= "h-bar" = Planck's constant/(2 Pi) = h /(2 Pi) 1 A = 1 Angstrom = 10^-10 m = 10^-8 cm (unit of length)
1 F = 1 Fermi = 1 femtometer = 10^-15 m = 10^-13 cm (unit of length)
1 T = 1 Tesla = 10^4 gauss (MKS unit of magnetic field, gauss is cgs)
1 amu = 1 atomic mass unit = 931.5 MeV = 1.49*10^-10 J = 1.49 * 10^-3 erg
m_e = electron mass = 0.511 MeV = 8.19 *10^-14 J = 8.19*10^-7 erg
c = speed of light = 3*10^8 m/s = 3*10^10 cm/s
= "h-bar" = Planck's constant/(2 Pi) = h /(2 Pi)
= 6.626*10^-34 J sec/(2 Pi) = 1.054*10^-34 J sec
c
(= hbar c) = 200 MeV fm = 2000 ev Angstroms. (Handy to remember!)
Charge of an electron, e = 1.602*10^-19 C (MKS) = 4.803 *10^-10 esu (cgs)
I'll try to stick to cgs units when dealing with E+M, so there are no
's,
and for example
(units on that last equation are
[dynes] = [esu * statvolts/cm] + [esu * (cm/sec) * gauss / (cm/sec)]
In cgs units, e^2/(
c)
= 1/137 =
,
the fine structure constant
We will make a lot of use of "unit analysis" in this class - we'll have nasty
formulas involving tons of fundamental constants like e and
and c and m_e: rather than plugging in cgs values everywhere (this inevitably
leads to calculator mistakes!) we try to massage things up in terms of a few
basic easy-to-remember combinations, especially
Example: The Rydberg constant is
(an energy scale) How big is the Rydberg? Don't plug in values for m_e,
e, and
yet! First, rewrite it as
Another example: Potential energy between two unit charges in an atom:
_____________________________________________________________
Syllabus Section I - PARTICLE PROPERTIES OF WAVES
In ordinary life, the concepts of particles and waves are not at all ambiguous, or even very mysterious. A pebble is a particle. It carries energy and momentum. The ripples in a pond it creates when you drop it are waves. These also carry energy and momentum. Beyond that, they seem like completely distinct beasts, each with their own chapters in 1110.
The underlying physics of both water waves and pebbles is that of atoms, molecules, electrons, and nuclei. In that world, we will discover there are neither "particles" nor "waves". Electrons may seem particle-like (I do often think of them as teensy billiard balls, with charge) They have mass, charge, travel along, following laws of mechanics. But they also *diffract*, they spread out like ripples in a pond, they interfere rather than bounce (sometimes). They have wave manifestations. This wave-particle duality is key to understanding modern physics, and we will spend much time learning about this important and fundamental idea. We begin with the most basic of waves, electromagnetic waves, and will first recall why we think of them as waves, and then learn how they sometimes manifest "particle properties".
Maxwell, in the mid 1800's, knew (e.g. from Faraday) that changing magnetic
fields induce currents in loops. (currents are easy to measure!) Maxwell
postulated that changing electric fields induce B fields, and if so,
then those changing B fields themselves make a changing E ->
B -> E -> ... This creates a self-propagating disturbance
of fluctuating E and B fields. From Maxwell's equations (in MKS
units), one discovers
(and same for B), i.e. the fields satisfy a wave equation, with velocity
=
2.998*10^8 m/s. This is c, the speed of light (!!) Conclusion: surely light is
an electromagnetic wave. (There are of course other types of EM wave - radio
waves, microwaves, x-rays, gamma-rays, UV, etc...)
E and B fields, and thus the waves they make, obey superposition. When 2 or more waves of the same nature travel past a point at the same time, the resulting value there is the arithmetic sum of the values of the 2 waves. Since E and B can be positive or negative, one can have waves which constructively interfere (two troughs, or two peaks, reach the point at the same time and add up), or they can destructively interfere (a trough and peak arrive at the same time, and CANCEL)
Water waves behave like this too (if their amplitudes aren't too high). Same for sound. This results in e.g. the classic Young's two slit interference pattern (seen first in 1801)
(Front view of diffraction pattern: see Beiser, or my photocopied notes - my gif file didn't come out!)
A useful (but not so well known trig identity) says that
cos a + cos b = 2 cos (a+b)/2 cos (a-b)/2, so
This means adding any 2 waves with frequency omega (but any phase) still gives a wave of frequency omega, but the amplitude is anything from 0 to 2, depending on the relative phases. So adding waves still gives a wave, but it can interfere constructively (coefficient is 2), destructively (coeff is 0), or anything in between.
