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.

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

– This is a direct extension of the one-dimensional Schrödinger equation from Chapter 40.

−h2

2m

∂2Ψ x, y, z,t( )∂x2

+∂2Ψ x, y, z,t( )

∂y2+

∂2Ψ x, y, z,t( )∂z2

⎛

⎝⎜⎞

⎠⎟

+U x, y, z( )Ψ x, y, z,t( )= ih∂Ψ x, y, z,t( )

∂t

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).

– The function (x, y, z) obeys the time-independent Schrödinger equation in three dimensions:

( ) ( ) ( )

( ) ( ) ( )

2 2 22

2 2 2

, , , , , ,

2

, , , , , ,

ψ ψ ψ

ψ ψ

⎛ ⎞∂ ∂ ∂− + +⎜ ⎟

∂ ∂ ∂⎝ ⎠+ =

h x y z x y z x y z

m x y z

U x y z x y z E x y z

( ) ( ) /, , , , ,ψ −Ψ = hiEtx y z t x y z e

Seek solutions that are a product of a function only of x, a function only of y, and a function only of z.

€

(x, y,z) = C sinnxπx

Lsin

nyπy

Lsin

nzπz

L

Ex =nx

2π 2h2

2mL2

E = Ex + Ey + E z

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

a) Trueb) False€

(x,y,z) =ψ x (x)ψ y (y)ψ z (z)

True or false:Every stationary state of the 3D cubical box may

be written as a product

a) Trueb) False

False. If we combine stationary states of the same energy (degenerate states), we get another stationary state. For example,

Is stationary (and properly normalized!)

€

(x, y,z) =ψ x (x)ψ y (y)ψ z (z)

€

(x, y,z) =8

L3

1

2sin

πx

Lsin

2πy

Lsin

πz

L+

1

2sin

2πx

Lsin

πy

Lsin

πz

L

⎡

⎣ ⎢ ⎤

⎦ ⎥

Q41.1

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

A particle in a cubical box is in a state of definite energy. The probability distribution function for this state

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 z-component 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

And you would be RIGHT!

You probably wouldn’t guess that the “total angular momentum” is

€

Lz = mlh

€

L = l(l +1)h

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 nml can take any integer value between -l and +l

“spectroscopic notation” spdfghij…. = 0,1,2,3,4…. for lAll 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)h

€

Lz = mlh

Angular momentum vectors for l=2.

These states can have a precise z-

component 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) Directed exactly along the x-axisb) Directed exactly along the y-axisc) Directed exactly along the z-axisd) Could have any direction

Suppose an electron were able to orbit in the xy plane.If that were so, its angular momentum would bec) Directed exactly along the z-axis

Continuing our thought experiment: what would be its z-component of linear momentum, pz ?

a) Exactly zerob) It would sinusoidally oscillate around zeroc) It would sinusoidally oscillate around a positive offsetd) It would sinusoidally oscillate around a negative offsete) 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 bec) Directed exactly along the z-axis

Continuing our thought experiment: what would be its z-component of linear momentum, pz ?

a) Exactly zero

Does quantum mechanics allow anything to have a zero momentum component?

a) Nob) 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 bec) Directed exactly along the z-axis

Continuing our thought experiment: what would be its z-component 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 Lx

2+Ly2+Lz

2 = 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.

– 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 I

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.

If a sample of gas atoms is placed in a strong, uniform magnetic field, the spectrum of the atoms changes. Why is this?

Q41.6

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

If a sample of gas atoms is placed in a strong, uniform magnetic field, the spectrum of the atoms changes. Why is this?

A41.6

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).

Which statement about electron spin is correct?

Q41.7

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

Which statement about electron spin is correct?

A41.7

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.

€

En = −Zeff

2

n2(13.6eV )

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 x-ray 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).

€

En = −Zeff

2

n2(13.6eV )

Let’s use this formula to derive Moseley’s law.

Let’s assume that both the L-shell electron and the K-shell orbital it falls into are screened by one electron.

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

Q41.9

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

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

A41.9

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

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

The Bohr model and the Schrödinger equation both make predictions about the hydrogen atom. For which of the following quantities are the predictions different?

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

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

Q43.1

A. 9.5 times larger.

B. 8 times larger.

C. 6.5 times larger.

D. 4 times larger.

E. 2 times larger.

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

A43.1

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).

Why do stable nuclei with many nucleons (those with a large value of A) have more neutrons than protons?

Q43.2

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.

Why do stable nuclei with many nucleons (those with a large value of A) have more neutrons than protons?

A43.2

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, beta-

plus, 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.

Which kinds of unstable nuclei typically decay by emitting an electron?

Q43.4

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.

Which kinds of unstable nuclei typically decay by emitting an electron?

A43.4

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” --

There is no problem creating “photons” to mediate the Coulomb force. There is no minimum about of energy required for a photon!

€

ΔEΔt < h

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] zeroB] smaller than an electronC] about as big as a strongly flavored quarkD] 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-MannProfessor of Physics(now) University of New Mexicoproposed 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 numberAnd 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 UniverseIf 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.

You are here

We think we understandphysics until here (or so)

thought

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