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HUMS4100: Wave Mechanics

A wavicle

Wave-Particle Duality

De Broglie 1924

Wave - particle duality: All fundamental (i.e small!) particles also act like waves (what is an electron?...) and waves act like particles.
You cannot ask: Is light a wave or a particle: answer is "yes"
Einstein/Planck suggest light (wave) has some particle properties,
"particle" of light is photon (γ : gamma)
so maybe electron (particle) has some wave properties
What is wave-length of electron? for photon
\color{red}{ E_\gamma = hf = \frac{{hc}}{\lambda }}
Momentum of photon
\color{red}{ E = \sqrt {p^2 c^2 + m^2 c^4 } \Rightarrow pc}
(if particle is massless) so
\color{red}{ \lambda = \frac{h}{p}}
If this is always true, then for electron p = m v and λ=h/mv: if v = 1000 ms^-1, what is λ ?

Thomson and Davisson-Germer

used crystal as diffraction grating: (2-D so pattern is more complicated)

A simpler experiment is now possible: the electron analog of Young's slits. Very low energy electrons pass through slits and hit detector (e.g. photo plate) and give 2-slit interference pattern

Thompson carried out series of experiments using weaker and weaker sources, until he had less than one electron in apparatus at any one time
Pattern unchanged: i.e. not one electron interfering with second, but one electron interferes with itself

A dramatic recent example uses a buckyball C₆₀
American Journal of Physics, Vol. 71, No. 4, p319, April 2003, Nairz, Arndt, and Zeilinger
Apparatus uses a diffraction grating:velocity v = 117 ms^-1
Circles are the experimental data. Line represent the model
A buckyball C₆₀ has a mass of 60*12*1.67x10^-27 = 1.2x10^-24 kg. The speed is 117 ms^-1. What is λ for these parameters?

Why don't ordinary objects look like waves?
e.g. a billiard ball: if v = 1 ms^-1, m_ball = 0.1 kg what is λ?
Problems
- Which slit did the ~~electron~~ buckyball go through??
- What waves??
But if an electron is a wave, how do we find out how it moves?

Schrödinger's equation

Schrödinger (1925)

Now need to describe what happens to an "interacting" wave. Simplest case is electron confined to a 1-D box
Newton:
\color{red}{ E = \frac{1}{2}mv^2 }
De Broglie: Standing wave
\color{red}{ \lambda = \frac{h}{p}}
Wave (like guitar string)
\color{red}{ \lambda = \frac{{2L}}{n}}

Combine these

\color{red}{ E_n = \frac{{h^2 n^2 }}{{8m_e L^2 }}}
i.e. only certain energies are allowed. Note lowest energy is not zero!

The hydrogen atom

(Bohr 1917) but we skipped that

Model for H. atom must explain

Spectrum
\color{red}{ \frac{1}{\lambda } = R\left( {\frac{1}{{n^2 }} - \frac{1}{{m^2 }}} \right)}

m > n, n = 1, 2, 3...

n = 1 m = 2, 3, 4...
n = 2 m = 3, 4, 5...
n = 3 m = 4, 5, 6...

This implies only photons of certain definite energies are emitted . . .
Rutherford's observation of massive nucleus
Stability

Let's build an electric solar system!

~~gravitational~~ electrostatic force = centripetal force

\color{red}{ F = \frac{{Gm_1 m_2 }}{{r^2 }} \Rightarrow \frac{{kq_1 q_2 }}{{r^2 }} = \frac{{mv^2 }}{{r^2 }}}

q is charge on electron/proton = 1.6x10^-19 Coulombs
k is the electrostatic force constant = 9x10⁹

However, v & r are not easy to find.

De Broglie suggested that allowed orbits have an integral number of waves fitted into one orbit
\color{red}{ \lambda = \frac{{2\pi r}}{n} = \frac{h}{p}}
A simulation.
Electrons exist as 3-D standing waves: this requires solution of Schrödinger's equation.
Unfortunately the real case of an electron is a 3-D differential equation, and hence hard to solve.