Also, waves going though a single slit act like each point of the slit is a source (Huygen's principle), and those (many) outgoing waves interfere and cause diffraction. (i.e, light "bends" around corners)
So, in the 1860's, Newton's corpuscular view of light was totally discredited. It was clear that light is a wave! It interferes, it diffracts, it refracts. It can be polarized. Intensities could have any values. Prisms, lenses, gratings, were all understood. And the underlying theory, Maxwell's equations, explained it all. But then...
PHOTOELECTRIC EFFECT (F+T 1.6, Beiser 2.2)
In the 1880's, Hertz was proving that light is an electromagnetic wave. (He put AC current across a gap, sparking once/cycle) Hertz measured wavelength, and velocity, he found E and B fields, etc., etc. He could refract, reflect, diffract the waves. Everything was fitting in just as predicted by Maxwell's equations. Later, he noticed that sparks are easier to get if one aims UV light (deep purple, "tanning" light) at the metal. Why? He didn't follow it up... which was ironic, because his experiment showing that light is a wave was thus also giving a hint that that's not all there is to it! The reason was that the UV light was creating (ionizing) electrons. Let's look at this effect in more detail, because so far it shouldn't be clear why this has anything to do with a "particle nature" of light...
If V is negative (flip the battery), electrons are naturally accelerated by the E field, and one measures a current. (If you move the plates closer: more and more current, until you reach a max => almost all the electrons are being collected) Curiously, even if V is positive (as shown), and you shine light, you still get a current. As V increases, you get less and less current, until finally at some V_0, you get nothing. (Interpretation: e V_0 is the maximum kinetic energy of the electrons)
(Plot of current vs. voltage in above setup)
Is this so surprising? Light carries energy, and some of that energy could knock into electrons. Like waves hitting a beach, rolling pebbles around, or even knocking them "uphill". If the electrons get enough energy (more than e V) then they can make it across to the cathode. So, maybe this result isn't so surprising...
BUT WAIT! Bright light has more energy, so why doesn't V_0 depend on the intensity of the light?? It's like a tidal wave jiggles pebbles exactly the same as little teeny waves do. How can the intensity affect the rate of ejection of electrons, (brighter light does give more current) but not their energy? (brighter light does not raise V_0)
ALSO, as soon as you turn on the light, electrons start to be emitted. Even if the light is very dim. Even if it is so dim, that the energy/unit area hitting the plate is tiny. Waves are generally spatially big, and deposit their energy over large areas. To knock out an electron, you need to give it lots of energy. Let's check this quantitatively: Suppose you have 10^-6 W/m^2 of light energy, hitting the whole anode. Say 10^20 atoms/m^2 are on the surface (this corresponds to about one atom/Angstrom) So each atom gets to absorb (10^-6 W/m^2)/(10^20 atoms/m^2) = 10^-26 W/atom. Now, to get 1 eV of energy = 1.6*10^-19 J, you need 1.6*10^7 sec = 0.5 years!! In 10^-9 sec (experimental accuracy) we've only given about 10^-16 eV/atom, which is not enough to get them out of the metal anode, let alone fly across the vacuum tube *against* a voltage drop. How can this be? Why do we see the electrons so quickly?
THERE'S MORE! This observed current (the "photo-electric" effect) also depends on the frequency (color) of the light used.
There is also a critical color, or frequency, nu_0, below which NO electrons at all are emitted, no matter how bright the light is! For higher frequencies (nu_1-3 in the figure) the emitted electrons have a range of energies up to some maximum (e V_0 in the figure) This maximum increases linearly with the frequency.
So, faint blue light gives electrons more energy than bright red light (although, the latter does yield more of them) The last plot is linear, so
e V_0 = Max KE of electron
Let's call the proportionality constant "h". So KE_max = h (nu-nu_0) Note that
different materials have different nu_0's, but the slope (h) is always the same
for all materials. Experimentally, h=6.626*10^-34 J sec. So we have several
puzzles: why are the photoelectrons emitted so quickly? Why doesn't higher
intensity light give the electrons more energy? Why does the maximum energy of
the electron grow linearly with the frequency of the light? Why is the slope of
the energy-frequency relation apparently independent of the type of metal?
Einstein resolved these puzzles (1905) using an idea originated by Max Planck in 1900 from a completely different phenomenon (we'll talk about it soon) - he postulated the Quantum Theory of Light. Namely, despite all the successes and experimental evidence that light is a wave, this seems irreconcilable with the photoelectric effect.
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