Download The Schrödinger equation in 3-D

The Schrödinger equation in 3-D
– Electrons in an atom can move in all three dimensions of space.
If a particle of mass m moves in the presence of a potential
energy function U(x, y, z), the Schrödinger equation for the
particle’s wave function (x, y, z, t) is
h2  2  x, y, z,t  2  x, y, z,t  2  x, y, z,t 



2
2


2m 
x
y
z 2
 x, y, z,t 
 U x, y, z  x, y, z,t   ih
t
– This is a direct extension of the one-dimensional Schrödinger
equation from Chapter 40.
The Schrödinger equation in 3-D: Stationary states
– If a particle of mass m has a definite energy E, its wave function
(x, y, z, t) is a product of a time-independent wave function
(x, y, z) and a factor that depends on time but not position. Then
the probability distribution function |(x, y, z, t)|2 = |(x, y, z)|2
does not depend on time (stationary states).
  x, y, z, t     x, y, z  eiEt /
– The function (x, y, z) obeys the time-independent Schrödinger
equation in three dimensions:
  2  x, y, z   2  x, y, z   2  x, y, z  





2
2
2m 
x
y
z 2

 U  x, y, z   x, y, z   E  x, y, z 
2
Seek solutions that are a product of a function only of x, a function
only of y, and a function only of z.
n yy
n zz
n xx
 (x, y,z)  C sin
sin
sin
L
L
L
n x2 2 2
Ex 
2mL2
E  Ex  Ey  Ez
Particle in a three-dimensional box
– For a particle enclosed in a cubical box with sides of length L
(see Figure 41.2 below), three quantum numbers nX, nY, and nZ
label the stationary states (states of definite energy).
– The three states shown here are degenerate: Although they
have different values of nX, nY, and nZ, they have the same
energy E.
Recall that the 1D box states have energies that vary as n2
Recall that the 3D energy is the sum of the 3 1D energies.
Which 3D box state has the higher energy?
(The box is cubical, Lx = Ly = Lz)
a) nx = 1, ny = 1, nz = 3
b) nx = 1, ny = 2, nz = 2
c) They have the same energy
Recall that the 1D box states have energies that vary as n2
Recall that the 3D energy is the sum of the 3 1D energies.
Which 3D box state has the higher energy?
(The box is cubical, Lx = Ly = Lz)
a) nx = 1, ny = 1, nz = 3
b) nx = 1, ny = 2, nz = 2
c) They have the same energy
Answer: a.
True or false:
Every stationary state of the 3D cubical box may
be written as a product
 (x, y,z)   x (x) y (y) z (z)
a) True
b)
False
True or false:
Every stationary state of the 3D cubical box may
be written as a product
a) True
b) False
 (x, y,z)   x (x) y (y) z (z)
False. If we combine stationary states of the

same energy (degenerate states), we get
another stationary state. For example,
 (x, y,z) 
8  1
x 2y z 1
2x
y z 
sin sin
sin 
sin
sin sin 
3 
L  2
L
L
L
L
L
L 
2
Is stationary (and properly normalized!)
Q41.1
A particle in a cubical box is in a state of definite energy. The
probability distribution function for this state
A. oscillates in time, with a frequency that depends on the
size of the box.
B. oscillates in time, with a frequency that does not depend
on the size of the box.
C. varies with time, but the variation is not a simple
oscillation.
D. does not vary with time.
E. answer depends on the particular state of definite energy
A41.1
A particle in a cubical box is in a state of definite energy. The
probability distribution function for this state
A. oscillates in time, with a frequency that depends on the
size of the box.
B. oscillates in time, with a frequency that does not depend
on the size of the box.
C. varies with time, but the variation is not a simple
oscillation.
D. does not vary with time.
E. answer depends on the particular state of definite energy
Q41.2
A particle is in a cubical box with sides at x = 0, x = L, y = 0, y = L,
z = 0, and z = L.
When the particle is in the state nX = 2, nY = 1, nZ = 1, at which
positions is there zero probability of finding the particle?
