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The Origins of
Quantum Mechanics
1900: Max Planck discovers that electromagnetic
waves deliver their energy in small “bundles”
The colored lines in the plot above show how much light intensity is given off in
the glow of an object heated to a given temperature, as a function of its
wavelength. The best “classical” theory of the time predicted the black curve—
which was clearly wrong! Planck realized that he could match the observed
spectra perfectly if he assumed that the light energy was being emitted in small
bundles, with each bundles carrying energy proportional to the frequency of the
light. Specifically:
34
E  hf
with
h  6.6260693(11)  10
“Planck’s constant”
Js
1905: Albert Einstein shows that electromagnetic
waves are composed of particles, “photons” (g),
each of energy E = hf.
He did this by explaining the
photoelectric effect, shown at right.
Light of a single frequency is
shined onto a metal plate, ejecting
electrons from the plate. The
experiment measures the rate of
electrons arriving, and their
maximum energy.
Classical Picture
At t = 0, light is shined on the plate.
The E field of the electromagnetic
wave causes the atomic electrons to
oscillate. The oscillations build up
until the electrons break way from
their atoms and leave the metal.
What is observed
When the light is turned on, even at low
intensity, electrons begin to emerge from
the plate—with no time delay. Their rate
depends on the light intensity, but their
maximum energy depends only on the
frequency of the light.
1905: …“photons” (g), each of energy E = hf.
Think of these now as individual photons
instead of a smooth plane wave!! This is at the
core of Einstein’s explanation. We now think of
light as being made of photons. Each one
carries a specific amount of energy, E = hf. It
also has other particle properties such as
“spin”, but its mass is zero. As we go further,
we’ll see that a photon is a particle, just as
an electron is a particle, but it is also a
wave, with well defined frequency. In the
photoelectric effect, a photon collides with an
electron, and sometimes the electron receives
all the incoming energy. But even then, work is
needed to free the electron from the metal, and
the amount is specific to each metal. Einstein’s
simple equation for the photoelectric effect
takes all of these effects into account, and
explains the data. In fact, we can use the data
to accurately measure Planck’s constant, h.
g
g
g
Einstein’s explanation:
E max electron  E photon  W freeelectron
Photoelectric effect in
standard notation:
Ee  Eg    hf  
Look at data, next slide
1916: R. A. Millikan, photoelectric effect experiment
Ee  Eg    hf  
g
g
g
For the plot shown here, the metal plate was sodium, with a frequency threshold of 683
nm and a work function of 1.82 eV. Each data point corresponds to a different light
source, at the frequency shown on the horizontal axis. The vertical axis is the maximum
energy of the electron, and the slope of the plot is Planck’s constant, h, in units of eV/Hz.
Millikan also did the “oil drop experiment”, finding that charge is in units of e.
Photoelectric effect on the moon
Moon dust
Light from the sun hitting lunar dust causes it to
become charged through the photoelectric
effect. The charged dust then repels itself and
lifts off the surface of the Moon by electrostatic
levitation. This manifests itself almost like an
"atmosphere of dust", visible as a thin haze and
blurring of distant features, and visible as a dim
glow after the sun has set. This was first
photographed by the Surveyor program probes
in the 1960s. It is thought that the smallest
particles are repelled up to kilometers high, and
that the particles move in "fountains" as they
charge and discharge.
1924: Louis de Broglie puts forth the hypothesis
that particles, such as the electron, are also
waves of wavelength l = h/p  p = h/l
Notice that de Broglie put forth this idea almost 20
years after Einstein’s postulate that electromagnetic
waves are photons. But we’re purposely taking this
out of historical order to make it easy to understand
the Bohr model of the atom (that appeared in the
intervening years). De Broglie announced this idea in
his Ph.D. thesis, perhaps the most spectacular Ph.D.
in all of physics! (Story.)
How was the hypothesis verified? By the
Davisson-Germer experiment (next slide)
1927: Davisson-Germer experiment
In this experiment, electrons of
known momentum are beamed
onto a crystal. Electron waves
diffract from the crystal lattice, just
as x-rays of the same wavelength
would, giving a diffraction pattern
measured by rotating the electron
detector and plotting intensity vs.
angle.
(Clinton Davisson,
and Lester Germer)
Quanta
Waves are particles, and particle are waves.
