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Overview of Modern Physics: Quantum Mechanics & Special Relativity
- Reading Memo Insights:
- Summary of Important Equations to understand for the HW:
- c = λ · f
- E = h · f
- λ = h/mv
- Δt' = Δt/√(1 - v2/c2)
- E = m · c2
- 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 something called interference -- Double Slit Movie and Great Java Applet.
- 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.
- This property, interference, only happens for waves! We can represent it graphically by plotting the big and small parts as a graph:
- Light was, at this time, 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 two slits? What does that look like? Just two separate curves...
- 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!
- 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, which 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...
- Three problems at the turn of the century were the first
clues that all was not how it seemed
- Ultra-Violet Catastrophe
- As temperature of a BlackBody goes up, more energy emitted per second at each wavelength and the peak wavelength shifts to smaller values
- Plot of UV catastrophe versus reality (reality is what we studied 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 (more intense) 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 e- energies
- Higher frequency light ejected e- with higher energy
- 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?
- 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?
- Ultra-Violet Catastrophe
- The 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!
- 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 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
- 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 h = 6.63 x 10-34J-s = 4.136 x 10-15eV/Hz
- 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)
- Einstein proposed that light is also quantized
and its energy is also determined by its frequency via E = hf
- 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)
- 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 → &lambda = hv/mv2 = h/mv (see http://www.chemistrycoach.com/BohrAssump.htm)
- Energy is quantized
- 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 waves; another way, they're particles!
- 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 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
- 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 f = ?Hz nf = 2 λ = ?m Ei = eV Ef = eV h = 6.63 x 10-34J-s = 4.136 x 10-15eV/Hz c = 3 x 108m/s
- 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)
- When white light hits an atom, only photons with the
right energy are absorbed!
- 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.)
- 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)
- Special Relativity
- Galilean relativity: throw ball at speeding truck
- Repeat with flashlight and light: you see same speed of light!
- Einstein noticed 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
- "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)
- "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!
- Consequence of reconciling them (i.e., invariance of Maxwell's
equations) leads to the constancy of the speed of
light
- Principle of Relativity: laws of physics are the same for all observers moving uniformly
- Special Theory of Relativity: two observers, in
uniform relative motion, perceive space and time differently
- Transparency #4: Fig. 12.4 on p. 450 (reproduce)
- Synchronize two light clocks
- Spaceship moves with v relative to observer on Earth
- Light path on spaceship, according to observer on Earth, follows longer zig-zag path
- Since d/t = v = c and c is same, longer d means longer t = d/c → time dilation (as measured by the people on Earth!)
- But to the people on the ship, 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)
- Time dilation only happens at high speeds:
- Transparency #5: Fig. 12.5 on p. 451
- Experimental confirmation: muon decay times (from
collision of cosmic rays with atmospheric molecules)
- In-Class Exercise #3: If the mean lifetime
of a muon, as measured in the laboratory, is
2.2 x 10-5sec, then what is its speed,
with respect to the laboratory? Note:
the mean lifetime of a muon at rest is 2.2 x
10-6sec. (see p. 451)
Known Unknown Δt' = 2.2 x 10-5sec v = ?m/s Δt = 2.2 x 10-6sec c = 3 x 108m/s
- In-Class Exercise #3: If the mean lifetime
of a muon, as measured in the laboratory, is
2.2 x 10-5sec, then what is its speed,
with respect to the laboratory? Note:
the mean lifetime of a muon at rest is 2.2 x
10-6sec. (see p. 451)
- Another consequence: length contraction
- ΔL' = ΔL • √(1 - v2/c2)
- Yet another consequence: rest energy is Eo = mo c2
- This means, according to Einstein himself, "[Mass and
energy] are only different expressions for the same
thing."
- In inelastic collisions, mass is transformed into energy and energy may be converted into mass because mass and energy are equivalent!
- Erel = KErel + mc2 = mc2/√(1 - v2/c2)
- Correspondence Principle: KErel reduces to ½ mv2 for low speeds (for higher speeds, higher the speed, higher the energy/mass)
- Forces and Particles:
Four Known Forces Gravitation ElectroMagnetic Strong Nuclear Force
(Nucleus)Weak Nuclear Force
(Neutron/β decay)