and fusion they make possible. Today we’ll look inside the

nucleons themselves. Fundamental particles in the

Standard Model are: Leptons Quarks Intermediate Gauge Bosons

Anti-matter

Each kind of elementary particle has a counterpart with the same mass, but the opposite electric charge, called its “anti-particle”. Electron: m= .0005 GeV, charge = +1, symbol e-

Positron: m = .0005 GeV, charge = -1, symbol e+

The anti-particle has a bar over its symbol: Anti-proton is written , anti-neutrino is

Anti-matter is rare in the explored universe It’s created in cosmic rays and particle accelerators

and some radioactive decays. When a particle and its anti-particle collide, they

“annihilate” one another in a flash of energy.

p

v

Where do the elements come from?

Stability diagramHeavy elements can fissioninto lighter elements.

Light elements can undergofusion into heavier elements.

Elements from helium to iron were manufactured in the cores of stars by fusion. Heavier elements are metastable and were made during supernovae explosions.

Chain reaction

For reaction to be self-sustaining, must haveCRITICAL MASS.

Nuclear reactors

Fission bomb

Fusion Light nuclei are more stable when

combined Tremendous energy released Hydrogen bombs and Fusion

power?