If we want to make this look more like the Bohr atom, we can write it in terms of the angular momentum
\color{red}{ L = mvr = \frac{h}{{2\pi n}}}
i.e. fitting in whole number of wavelengths ≡ angular momentum being quantised
\color{red}{ L = n\hbar }
Where
\color{red}{ \hbar = {\rm{1}}{\rm{.05}} \times {\rm{10}}^{{\rm{ - 34}}} {\rm{Js = 6}}{\rm{.56}} \times {\rm{10}}^{{\rm{ - 16}}} {\rm{eVs}}}
(say h-bar).
This was Bohr's quantization condition.
This means only certain orbits allowed:
why?
because!

Sizes of Things

We can use this to find the radius of orbits
\color{red}{ r_n = n^2 r_0 ,r_0 \approx 0.05nm}
in agreement with prejudice about size of atoms
These levels have energies
\color{red}{ E_n = -\frac{{13.6}}{{n^2 }}eV}
So how does this explain the spectra?

Radiation

Energies of these levels:
```
E_n = = - 13.6   eV
         n²
```
Since these are the only allowed levels, energy can only be emitted when electron jumps from one to the other
e.g. n = 3 ⇒ n = 2
δE = E₃- E₂ = 1.9 eV
what wavelength light does this correspond to?
This transition gives 1.9 eV photon which is red line in H.

In general
\color{red}{ E_\gamma = E_n - E_m = 13.6\left( {\frac{1}{{m^2 }} - \frac{1}{{n^2 }}} \right)}
and
\color{red}{ E_\gamma = hf = \frac{{hc}}{\lambda }}
so
\color{red}{ \frac{1}{\lambda } = \frac{{13.6}}{{hc}}\left( {\frac{1}{{n^2 }} - \frac{1}{{m^2 }}} \right)}
i.e. we get all the lines in the spectrum
n = 1 Lyman series
n = 2 Balmer Series
n = 3 Paschen Series

But why do we see dark lines in the sun?

Emission and absorption Spectra

If electron makes transition from one level to another, we will get emission line of definite energy

However, if we have photons of all energies, one may have exactly the energy to raise the energy of an electron
Note that this will just remove one energy of photon from continuous spectrum.

With care, can see both absorption and emission at the same time.
Why do you see absorption spectra in the sun?
The atoms in the chromosphere (the solar atmosphere) absorb the radiation from the solar "surface".

Are these energy levels so weird?

e.g. energy levels for block of wood. "Transitions" from one to another ⇒ fixed amount of energy as sound.
That's why only get certain energies as photons in atoms

Heisenberg's Uncertainty Principle

Heisenberg 1927

If an electron is a wave, how can we define its position?

Suppose we try to measure position of electron by confining it to box
Uncertainty in position
δx = L
but there is also an uncertainty in momentum
δp~2p~2h/λ=h/L
Hence
```
δxδp = L h/L = h
```
if we squeeze walls together to measure position better, momentum becomes more uncertain.

e.g. Suppose we try to measure position of electron with microscope:

if we could do it with one photon then the position uncertainty ~ wavelength
δx >λ
So decrease wavelength to get position better, but photon carries momentum
p=h/λ
and some of it gets transferred
```
δx δp >λ(h/λ) >h
```

Done properly:

(uncertainty in position.) x (uncertainty in mom.) > ħ
e.g. Ball bearing, m = 1 g, position measured to 100 nm: what is δP?
e.g. electron, m = 9x10^-31 kg, position measured to 100 nm: what is δP?
This is a fundamental limitation on human knowledge: can always do worse but cannot do better!
There is another form of H's uncertainty principle, which is going to become important later
This says that to measure the energy of anything perfectly, you need to take an infinite time.
```
δE δt > h
```

Atoms

Atoms and the Periodic Table

Mendeleev had found the periodic table in ~ 1850. Very complex pattern: e.g Very reactive acid-forming element (e.g. Cl = Chlorine) is always followed by an inert gas (e.g. Ar = Argon) is followed by a reactive alkaline metal (e.g. K = Potassium)

Bohr-De Broglie-Schrödinger's equation works for all atoms , with some subtleties since we have several electrons in each atom. Nucleus of each atom has charge Z = 1 for H, 2 for He etc.

