PHYS2903: Old Quantum Theory

Orion Nebula: the red light is (mostly) hydrogen Balmer series

Summary of Classical Physics

• Haven't we learned a lot!

• Jakob stepped back to admire what he had written last: the concise equations of an ideal mass point moving under the influence of elastic forces. How beautiful classical physics is! Night Thoughts of a Classical Physicist McCormach

• but note the hubris!

• Jakob stepped back to admire what he had written last: the concise equations of an ideal mass point moving under the influence of elastic forces. How beautiful classical physics is! (Was! He meant before the Quantum Theory) Night Thoughts of a Classical Physicist McCormach

• Basis of Success: Newton's Laws of Motion valid for everyday objects, but also for very large
• Falling Apple ⇒ Planets ⇒ Galaxies
• and very small
• Conservation of Momentum & Energy ⇒ Kinetic Theory of gases
• Or Common Sense

  The layer of prejudices we acquire before we are sixteen" A. Einstein

• So what could go wrong?

• Size of atoms ∼ 0.1 nm = 10^-10m
• Size of nucleus ∼ 5 fm = 5x10^-15m
• Times ∼ 1 ps = 10^-12s
• In what follows, we'll be using electron-Volts (eV) to measure energy:
• 1 electron Volt (eV) ∼ 1.6x10^-19J: better for atomic sized objects

Line Spectra

• Found black lines superimposed on sun's spectrum. This is the sun's spectrum "folded up"

• Heated gases give characteristic wavelengths, and can match dark lines in solar spectrum to bright emission lines: e.g. Sodium

Hydrogen is simplest

• Balmer (1885) Found
• \color{red}{\frac{1}{\lambda } = R\left( {\frac{1}{{2^2 }} - \frac{1}{{n^2 }}} \right),R = 1.1 \times 10^7 m^{ - 1} } n = 3,4...
• Extends down into U.V., so only 4 lines can be seen easily.
• Subsequently Paschen found lines in the UV \color{red}{\frac{1}{\lambda } = R\left( {\frac{1}{{1^2 }} - \frac{1}{{n^2 }}} \right)}
• and Lyman found lines in the IR \color{red}{\frac{1}{\lambda } = R\left( {\frac{1}{{3^2 }} - \frac{1}{{n^2 }}} \right),}
• Problems
  • What gives the formulae?
  • Why do atoms only emit certain wavelengths?
  • How are absorption and emission related?

Photo Electric Effect

• Light falls on metal, ejects negative charge, which we now know consists of electrons: can measure energy of electrons by stopping them in an electric field

• No problem, since light carries energy?

• Problems for the PE effect:
  • The energy of electrons does NOT depend on intensity of light,
  • DOES depend on frequency.
  • No electrons emitted below critical frequency f₀, which depends on metal

Discovery of Electron

• Millikan (1909) measured (indirectly) mass of electron = 9.1 x 10^-31 kg
• mass of H. atom = 1.67 10^-27 kg

• (Thomson) proposed currant bun model of atom,: electrons imbedded in positively charged material.

• Problems:
  • Why is the electron so much lighter than an atom?
  • What is the positive charge that must be in the atom?

Black-Body Radiation

• Black-body radiation: the radiation emitted by all hot bodies is (almost) exactly the same. Must measure temperature in degrees absolute (K).

  T(K) = T(oC) + 273
  

• so that room temperature (~20^oC) is ~ 300 K. For stars T ~ 6000 K ⇒ 20,000 K so doesn't matter whether we use K or ^oC!

  Two fundamental laws:

• Total Energy \color{red}{U = \sigma T^4 ,\sigma = 5.67 \times 10^{ - 8} Jm^{ - 2} K^{ - 4} }
• Wavelength of peak \color{red}{\lambda _{\max } = \frac{B}{T},B = 2.9 \times 10^ {- 3} mK^{ - 1} }
• What does this look like??

• To derive a theory of Black body radiation put E.M. standing waves in cavity. no particular reason to assume that any one wavelength preferred to any other: short waves dominate (since there are more of them)

• this gives Rayleigh-Jeans curve. The "ultra-violet catastrophe": all radiation should be very short λ. and should not depend on temp. Wien noticed that experimental curve looked like Maxwell-Boltzmann distribution. for molecules

• Where does the B-B curve come from?
• Why does it depend on temperature?

X-Rays

Röntgen (1895)

• Very penetrating radiation produced by vacuum tube (requires pot. diff ~ 20 kV), which passes through solids, fogs photographic plates

• First picture was of his wife's hand
• Problem:
  • What are X-rays?

The Nucleus

Lead block with radium salt: α-particles are produced by radium, collimated by screen to narrow beam, pass through gold foil and are detected by scintillator (produces spark of light when hit by charged particle

• Expect: rather small angle of deflection due to many small scatters

• Discover some deflected by more than 90^o
• How could this happen?

• Close encounter with electron? me ~ 1 mα 8000

• Cannonball bouncing backwards off tissue paper. Rutherford Need small, massive, positive charged nucleus. Electrons irrelevant!