A. on the plane x = L/2
B. on the plane y = L/2
C. on the plane z = L/2
D. more than one of A., B., and C.
E. none of A., B., or C.
A41.2
A particle is in a cubical box with sides at x = 0, x = L, y = 0, y = L,
z = 0, and z = L.
When the particle is in the state nX = 2, nY = 1, nZ = 1, at which
positions is there zero probability of finding the particle?
A. on the plane x = L/2
B. on the plane y = L/2
C. on the plane z = L/2
D. more than one of A., B., and C.
E. none of A., B., or C.
The hydrogen atom: Quantum numbers
–
–
The Schrödinger equation for the
hydrogen atom is best solved using
coordinates (r, , ) rather than (x, y,
z) (see Figure 41.5 at right).
The stationary states are labeled by
three quantum numbers: n (which
describes the energy), l (which
describes orbital angular momentum),
and ml (which describes the zcomponent of orbital angular
momentum).
In quantum mechanics,
quantization and quantum numbers
come from CONFINEMENT.
What is confining the electron in the
angular coordinates 
Answer: if you go around the circle,
you have to come back to the same
value! (Just as Bohr said!)
What do the  solutions look
like? What is ml ?
You would guess that the
angular momentum about
the z-axis is
Lz  ml
And you would be RIGHT!
You probably wouldn’t
guess that the “total angular
momentum” is

L  l(l 1)
The stationary state solutions to the SE for the Coulomb potential
are characterized by 3 quantum numbers (of course!)
n
principal quantum number
l
orbital angular momentum quantum number
ml
magnetic quantum number
The energy E is determined solely by n.
Note that En varies just as in the Bohr model! BUT… this n doesn’t seem to have
anything to do with angular momentum!
n can take any integer value from 1 to infinity.
l can take any integer value less than n
ml can take any integer value between -l and +l
“spectroscopic notation” spdfghij…. = 0,1,2,3,4…. for l
All l=0 states have perfect spherical symmetry
(MEMORIZE THESE RULES.)
Just as with square wells, increasing n increases the number of zero-crossings of
the wave (which increases the number of nodes in the probability distribution.)
The hydrogen atom: Degeneracy
– Hydrogen atom states
with the same value of n
but different values of l
and ml are degenerate
(have the same energy).
– The higher l the fewer
the probability nodes.
(Compare 3s, 3p, 3d)
– BUT, to really “see” the
higher l waves, we do
better with 3D pictures…
Last time:
The electron stationary states (“orbitals”) for the hydrogen
atom are characterized by three quantum numbers.
All orbitals with the same n have the same energy (only in
hydrogen)
Higher n  more wiggles in (r) (l fixed)
Higher l  fewer wiggles in (r) (n fixed)
l, ml give the angular momentum and the z-component of
the angular momentum. (We often use s, p, d, f, g… to
represent l = 0,1,2,3,4…)
L  l(l 1)
Lz  ml
Angular momentum vectors for l=2.
These states can have a precise zcomponent of their angular
momentum, but they cannot have a
precise x and y component.
Just as a wave packet has a
mixture of different
wavelengths, l=2 states have a
mixture of different Lx and Ly.
Lx and Ly are “uncertain”, just as
the momentum of a wavepacket
is uncertain.
Suppose an electron were able to orbit in the xy plane.
If that were so, its angular momentum would be
a)
b)
c)
d)
Directed exactly along the x-axis
Directed exactly along the y-axis
Directed exactly along the z-axis
Could have any direction
Suppose an electron were able to orbit in the xy plane.
If that were so, its angular momentum would be
c) Directed exactly along the z-axis
Continuing our thought experiment: what would be its zcomponent of linear momentum, pz ?
a)
b)
c)
d)
e)
Exactly zero
It would sinusoidally oscillate around zero
It would sinusoidally oscillate around a positive offset
It would sinusoidally oscillate around a negative offset
It would be a non-zero constant value
Suppose an electron were able to orbit in the xy plane.
If that were so, its angular momentum would be
c) Directed exactly along the z-axis
Continuing our thought experiment: what would be its zcomponent of linear momentum, pz ?
a) Exactly zero
Does quantum mechanics allow anything to have a zero
momentum component?
a) No
b) Yes, but only if the object is infinitely spread out!
c) Yes, as long as the object doesn’t have a precise x.