Charge on nucleus is no longer 1, so
```
E_n = -13.6Z²  eV
      n²
```
where Z is the atomic number.
3 Subtleties:
Each state is described by 3 quantum numbers (because it is 3-D!)
- n = 1,2,3 ....... (which has the same meaning as before)
- L = 0,1,.....(n-1) (which is angular momentum)
- m = -L,-(L-1),..-1,0,1,...L-1,L

Note that the energy depends only on n.
electron has internal angular momentum (earth spins on its own axis in addition to orbiting) s = +1/2, -1/2
Exclusion Principle Can two objects occupy same space at same time?
Classically? No
Quantum mechanically? Maybe
Pauli showed particles with same properties (e.g. two electrons) cannot be in same state (if they have spin 1/2: fermions). i.e. for each level, we can put in 2 electrons

Allows us to understand periodic table:

must have number of electrons = Z = charge on nucleus, and fill lowest energy levels first.

X-rays:

Electron accelerated
Electron collides with atom, knocks out electron in lowest energy level, leaves vacancy for electron in higher level to fall into.e.g. for e.g Chromium: (Z = 24)

Why are X-rays and UV bad for you?

Typical energy of chemical bond 1 - 10 e.V. cannot be broken by long-wave, but can be broken by U-V
e.g. DNA is two interlocked coils of amino-acids
X-rays (1000 e.V) break chain
U-V (~ 10 e.V) causes thiamin to bond to other coil (dimer) so cannot replicate.

Nuclear Physics

Just enough so we understand the basics.

Proton

discovered as nucleus of H by Rutherford and Blackett (1921) in

α + N ⇒ O + p

M_proton = 1.67x10^-27 kg

Neutron

discovered by Chadwick (1932) as penetrating massive neutral particle (i.e., not a γ)
M_neutron = 1.68x10^-27 kg. since neutrons and protons are so similar in mass, convenient to think of them as the same particle (a nucleon) which comes in two flavours!

Atomic Number Z 
= charge on nucleus
 = N_protons

This defines chemistry

Mass number A = N_protons + N_neutrons
Isotopes: nuclei with different A but same Z.
Usual notation is ^A_ZX_N but often just write a nucleus as (e.g) ³⁵S, since the name implies Z. In this notation:

Conservation laws:

Much of nuclear physics is governed by conserved by conservation laws.

Mass-energy conservation

```
E = mc²
```
Usually easier to quote elementary particle masses in terms of energy, measured in eV

e.g. "mass" of electron

= mc² = 9.1x10^-31x(3x10⁸)²/1.6x10^-19 
  = 511 keV = .511 MeV

Conservation of charge
n ⇒ p + γ
is forbidden: algebraic sum of charges at end of reaction = sum at start

The sizes of things:

a proton is about 1 fm (=10^-15m) in diameter.

A nucleus is a close-packed array of protons and neutrons, so radius
```
R ~ R₀ A^1/3 where R₀~ 1.4 fm
```
In turn, this allows us to estimate the energies involved: if a proton is trapped in a nucleus 5 fm in radius
```
δxdp ~ h 
```
and δx ~10^-14 m
\color{red}{ E = \frac{{p^2 }}{{2m}} \approx \frac{{h^2 }}{{2m\left( {\delta x} \right)^2 }} \sim 10MeV}
Typical energy levels in nuclei tend to be separated by 10 MeV, compared to 10 eV in atoms.

Nuclei build up in much the same way as atoms in the periodic table: force are much more complicated, so cannot really solve for the energy. Light nuclei:
Note that there are no stable elements of A = 5 or A = 8: if you make ⁸Be it will decay instantly into 2 ⁴He.

Roughly speaking, need equal numbers of p and n: if a nucleus is badly out of balance, it will undergo a decay.
Many nuclei can be created but only a few are stable: this shows nuclei up to oxygen.

Whole pattern shows N ~ Z for light, N > Z for heavy.

Radio-Activity and Decays

Becquerel:

Radio-activity and nuclear decays: have already seen 3 varieties.

Simplest conceptually is γ decay: just as atoms have energy levels, so do nuclei. . However, they are more complicated:

two kinds of particles
protons and neutrons each have their own levels
Forces are more complicated (often use SHO)

One of the protons (or neutrons) can make a transition if there is a gap.Energy is much higher than in atoms: ~ 10 MeV.

β-decays

β-decays: an isolated neutron will decay.
```
n ⇒ p + e^-
```
This led to a huge problem:the electron came out with varying energy.
Fermi (1933) proposed that the missing energy is carried off by an invisible particle, the neutrino, so the real reaction is:
```
n ⇒ p + e^- + ν
```
This can also happen in a nucleus if the energies are favorable: e.g. could have
¹²B ⇒ ¹²C + e^- + ν

This allows us to do radioactive dating: e.g. carbon dating.