They are really all “one type” of object, that we call “quanta”.
All quanta have energy E = hf and momentum p=h/l. What
makes them differ as particles is “additional” properties: spin,
mass, electric charge, weak charge, strong charge, parity, etc.
(Photons are an example of a mass-less particle.)
Nature forces us to the conclusion that quanta are real, but offers
no additional “guidance” to help us create a mental picture of how
quanta act as both waves and particles. The best we can do
currently is to label this two-sided behavior as “wave particle
duality”. Quantum mechanics still fascinates and mystifies the
people who work most closely with it. (Feynman quotes.)
1911: Ernest Rutherford “sees” inside the atom
Geiger and Marsden experiment, 1909
“It was almost as incredible as
if you fired a fifteen-inch shell
at a piece of tissue paper and
it came back and hit you".
Rutherford supervised Geiger and Marsden. Alpha particles
were beamed through gold atoms. The large-angle recoils
showed that there was a very small object at the center of the
atom, containing most of its mass: the nucleus.
The “classical” model of the hydrogen atom
The Rutherford scattering experiment showed that
atoms have nuclei. Physicists immediately tried
modeling the simplest one, hydrogen, as a small
“solar system”, with an electron orbiting a single
proton, using classical mechanics and E&M. This
we can do ourselves!
P
e
Assume the electron is in circular motion, and
equate the radial force to the Coulomb force:
m 2 kqP qe ke2
Fr  mar 
 2  2
r
r
r
Solve for the speed as a function of r:
2
ke
2 
mr
k
 e
mr
This is a perfectly good classical result that we shall use as a starting point. But this
model is “too classical”, and too underspecified, to describe the hydrogen atom.
The classical model of hydrogen is unstable!
Our starting point for the classical
analysis was a particle in circular
motion—meaning that it is
accelerating. But charged particle
that are accelerating will radiate
photons. This means that they are
losing kinetic energy on each
revolution. As the speed
decreases, so does the radius.
This model predicts that electrons
will spiral into the nucleus (in a tiny
fraction of a second) and that all
atoms will collapse! Poof…
I think we can all agree that this
hasn’t happened.
The long-standing mystery of dark lines in solar spectra
The English chemist William Hyde Wollaston was in 1802 the first person to note the
appearance of a number of dark features in the solar spectrum. In 1814, Fraunhofer
independently rediscovered the lines and began a systematic study and careful
measurement of the wavelength of these features. In all, he mapped over 570 lines, and
designated the principal features with the letters A through K, and weaker lines with other
letters.
It was later discovered by Kirchoff and Bunsen that each chemical element was associated
with a set of spectral lines, and deduced that the dark lines in the solar spectrum were
caused by absorption by those elements in the upper layers of the sun. Some of the
observed features are also caused by absorption in oxygen molecules in the Earth's
atmosphere.
1915: Niels Bohr “figures out” the hydrogen atom, and
“explains” absorption and emission lines
Bohr started from the classical
model, but then added what was
known from solar spectrum lines:
(1) all hydrogen atoms must be
identical, and (2) they absorb and
emit light only at specific
frequencies (corresponding to
specific energies, Eg = hf).
His major insight was that different electron orbits
correspond to different electron total energy. The
atom would emit and absorb at specific frequencies
if (1) orbits would be “allowed” only at certain radii,
(2) photons were emitted or absorbed when
electrons “jump” from one orbit to another. An
electron jumping from large radius to small would
emit a photon of energy equal to the energy
difference between orbits. Absorption is the inverse
process.
1915: Niels Bohr “figures out” the hydrogen atom, but
we’ll calculate it as de Broglie would.
Bohr fished around and found that, by restricting the
angular momentum of the orbits to certain values, he could
reproduce the hydrogen spectrum. But we are going to
start out with de Broglie’s equation for momentum, because
it is easy to see why only certain orbits are allowed.
In de Broglie’s picture, we should treat the electrons as
particle waves traveling in a circle around the nucleus.
Each orbit, these waves return on themselves, and if they
are not in phase, they will destructively interfere.
Constructive in interference will occur only when the waves
are in phase after one revolution. This means for an orbit of
a given radius, an integral number of wavelengths must fit
into one circumference:
nl  2r
h
hn
p 
 m
l 2r
hn