Suppose an electron were able to orbit in the xy plane.
If that were so, its angular momentum would be
c) Directed exactly along the z-axis
Continuing our thought experiment: what would be its zcomponent of linear momentum, pz ?
a) Exactly zero
Does quantum mechanics allow anything to have a zero
momentum component?
b) Yes, but only if the object is infinitely spread out!
Electrons in atoms are not infinitely spread out. So we
can’t have an angular momentum with no Lx and Ly
(but you CAN have Lz = 0.)
What does this figure represent?
a) Electrons with different ml travel in
circular orbits offset from z=0 (along
the red circles at the end of each
cone)
b) In the atom, the total angular
momentum L precesses around z
like a top.
c) Although a quantum state has a
mixture of different Lx ‘s and Ly ‘s, it
remains true that Lx2+Ly2+Lz2 = L2
d) None of these statements is true; the
diagram shows something else.
The hydrogen atom: Quantum states
– Table 41.1 (below) summarizes the quantum states of the
hydrogen atom. For each value of the quantum number n, there
are n possible values of the quantum number l. For each value of
l, there are 2l + 1 values of the quantum number ml.
The hydrogen atom: Probability distributions I
– States of the hydrogen atom with l = 0 (zero orbital angular
momentum) have spherically symmetric wave functions that
depend on r but not on  or . These are called s states. Figure
41.9 (below) shows the electron probability distributions for three
of these states.
The hydrogen atom: Probability distributions II
– States of the hydrogen atom with nonzero orbital angular
momentum, such as p states (l = 1) and d states (l = 2), have
wave functions that are not spherically symmetric. Figure 41.10
(below) shows the electron probability distributions for several of
these states, as well as for two spherically symmetric s states.
Chemists like to refer to orbitals like px, py, and pz.
Their pz orbital is the same as the l=1 ml=0 orbital.
Note also: the probability distribution for both the ml = ± 1 states is
EXACTLY THE SAME!
Px is a combination of ml = ±1
Py is a different combination of ml = ±1
Zeeman Effect
•
•
Electron states with nonzero orbital angular momentum (l = 1, 2, 3,
…) have a magnetic dipole moment due to the electron motion.
Hence these states are affected if the atom is placed in a magnetic
field. The result, called the Zeeman effect, is a shift in the energy of
states with nonzero ml. This is shown in Figure 41.13 (below).
Follow Example 41.5.
Zeeman effect is used to
measure magnetic fields
on the surface of the sun.
Energy
The Zeeman effect and selection rules
•
An atom in a magnetic field can make transitions between different
states by emitting or absorbing a photon A transition is allowed if l
changes by 1 and ml changes by 0, 1, or –1. (This is because a
photon itself carries angular momentum.) A transition is forbidden if it
violates these selection rules. See Figure 41.15 (lower right).
Note that n doesn’t have to change!
The anomalous Zeeman effect and electron spin
•
For certain atoms the Zeeman effect does not follow the simple
pattern that we have described (see Figure 41.16 below). This is
because an electron also has an intrinsic angular momentum,
called spin angular momentum.
Q41.6
If a sample of gas atoms is placed in a strong, uniform magnetic
field, the spectrum of the atoms changes. Why is this?
A. Electrons have magnetic moments due to their spin and their
orbital motion.
B. The nucleus and the electrons are pushed in opposite
directions by a magnetic field.
C. Electrons are drawn into regions of strong magnetic field.
D. Electrons are repelled from regions of strong magnetic field.
E. none of the above
A41.6
If a sample of gas atoms is placed in a strong, uniform magnetic
field, the spectrum of the atoms changes. Why is this?
A. Electrons have magnetic moments due to their spin and their
orbital motion.
B. The nucleus and the electrons are pushed in opposite
directions by a magnetic field.
C. Electrons are drawn into regions of strong magnetic field.
D. Electrons are repelled from regions of strong magnetic field.
E. none of the above
Last time: Stern-Gerlach Experiment:
Directly measure the magnetic moments of atoms,
using a magnetic field gradient.