• Why don't the electrons fall into nucleus?

• Maybe electrons are in orbit but an accelerating charge emits radiation so orbiting electrons should emit radiations and lose energy.

• Problems:
  • Why don't the electrons fall into nucleus?
  • Why are all atoms the same?

The End of Classical Physics

• Spectra Problems
  • What gives the formulae?
  • Why do atoms only emit certain wavelengths?
  • How are absorption and emission related?
• P-E effect Problems
  • The energy of electrons does NOT depend on intensity of light,
  • DOES depend on frequency.
  • No electrons emitted below critical frequency f₀, which depends on metal
• Radioactivity Problems:
  • What is the positive charge that must be in the atom?
  • Where does the radioactivity come from?
• Black Body Problems:
  • Where does the B-B curve come from?
  • Why does it depend on temperature?

• X-ray Problems:
  • What are X-rays?
• Rutherford's Problems:
  • Why don't the electrons fall into nucleus?
  • Why are all atoms the same?
• We would like an explanation of all of this!......

Planck's Radiation Law

• Consider box of Z different atoms: velocities of each will be random, but will "average out" to give smooth curve. Energy is equally divided between different molecules

  E = 1/2 m v²
  

• on the average: also average energy increases with temperature.
• But suppose light is a particle...
• Planck (1900) suggested that E.M. radiation is emitted in lumps (quanta) which became known as "photons"

  f λ = c 
  

• (frequency x wavelength = speed)

  E = h f
  

• (energy of photon = Planck's constant x frequency)
• Energy contained in quantum increases with frequency so costs more energy to emit short wavelength light.

• Planck's constant

  h = 6.6 x 10-34 Js = 4.1 x 10-15 eV s
  

• Photons are also particles with a difference: they always travel at c and can easily be created and destroyed.
• At room temperature average photon has E ~1/40 eV
• The photo-electric effect also involves light: maybe that can have a photon explanation.

Einstein and the Photo-electric Effect

Since particles have energy, photons must carry energy (obviously light has energy: that's how you get a sun tan!)

• Einstein (1903): (note this was what he actually got the Nobel prize for) adapted Planck's hypothesis to say that light is absorbed in quanta (photons) most energetic electrons in metal need extra boost to escape:

• K.E. = 1/2mv²= hf-φ where φ is work function: differs for different metals. Slope is universal ⇒ Planck's constant

• So we have to take photons seriously!

• Electrons are ejected as soon as light strikes (no need for energy to accumulate), since photons arrive randomly

"Classical" corks bob up and down

• "Quantum" corks are either stationary or ejected.

What is light?

• Particle? Newton, Descartes

• Kerner: Look at the edge if the shadow. It is straight like the edge of the wall that makes it. This means light is ..little bullets. Bullets go straight.
  Hapggod (Tom Stoppard)

• Wave? Young, Huyghens

• Kerner: When you shine a light through two little gaps, side by side, you don't get particle patterns like for bullets, you get wave patterns like for water. The two beams of light mix together
  Hapggod (Tom Stoppard)

• Yes? Einstein
• Light travels as wave, but arrives and departs as particle

• frequency of orange light 4x10¹⁴
• Energy of photon 2eV
• A 60W bulb produces 2x10¹⁹ photons each second
• Why are we not aware of discrete arrival of photons?

• Note that the shorter the wavelength, the more like a particle, so γ-rays (produced in nuclei) are usually treated only as particles.

The Bohr atom

Model for H. atom must explain

• Spectrum : This implies only photons of certain definite energies are emitted . . .
• Rutherford's observation of massive nucleus
• Stability
• Bohr was able to construct model of electron in orbit round nucleus lets us explain spectrum of H and ionized He⁺ but not other atoms

Wave-Particle Duality

• 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 (massless particle)
  \color{red}{ E = pc}
  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, λ ∼ 500 nm (like yellow light!) ?
• Wave - particle duality: All fundamental (i.e small!) particles also act like waves (what is an electron?...) and waves act like particles.

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

• you can even watch how it builds up

• 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 λ ∼ 10-11m, but its "size" is 100 times bigger (∼ 10-11m)

• 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!

The hydrogen atom

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!

• 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.
• This means only certain orbits allowed:
• why?

• because!

• Unfortunately the real case of an electron is a 3-D differential equation, and hence hard to solve.

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:

  En = = - 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

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 ²

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.

• 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
  PHYS 5602
• Solids and liquids: e.g why copper is a good conductor and plastic is a lousy one PHYS 4508
• Nuclear forces (why don't they simply fall apart, what makes uranium radio-active, but not lead) PHYS 3606
• Transistors and hence integrated circuits PHYS 4508
• Light in fibres PHYS 4204
• Stars (how long will the sun last,and what will happen to it) PHYS 4203
• Superconductors (why some materials conduct electricity perfectly) PHYS 4508
• Lasers (another idea that started with Einstein) PHYS 4208
• Magnetic Resonance Imaging (MRI) PHYS 5203
• and a whole lot more

  such as Giant Magneto_Resistance (GMR) Remember?

But

• 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.....