2mr
n = 1,2,3,…
Solving for the Bohr orbits of hydrogen
Standing waves
of the electron
in circular orbit:
n = 1,2,3, …
Take the two
formulas above,
square the velocity,
and equate:
hn

2mr
ke  hn 


mr  2mr 
2
2
Classical circular
orbits of the electron
in the Coulomb field
of the proton
k
mr
 e
2
n2  h 
2
2
r n

  n r1  n a0
km  2e 
“Bohr radii” of hydrogen, n = 1,2,3 …
2
Bohr radius of the
“ground state” of
hydrogen (n = 1):
1  h 
10
a0 

  .5291772108(18) 10 m  .529 A
km  2e 
Insert the Bohr radii,
rn, into the first
equation for v:
hn
 hn  km  2e 
n 

 2 

2mrn  2m  n  h 
o
2
2ke2
 n 
hn 2
“Bohr velocities” of hydrogen, n = 1,2,3 …
(Note how the Bohr radii and velocities change with n.)
The emission and absorption spectrum of hydrogen
Light is emitted when an electron jumps
from an orbit of large radius to a smaller
one. It is absorbed in jumps from a small
radius to a larger one. The photon
energy equals the magnitude of the
electron energy difference before and
after the jump, which in turn sets the
photon wavelength. The pattern of
wavelengths for hydrogen was measured
in the early 1800’s, and is summarized by
the formula at the right, found by Rydberg
in the 1890’s. (Also Balmer.)
With n > m:
91.18 nm
l
1 
 1

 2
2 
n 
m
m=1: Lyman series (ultraviolet)
m=2: Balmer series (visible)
m=3: Paschen series (infrared)
Spectrum of hydrogen: first we
need the electron energy levels
Before calculating the spectrum of hydrogen, we
need to find the “electron energy levels” of
hydrogen. This calculation must include both the
electron’s kinetic energy and its potential energy.
E=K+U
unbound electron
r
0.0
E2
E3
U
-13.6 E “the ground state”
1
The electron total energy is the sum of
its kinetic energy and its potential energy
in the electric field of the proton:
2
1
ke
En  K n  U n  m n2 
2
rn
Use the equation for
speed vs. radius to
rewrite K in terms of r:
1  ke2  ke2
ke2
 
En  m

2  mrn  rn
2rn
Bring in the expression
for the Bohr radii from
the previous slide:
k
 e
mr
ke2
En  
2
 km  2e  2  1  2 2 k 2 e 4  E1
   2 
 2
 2
2
h
 n
 n  h   n 
We can calculate the ground state energy, and then put all other energy levels in terms of it:
2 2 k 2e 4
18
E1  


13
.
6
eV


2
.
18

10
J
h2
then,
En 
E1
n2
with
n  1, 2, 3, ...
Now to derive the spectrum of hydrogen
l
Light is emitted or absorbed when an electron jumps from one
Bohr orbit to another. The energy of the photon equals the
difference in electron total energy between the two orbits:
91.18 nm
1 
 1
 2 2
n 
m
Eg  En  Em  hf  hc / l
Bring in the electron energy levels from the previous slide:
2 2 k 2e 4
18
E1  


13
.
6
eV


2
.
18

10
J
2
h
So that:
Eg 
then,
En 
E1
n2
with
n  1, 2, 3, ...
E1 E1
1  hc
 1


E


1 2
2
2
2 
n
m
m  l
n
Solving for the wavelength and putting in the numbers, we match the experimental result!
1


 hc  1
 6.63  10 34 Js 3.00  108 m/s
1 
 2  2   
l  

E
n 
2.18  10 -18 J

1  m


1
1
1 
1 
 1


(
91
.
2
nm)



 2
2
2
2 
m
n
m
n




It is amazing, and somewhat lucky, that Bohr’s model—containing a mixture of
classical and quantum ideas—actually got the right result! But it gave people the
courage to press forward with a more fundamental description (2 slides away).
1
How does the pattern of jumps yield the observed spectrum?
Since some of the Balmer series
wavelengths lie in the visible region,
we’ll look at the ones for which we
have a picture. The Balmer
emission lines correspond to an
electron jumping from a level above
n = 2 down to n = 2. The smallest
energy difference E32, leads to the
lowest energy photon for this
series, so it has the longest
wavelength. Larger energy
differences come from electrons
starting at higher levels, and since
the levels bunch up as n gets large,
the lines get closer together in
wavelength. The maximum energy
(“series limit”) occurs when the
electron energy goes to its
maximum (zero!), since above this
point, the electron is no longer
bound to the proton.
l
91.18 nm
1 
 1
 2 2
n 
m
More colorful diagrams of the energy levels
and transitions of hydrogen
Another diagram, this time with the
photon wavelengths labeled
1926: Max Born figures out how to interpret quantum waves.
Max Born (1882 - 1970) was a German mathematician and physicist. He
won the 1954 Nobel Prize in Physics, and was also the maternal grandfather
of the British-born Australian singer and actress Olivia Newton-John.
The picture above is an example of a particle “wave packet” or “wave function”. It has a
length—the region where we might find the particle—and a wavelength that determines its
momentum (from the de Broglie equation). In general, wave functions can be complex, with
imaginary parts. The plot above is of the real part only, and there is some imaginary part
which might look similar, but offset in phase. Max Born realized that if one takes the “square”
of the wave function, in the complex sense, what results is a probability density, telling
2
where the particle might be found, and with what probability:


    P( x)
Notice how similar this procedure is to squaring a wave amplitude to find the intensity of the
wave. In this case, the amplitude has no reasonable “meaning” until it is squared. This
mathematical procedure always yields a positive probability density, which is essential!
1926: Erwin Schrodinger figures out the wave equation for
particles, using the probabilistic wave function of Max Born
Erwin Schrödinger (1887 -1961) was an Austrian physicist who
achieved fame for his contributions to quantum mechanics, especially
the Schrödinger equation, for which he received the Nobel Prize in
1933. In 1935, he proposed the Schrödinger's cat thought experiment.
To cut down on so
many factors of 2:

h
2
Schrodinger started with conservation of energy:
E  K U
He created an operator equation, where space and time
derivatives act on the wave function. The left hand side
has units of E = hf, and the first term on the right has units
of (p = h/l) squared. This is the one dimensional version:
3D version adds y and z derivatives.
2
2
2
2
One can now describe quantum waves   x 2  y 2  z 2
traveling or standing in 3D:
p2
E
V
2m

 2  2
i

 V
2
t
2m x

2 2
i

   V
t
2m
When this is applied to the hydrogen atom—by making V the Coulomb potential
energy—the result is standing waves in 3D, the “orbitals” of hydrogen.
Probability density of the orbitals of hydrogen,
for zero angular momentum
n=1
n=2
     P
2
n=3
ground state
Electron waves with zero angular momentum don’t “orbit” the proton, they
just vibrate in and out (“radially”). The denser the color, the greater the
probability that the electron will be found at that point!
Probability density of the orbitals of hydrogen,
when angular momentum is included
Perhaps you’ve seen such patterns pictured in chemistry books!
Notice how the wave-particle duality has been
dealt with mathematically
(1) The wave function, , is a smooth wave
(not “grainy”), with real and imaginary parts
that may oscillate just like any classical
wave.
(2) Wave functions propagate according to
Huygen’s principle, and they may be
superposed to give all the interference and
diffraction effects we have studied in optics.
So they can describe the intensity patterns
of photons, electrons, and other particles.
As we’ve seen, quantum waves may be
traveling, or standing.
(3) To find out where a given particle might be,
we square the wave function to find the
probability density. Until we interact with it,
and measure its location, it could be
anywhere that this density is non-zero.
     P
2
Solvay Conference, 1927
A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrödinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin,
P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr,
I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. Richardson
A footnote on quantum mechanics…
Einstein could never accept some of the revolutionary ideas of quantum mechanics ("God does not
play dice"). When reminded in 1927 that he revolutionized science 20 years earlier, Einstein replied,
"A good joke should not be repeated too often."
A
T
A
T
Light as a Particle
•Light of very low intensity – can see
single “particles” that DO have
momentum hit screen
•Eventually the expected double slit
diffraction pattern emerges
•Light consists of photons (massless
particles) – originally postulated by
Einstein
•Ephoton=hf=hc/l
•h=6.63x10-34 Js (Planck’s constant)
T
Matter as a Wave
• Originally postulated by de Broglie in 1924
– NO evidence at this point
– He surmised that l=h/p: E=1/2 mv2=p2/2m
• Davison-Germer (1924)
– Electron diffraction
– Found by accident (“faulty” nickel target)
– “ruined” Germer’s second honeymoon
• G.P. Thompson – Went through the crystal
Particle Waves
ln=2L/n
Since l=h/mv,
vn=nv1
v1=h/(2Lm)
En=½mvn2
En=n2E1
E1=h2/(8mL2)
Me: l=10-34 m, E1=10-38 J,
n=1035 (n BIG, classical)
Electrons not!!!
Spectrometer
The end of the 19th century
– towards the slow death of
classical physics