Some atoms have magnetic moments that are not
consistent with integer multiples of h-bar for angular
momentum… there appears to be half integer
angular momentum.
This is attributed to electron spin angular momentum.
Spin vs orbit: Earth’s rotation vs revolution.
(But it’s not clear that anything in the electron really
spins… if it did, it would have to spin faster than c!)
Electron spin and the Stern-Gerlach experiment
•
The experiment of Stern and Gerlach demonstrated the existence
of electron spin (see Figure 41.17 below). The z-component of the
spin angular momentum has only two possible values
(corresponding to ms = +1/2 and ms = –1/2).
Q41.7
Which statement about electron spin is correct?
A. The spin angular momentum has two possible magnitudes
and two possible values of its z-component.
B. The spin angular momentum has only one possible
magnitude but two possible values of its z-component.
C. The spin angular momentum has two possible magnitudes
but only one possible value of its z-component.
D. The spin angular momentum has only one possible
magnitude and only one possible value of its z-component.
E. none of the above
A41.7
Which statement about electron spin is correct?
A. The spin angular momentum has two possible magnitudes
and two possible values of its z-component.
B. The spin angular momentum has only one possible
magnitude but two possible values of its z-component.
C. The spin angular momentum has two possible magnitudes
but only one possible value of its z-component.
D. The spin angular momentum has only one possible
magnitude and only one possible value of its z-component.
E. none of the above
Energy
Quantum states and the Pauli exclusion principle
•
•
The allowed quantum numbers for an atomic electron
(see Table 41.2 below) are n ≥ 1; 0 ≤ l ≤ n – 1; –l ≤ ml ≤ l;
and ms = ±1/2.
The Pauli exclusion principle states that if an atom has
more than one electron, no two electrons can have the
same set of quantum numbers.
A multielectron atom
•
•
•
Figure 41.21 (at right) is a sketch
of a lithium atom, which has 3
electrons. The allowed electron
states are naturally arranged in
shells of different size centered
on the nucleus. The n = 1 states
make up the K shell, the n = 2
states make up the L shell, and
so on.
Due to the Pauli exclusion
principle, the 1s subshell of the K
shell (n = 1, l = 0, ml = 0) can
accommodate only two electrons
(one with ms = + 1/2, one with ms
= –1/2). Hence the third electron
goes into the 2s subshell of the L
shell (n = 2, l = 0, ml = 0).
Why is the 2s state lower in
energy than the 2p?
Why is the 2s state lower in
energy (in multielectron
atoms) than 2p?
In hydrogen, these states
have the same energy.
The 2p orbital overlaps more
with the 1s orbital than does
the 2s orbital. Electrons repel
each other, even if they ARE
waves!
Screening in multielectron atoms
– An atom of atomic number Z has a nucleus of charge +Ze and Z
electrons of charge –e each. Electrons in outer shells “see” a
nucleus of charge +Zeffe, where Zeff < Z, because the nuclear
charge is partially “screened” by electrons in the inner shells.
– Energies depend on the square of the effective nuclear charge.
This is because increasing nuclear charge makes the orbit radius
smaller and thus the average Coulomb (centripetal) force even
larger than you would expect just on the basis of changing Z.
2
eff
2
Z
En  
(13.6eV )
n
Potassium has 19 electrons. It is relatively easy to remove one
electron but substantially more difficult to then remove a
second electron. Why is this?
A. The second electron feels a stronger attraction to the
other electrons than did the first electron that was removed.
B. When the first electron is removed, the other electrons
readjust their orbits so that they are closer to the nucleus.
C. The first electron to be removed was screened from more
of the charge on the nucleus than is the second electron.
D. all of the above
E. none of the above
Potassium has 19 electrons. It is relatively easy to remove one
electron but substantially more difficult to then remove a
second electron. Why is this?
A. The second electron feels a stronger attraction to the
other electrons than did the first electron that was removed.
B. When the first electron is removed, the other electrons
readjust their orbits so that they are closer to the nucleus.
C. The first electron to be removed was screened from more
of the charge on the nucleus than is the second electron.