In the atmosphere, some of the ¹⁴N ⇒ ¹⁴C by cosmic rays. This gets incorporated in living things, substituting for ¹²C .
When the object dies, no more ¹⁴C is absorbed, and what is already there decays back to ¹⁴N, with a half-life of 5700 years.

e.g. the Turin Shroud, (supposedly used to wrap Christ in when he was lowered from the Cross)
Proportion of ¹⁴C which is 89.5% of that of current materials.
How old is the shroud?

So we now understand almost everything...

Spectra Problems
- ~~What gives the formulae?~~
- ~~Why do atoms only emit certain wavelengths?~~
- ~~How are absorption and emission related?~~
Thomson-Millikan Problems:
- ~~Why is the electron so much lighter than an atom?~~
- ~~What is the positive charge that must be in the atom?~~
Radioactivity Problem:
- ~~Where does the radioactivity come from?~~
Black Body Problems:
- ~~Where does the B-B curve come from?~~
- ~~Why does it depend on temperature?~~
Planck-Einstein Problem:
- ~~How can something be a wave and a particle at the same time?~~

X-ray Problems:
- ~~What are X-rays?~~
Rutherford's Problems:
- ~~Why don't the electrons fall into nucleus?~~
- ~~Why are all atoms the same?~~
Bohr-atom Problems:
- ~~How does the He atom work?~~
- ~~Where does the Periodic Table come from?~~

So with the (in principle) simple assumption that waves have particle-like properties and particles have wave-like properties, we have understood all of the problems that arose at the turn of the century.

But this is only part of quantum mechanics: we can also understand
Antiparticles: For every particle with given properties, there is a corresponding anti-particle with the properties flipped:
- e.g. electron has charge -1.6x10^-19 C
- positron has same mass, charge = 1.6x10^-19 C
Solids and liquids: e.g why copper is a good conductor and plastic is a lousy one
Nuclear forces (why don't they simply fall apart, what makes uranium radio-active, but not lead)
Transistors and hence integrated circuits
Light in fibres
Stars (how long will the sun last,and what will happen to it)
Superconductors (why some materials conduct electricity perfectly)
Lasers (another idea that started with Einstein)
and a whole lot more
such as Magnetic Resonance Imaging (MRI)

Cold Fusion

"Discovered" in March 1989 by Stanley Pons and Martin Fleischmann: confirmed by Steve Jones et al shortly afterwards
Electrolysis separates water into oxygen and hydrogen

Palladium electrodes absorb deuterium for several days.
When electric current is stopped, electrodes continue to release heat: up to 100 W. Helium is found in residual gas. Only possible interpretation is
D + D --> ⁴He+ energy.

What is the exact reaction?
Is not allowed in free space: instead we should get
So should see neutrons: can detect these with n detector
So should see tritium: can be detected in residual gases

Why does it work?

All sorts of reasons.
a) Obviously what the Pd lattice does is to compress the deuterons together to allow the process to run much faster rate ~ 10^-9 s^-1
Pressure in Pd can be calculated at 10²⁶ atmospheres (Centre of sun is 10⁹ atmos)
b) Electron screening:
c) Many body effects (sometimes 10⁶ electrons do not behave like 10⁶ times one electron: e.g. in superconductors)
SO "It is no longer possible to lightly dismiss the reality of cold fusion" Julian Schwinger (Nobel laureate)

Why is the scientific establishment suppressing the most exciting scientific discoveries of the 20^th century?
Could it be that it is being done by the oil companies?!?!?!?!?
The reality seems to be
1. Not nearly enough neutrons are seen (if any!)
2. Tritium shows up as a contaminant almost everywhere (and the quantities are tiny)
3. The energy released (if it is real) is probably just caused by distortion of the lattice of atoms
and finally: is this a success or failure of the scientific method?
A resounding success. a very provocative and exciting idea caused hundreds of people (including myself!) to drop everything to investigate it. The cumulative work showed that there is NO appreciable effect: it probably cost M$10 to show this.

But

Problems

Which slit did the electron go through??
What waves??

Since quantum mechanics works so well, maybe we shouldn't worry about what it actually means.....