PHYS1008: Wave Mechanics

A wavicle

Objectives: by the end of this you will be able to answer the following Are particles really just solid objects? How do we describe a particle as a wave? What is Schrodingers equation? How accurately can we measure things? How do complex atoms arise How do spectra arise? What are X-rays

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"
• "particle" of light is photon (γ : gamma)
• Einstein/Planck suggest light (wave) has some particle properties, 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 ?

1. .730 m
2. 730 nm
3. 7.3 nm
4. 73 μ

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 What is size of a buckyball? What is λ for these parameters?

1. .5x10^-10m
2. .5x10^-12m
3. .5x10^-15m
4. .5x10^-6m

• 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

• 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}}

• \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!

1. .1 eV
2. 37 eV
3. 6x10^-18 eV
4. 3.7x10⁶ eV

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.

• 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
  

• (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
  

Bohr Atom Revisited

• De Broglie suggested that allowed orbits have an integral number of waves fitted R = nh 2πp in 2πR = nλ= nh/p

• Hence

  pR = mvR = nh/2π = nħ = L
  

• which is exactly Bohr's quantization condition.
• 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.
• ⇒ Only certain energies allowed. Bohr was half right:

  En = -13.6  eV
        n²
  

Atoms

• Charge on nucleus is no longer 1, so

  En = -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

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

• Shells are specified by n and L, but L is written as
  0. s (L = 0) for sharp
  1. p (L = 1) for principal
  2. d (L=2) for diffuse
  3. f (L=3) for faint
  (comes from old spectroscopy notation)
• Hence the lowest shell is written as 1s, next shell has 2 subshells 2s and 2p, etc. The number of electrons k in each shell is written as ^k
• Hence we write the electronic structure of Li as 1s² 2s¹. We usually ignore full shells, since they play no role in chemistry or physics, so write Li as 2s¹

1. 1s² 2s² 2p⁶ 3s³
2. 1s² 2s² 2p⁶ 2s² 3p¹
3. 1s² 2s² 2p⁶ 3p³
4. 1s² 2s² 2p⁶ 3s² 3p¹

Radiation

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".

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)

• i.e. X-rays are just very energetic photons.
• Levels have energies given by Bohr formula: E_n = 13.6 Z²/n²
• hence most energetic X-ray will always have energy E = 10.2(Z-1)²
• why Z-1?
• Because there is one electron left in the innermost orbit, and this shields the nucleus)

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.

• 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?
  • Where does Moseley's formula come from?
• 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?

• 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
• A couple of applications:

Electron Microscope

• Can use electrons focussed by magnetic fields to produce a highly magnified image of object. Either

  SEM (Scanning Electron Microscope

• e.g ant with microchip (from the Science Museum, London)

• TEM (Transmission Electron Microscope)

• Limited by abberation in lenses. Limit is ultimately defined by wavelength of electrons: 100 keV gives magnification of about 10⁵.

Lasers

• Usually excited states decay producing photons. All photons have same energy but are produced at random by "spontaneous emission". lifetime ~ 10^-9 s.
• Phase of photons is also random.

• However, if we have an atom in an excited state, a second photon γ will cause it to decay, and then the new photon will be added to the first.

• This is "stimulated emission", and amplifies original signal.. Hence "Light Amplification by Stimulated Emission of Radiation" or Laser. In this case the outgoing photons are all in phase.

• Some states in atoms are much more long lived than this: these are "metastable".

  Hence find an atom with 3 levels, E₁,E₂,E₃, one of which is metastable. Get the atom into the 3rd state (by collision or excitation). Then (e.g) it drops into E₂ by spontaneous emission and then remains there until a second photon hits it

• Advantages:
  • Can pump slowly, but de-excite rapidly. e.g. CO₂ lasers run at about 100W, but pulse can be 10^-8 s, so instantaneous power ~ 10¹⁰ W.
  • Beam is very tightly focussed (best conventional methods give spread of about 1/2^o, lasers can be 10^{-4 o}
  • Beam is "coherent": i.e. all light is in phase

But

• Which slit did the electron go through??
• What waves??
• So maybe we shouldn't worry about what it actually means.....