D. all of the above
E. none of the above
X-ray spectroscopy
•
•
When atoms are bombarded
with high-energy electrons, x
rays are emitted. There is a
continuous spectrum of x rays
(described in Chapter 38) as
well as strong characteristic xray emission at certain definite
wavelengths (see the peaks
labeled K and K in Figure
41.23 at right).
Atoms of different elements emit
characteristic x rays at different
frequencies and wavelengths.
Hence the characteristic x-ray
spectrum of a sample can be
used to determine the atomic
composition of the sample.
•
•
X-ray spectroscopy: Moseley’s law
Moseley showed that the square root of the x-ray frequency in K
emission is proportional to Z – 1, where Z is the atomic number of
the atom (see Figure 41.24 below). Larger Z means a higher
frequency and more energetic emitted x-ray photons.
This is consistent with our model of multielectron atoms.
Bombarding an atom with a high-energy electron can knock an
atomic electron out of the innermost K shell. K x rays are produced
when an electron from the L shell falls into the K-shell vacancy. The
energy of an electron in each shell depends on Z, so the x-ray
energy released does as well.
•
The same principle applies to
K emission (in which an
electron falls from the M shell
to the K shell) and K emission
(in which an electron falls
from the N shell to the K
shell).
Let’s use this formula to derive Moseley’s law.
2
eff
2
Z
En  
(13.6eV )
n
Let’s assume that both the L-shell electron and the K-shell
orbital it falls into are 
screened by one electron.
Q41.9
Ordinary hydrogen has one electron and one proton. It requires
10.2 eV of energy to take an electron from the innermost (K)
shell in hydrogen and move it into the next (L) shell.
Uranium has 92 electrons and 92 protons. The energy required to
move an electron from the K shell to the L shell of uranium is
A. (91)(10.2 eV).
B. (92)(10.2 eV).
C. (91)2(10.2 eV).
D. (92)2(10.2 eV).
E. none of the above
A41.9
Ordinary hydrogen has one electron and one proton. It requires
10.2 eV of energy to take an electron from the innermost (K)
shell in hydrogen and move it into the next (L) shell.
Uranium has 92 electrons and 92 protons. The energy required to
move an electron from the K shell to the L shell of uranium is
A. (91)(10.2 eV).
B. (92)(10.2 eV).
C. (91)2(10.2 eV).
D. (92)2(10.2 eV).
E. none of the above
Q41.5
The Bohr model and the Schrödinger equation both make
predictions about the hydrogen atom. For which of the following
quantities are the predictions different?
A. the energy of the lowest (n = 1) energy level
B. the difference in energy between the n = 2 and n = 1
energy levels
C. the orbital angular momentum of the electron in the
lowest (n = 1) energy level
D. more than one of A., B., and C.
E. none of A., B., or C.—the predictions are identical for all
of these
A41.5
The Bohr model and the Schrödinger equation both make
predictions about the hydrogen atom. For which of the following
quantities are the predictions different?
A. the energy of the lowest (n = 1) energy level
B. the difference in energy between the n = 2 and n = 1
energy levels
C. the orbital angular momentum of the electron in the
lowest (n = 1) energy level
D. more than one of A., B., and C.
E. none of A., B., or C.—the predictions are identical for all
of these
Nuclear Physics
Nuclei are made up of protons and neutrons
(which are in turn made up of quarks… but you can’t get a free quark.)
There has to be a strong force binding protons together, because
otherwise their electrical repulsion would cause them to fly apart.
Being highly creative, we physicists call this strong force
“The strong force”.
The strong force is short range, unlike electricity, magnetism, gravity.
If you split a nucleus in half and move the halves a few fm apart, the
electric force takes over and the halves fly apart with tremendous energy.
We call this an “atom bomb”, but it’s really an electrical bomb.
Properties of nuclei
• The nucleon number A is the total number of protons
and neutrons in the nucleus.
• The radius of most nuclei is given by R = R0A1/3.
R0 = 1.2 fm = 1.2 x10-15 m.
• All nuclei have approximately the same density.
• Mass is approximately 1u x A.
1u = 1.66054 x 10-27 kg
Q43.1
A nucleus of neon-20 has 10 protons and 10 neutrons.
A nucleus of terbium-160 has 65 protons and 95 neutrons.
Compared to the radius of a neon-20 nucleus, the radius of a
terbium-160 nucleus is
A. 9.5 times larger.
B. 8 times larger.
C. 6.5 times larger.
D. 4 times larger.
E. 2 times larger.
A43.1
A nucleus of neon-20 has 10 protons and 10 neutrons.
A nucleus of terbium-160 has 65 protons and 95 neutrons.
Compared to the radius of a neon-20 nucleus, the radius of a
terbium-160 nucleus is
A. 9.5 times larger.
B. 8 times larger.
C. 6.5 times larger.
D. 4 times larger.
E. 2 times larger.
Nuclides and isotopes
• The atomic number Z is the number of protons in the nucleus. The
neutron number N is the number of neutrons in the nucleus.
Therefore A = Z + N.
• A nuclide is a single nuclear species having specific values for both
Z and N.
• The isotopes of an element have different numbers of neutrons.
• Protons, like electrons, have spin angular momentum and a
magnetic moment. Thus, the nucleus may have a magnetic moment.
(Not all nuclei do… with lots of protons the magnetic moments can
cancel… but hydrogen has only one proton as its nucleus, so it
always has a magnetic moment.
NMR and MRI
Nuclear binding energy
• The binding energy EB of a nucleus is the energy that must be added to
separate the nucleons.
• The nearly constant energy per nucleon for large nuclei is a
consequence of the limited range of the strong force! Each nucleon only
binds its neighbors!
• Why is there a maximum size for a nucleus? Too many protons -> too
much repulsion. How about neutrons… more later.
The nuclear force
• The nuclear force binds protons and neutrons together. It is
an example of the strong interaction.
• Important characteristics of the nuclear force:
 It does not depend on charge. Protons and neutrons are
bound. It has a short range, of the order of nuclear
dimensions.
 Because of its short range, a nucleon only interacts with
those in its immediate vicinity.
 It favors binding of pairs of protons or neutrons with opposite
spins and with pairs of pairs (a pair of protons and a pair of
neutrons, each pair having opposite spins).
Q43.2
Why do stable nuclei with many nucleons (those with a large
value of A) have more neutrons than protons?
A. An individual nucleon interacts via the nuclear force
with only a few of its neighboring nucleons.
B. The electric force between protons acts over long
distances.
C. The nuclear force favors pairing of both neutrons and
protons.
D. both A. and B.
E. all of A., B., and C.
A43.2
Why do stable nuclei with many nucleons (those with a large
value of A) have more neutrons than protons?
A. An individual nucleon interacts via the nuclear force
with only a few of its neighboring nucleons.
B. The electric force between protons acts over long
distances.
C. The nuclear force favors pairing of both neutrons and
protons.
D. both A. and B.
E. all of A., B., and C.
Nuclear stability
and radioactivity
• Radioactivity is the
decay of unstable
nuclides by the
emission of particles
and electromagnetic
radiation.
• Figure 43.4 (right) is
a Segrè chart
showing N versus Z
for stable nuclides.
Why isn’t there a nicely
stable isotope of Hydrogen
with 57 neutrons and 1
proton?
We need to talk about
radioactive decay
mechanisms to understand
this.
• An alpha particle  is a 4He nucleus.
• Alpha decay of the 226Ra nuclide.
Beta and gamma decay
• There are three types of beta decay: beta-minus, betaplus, and electron capture.
• A beta-minus – particle is an electron.
• A gamma ray is a photon.
Beta decay requires a new (fourth) force… it doesn’t happen frequently, so
it’s a weak force or interaction.
Being highly creative, we physicists call this weak force…
The “weak force”.
Beta decay (and the Pauli exclusion principle) explains
why there is an approximate equivalence between # of
protons and neutrons in the nucleus!
Natural
radioactivity
• Figure 43.7 (right)
shows a Segrè chart
for the 238U decay
series.
Q43.4
Which kinds of unstable nuclei typically decay by emitting an
electron?
A. those with too many neutrons
B. those with too many protons
C. those with too many neutrons and too many protons
D. Misleading question—the numbers of neutrons and
protons in a nucleus are unrelated to whether or not it
emits an electron.
A43.4
Which kinds of unstable nuclei typically decay by emitting an
electron?
A. those with too many neutrons
B. those with too many protons
C. those with too many neutrons and too many protons
D. Misleading question—the numbers of neutrons and
protons in a nucleus are unrelated to whether or not it
emits an electron.
Activities and half-lives
• The half-life is the time for
the number of radioactive
nuclei to decrease to one-half
of their original number.
• The number of remaining
nuclei decreases exponentially
• The rate of emission is always
proportional to the number
remaining… it goes down by
1/2 in one half life also.
Nuclear reactions
• A nuclear reaction is a rearrangement of
nuclear components due to bombardment
by a particle rather than a spontaneous
natural process.
• The difference in masses before and after
the reaction corresponds to the reaction
energy Q.
Nuclear fission
• Nuclear fission is a decay
process in which an unstable
nucleus splits into two
fragments (the fission
fragments) of comparable mass.
• Figure 43.11 (right) shows the
mass distribution of the fission
fragments from the fission of
236U*. (This is what you get if
you bombard 235U with a
neutron… an excited state of
236U.)
• MOST importantly for bombs
& reactors -- this fission also
yields > 1 neutron, on average.
Chain reactions
• The neutrons released by fission can cause a chain reaction (see
Figure 43.14 below).
Nuclear reactors
• A nuclear reactor is a system in which a
controlled nuclear chain reaction is used
to liberate energy.
While splitting heavy nuclei releases energy, so does fusion of light nuclei. This
powers the sun & stars*, and is the energy source of the future (and always will be?)
*The energy that powers a supernova explosion and makes heavy elements is gravitational
potential energy.
Fusion requires tremendous kinetic energy of the protons, in order to overcome the
Coulomb barrier. You can get hydrogen isotopes hot enough to fuse if you put them
in a box with an atom bomb, with reflectors to reflect and focus the gamma/ x-rays
rays.
There are (we think) four fundamental forces in nature.
What force holds a neutral oxygen atom to another in O2?
By the way, the same force binds water molecules together
into a liquid.
A] gravity
B] electromagnetic force
C] strong nuclear force
D] weak nuclear force
We have seen that waves have particle-like behavior.
Electromagnetic waves are just “modes” of the electromagnetic field.
The quantum view is that all fields have some particle-like behavior.
With the strong force / field, we usually use the particle language:
The strong nuclear force occurs because of the exchange of massive
“mesons” between protons/neutrons.
To “exchange” a particle, you first need to make it. To make a meson
from a proton, you need to violate energy conservation.
Quantum mechanics says that’s okay, as long as the energy is
“imprecise” --
Et 
There is no problem creating “photons” to mediate the Coulomb force.
There is no minimum about of energy required for a photon!

Through careful experimentation, you discover a force of
attraction between two people.
It only acts over a short range! There is no force if the
separation is > 1 m.
In keeping with physical nomenclature, you call this force
“smell”. There are apparently two smells, “nice” and
“nasty”. Like attracts like.
What is the mass of the odoron, the quantum of the smell
field?
A] zero
B] smaller than an electron
C] about as big as a strongly flavored quark
D] the same as a ton of bricks
Inside the proton and neutron are quarks. They have
“color” and any object with color experiences the strong
force very strongly. The mediator of this color (strong)
force is called the gluon. The gluon itself has both color
and mass.
Because the gluon carries color “charge”, a gluon field
emits more gluons.
This is why
a) the color force is finite range and
b) color cannot be isolated.
All observable hadrons are colorless.
(Pulling a red quark out of a proton would require so much energy that you
would make an anti-red and a red, ending up with the proton and a meson.)
Murray Gell-Mann
Professor of Physics
(now) University of New Mexico
proposed Quarks in 1964
How is a nucleus like an O2 molecule?
•
The neutral oxygen atoms bind together to form a molecule because of
the “residue” of their electromagnetic fields. Another way to think about
this is that, when close together, distorting the orbitals of one O atom
creates a net electric field, this field distorts the orbitals of the second
atom so as to lower the overall energy. The O atoms “work together” to
find a lower energy configuration for their electrons. But the
fundamental interaction is electromagnetic (plus some quantum
mechanics…)
•
Color-neutral neutrons and protons bind together because of the
“residue” of the strong force. As strong as the force between nucleons
is, it is still quite weak compared with naked color.
Quarks
(Baryons)
Leptons
Antiparticles
Top & Bottom rows are different “flavors”
Proton decay is u d + e+ + e
This conserves lepton number & baryon number
And also keeps the “flavor difference”* constant in the universe
(adding one net particle of each flavor.)
* Electrons, muons, taus are also said to have different flavor. This is not conserved.
Antiparticles
What’s the deal with antiparticles?
Dirac wrote down a relativistic version of the Schroedinger Eqn.
He found it had both positive and negative energy solutions for e-.
(Not surprising, since it had to be second order in time!)
Rather than discard the negative energy solutions as spurious,
(as any sane or modest person would do)
he hypothesized that the negative energy states were filled with
unobservable electrons. Moving an e- up out of this sea would leave a hole
of + charge, that acts just like an antielectron.
Feynman showed that antiparticles are just regular particles going backward
in time. This eliminates the need to postulate an “unobservable” e- sea…
perhaps at the psychic expense of worrying about “backward time”.
The Big Bang & The Expanding Universe
Distant galaxies are receding (Doppler shift of spectral lines)
The Big Bang & The Expanding Universe
There is (probably) no center for expansion.
To have a center, you need a boundary.
The Big Bang & The Expanding Universe
If the universe is expanding, it must have been denser and hotter in the past. We
can sample physics at higher temperatures (higher energies) in accelerator
experiments, and thus we can extrapolate the current universe back to very
nearly its beginning.
<380,000 years after the bang, the average energy per particle is > electronic
binding energy. There are no atoms, only ions, which scatter light. The universe is
opaque.
A hundred seconds after the bang, the average kinetic energy per particle is
greater than nuclear binding energy: there are no nucleii
A microsecond after the bang, there is quark soup… no neutrons, or protons.
Quark Soup - Hard to come by. They make it in New York.
What happened before
the universe was quark
soup is mostly
speculation. We don’t
have experiments to test
different ideas. We think
the forces should be
unified.
thought
We think we understand
physics until here (or so)
You are
here
Gravitational Lensing & Dark Matter
The bending of light by
massive galaxies,
And the rotation rates of
galaxies
Both indicate A LOT more
mass than can be accounted
for by stars & dust.
Accelerating Expansion - “Dark Energy”
When we look carefully at the Doppler redshifts of distant galaxies,
they are smaller than Hubble predicts.
The expansion of the universe used to be slower!
Gravitational attraction should slow the expansion down!
Maybe the universe has stuff in it pushing the expansion - “dark
energy”
Maybe general relativity is wrong.
Maybe there is large-scale structure in the universe we are unaware
of.
A few things we don’t understand
•
•
•
•
•
•
•
We haven’t been able to write a sensible & complete theory combining General
Relativity and QM
Although the quantization of fields seems to work well (and makes verified
predictions), it also predicts an infinite energy density (recall that even the
ground state of a quantum system has some energy!)
Although electrons appear to be pointlike particles, that would give them infinite
“self energy” in their Coulomb field, and so infinite mass.
We don’t understand why charge is quantized, and mass isn’t.
We don’t understand WHY there are three generations of matter.
We don’t understand the accelerating expansion of the universe
We don’t know what dark matter is.
What is a perfect theory of everything?
In physics, a perfect theory (IMO) shows how all physical laws and behaviors
arise from the smallest set of postulates.
In a perfect theory, we look at the small set of postulates and say, “given that
these are true, the universe could not be other than it is.”
(As an example, given that the spacetime has its geometry, + Coulomb’s law,
all of electromagnetism must follow!)
There is a regression to smaller and smaller sets of postulates, and more
basic (quasi-philosophical) questions. For example, why does spacetime
have Minkowski geometry? We don’t know.
BUT WE CAN KNOW!
“The most incomprehensible thing about the universe
is that it is comprehensible!” - A Einstein