Quantum Mechanics

Quantum Mechanics
The predictions of quantum mechanics are rather bizarre from classical
point of view:
Propositions of classical physics:
The physical universe is deterministic, i.e., given enough information
about a physical system, its future evolution can be predicted exactly.
The entire function of classical mechanics is to derive such predictions.
Light consists of waves, which ordinary matter is composed of particles.
The former statement is one of the triumphs of classical
electromagnetism, while the latter seems self-evident.
Physical quantities, such as energy and angular momentum, can be
treated as continuous variables, Again, this assumption is built into the
structure of classical mechanics.
There exists an objective physical reality independent of any observer.
If tree falls in the woods, of course it makes a sound.
Propositions of Quantum Mechanics:
The physical universe is not deterministic. At the subatomic level, we
can assign probabilities to the outcome of certain experiments but never
predict the exact result with certainty. Uncertainty is an intrinsic property
of matter at this level.
Both light and matter exhibit behavior that seems characteristic of both
particles and waves.
Under certain circumstances, some physical quantities are quantized,
i.e., they can take on only certain discrete values.
Finally, it appears that the observer always affects the experiment; it is
impossible to disentangle the two.
Why would anyone believe such a preposterous set of ideas? For
the only reason that any theory in physics is given credence:
because it works!!!
What is quantum mechanics?
Quantum mechanics is a quantum theory that
supersedes classical mechanics at the atomic
and subatomic levels. It is a fundamental
branch of physics that provides the
underlying mathematical framework for many
fields of physics and chemistry, including
condensed matter physics, atomic physics,
molecular physics, computational chemistry,
quantum chemistry, particle physics, and
nuclear physics. It is a pillar of modern
physics, together with general relativity.
Electron waves imaged by STM
You see electron waves!!
Quantum Corral
Writing with atoms
Lecture 1
The emergence of quantum mechanics
Best historical review: 3 Hans Bethe’s lectures: http://bethe.cornell.edu/
1.1 Collapse of Classical Physics
• Wave-particle duality
• Stability of atoms
1.2 Historical view on the emergence of quantum physics
1911 Solvay Conference
Seated (L-R): W. Nernst, M. Brillouin, E.
Solvay, H. Lorentz, E. Warburg, J. Perrin,
W. Wien, M. Curie, and H. Poincaré.
Standing (L-R): R. Goldschmidt, M.
Planck, H. Rubens, A. Sommerfeld, F.
Lindemann, M. de Broglie, M. Knudsen, F.
Hasenöhrl, G. Hostelet, E. Herzen, J.H.
Jeans, E. Rutherford, H. Kamerlingh Onnes,
A. Einstein, and P. Langevin.
1927 Solvay Conference
Seated (L-R): Walther Nernst, Marcel Brillouin, Ernest Solvay, Hendrik Lorentz, Emil
Warburg, Jean Baptiste Perrin, Wilhelm Wien, Marie Curie, and Henri Poincaré.
Standing (L-R): Robert Goldschmidt, Max Planck, Heinrich Rubens, Arnold Sommerfeld,
Frederick Lindemann, Maurice de Broglie, Martin Knudsen, Friedrich Hasenöhrl, Georges
Hostelet, Edouard Herzen, James Hopwood Jeans, Ernest Rutherford, Heike Kamerlingh
Onnes, Albert Einstein, and Paul Langevin.
1.3 The birth of photon: photoelectric effect
hνth = Φ
What are not understood by classical mechanics?
E does not depend on light intensity but
depend on frequency
A minimum frequency (not intensity) is
required for the photoelectron emission
Einstein hypothesized that light is composed of localized bundles
of electromagnetic energy called photon (hν)
E = hν - Φ
Sommerfeld Fermi gas model
Vacuum level
Φ (work function)
Fermi sea of electrons
Electrons are excited by quanta of light instead of classical
electromagnetic wave !
1.4 Compton scattering
Arthur H. Compton observed the scattering of x-rays from electrons in a
carbon target and found scattered x-rays with a longer wavelength than those
incident upon the target. The shift of the wavelength increased with scattering
Compton explained and modeled the data by assuming a particle (photon) nature
for light and applying conservation of energy and conservation of momentum to the
collision between the photon and the electron.
Photon: (hc/λi, Pi) → (hc/ λf, Pf)
Electron: (mec2, 0) → (√[pec2 + (mec2)2], pe)
Compton formula
Another direct
proof of
quantization of
light: photon
1.5 Blackbody radiation: birth of Planck constant
Any theory based on continuous
radiation of electromagnetic waves
failed to reproduce the observed
Planck assumed that the energy
radiated from radiation oscillators
could be only some whole number
multiple of hν where is the planck
ε = nhv, n = 1,2,3, …
Radiation energy density: the energy per unit volume per unit frequency
Application of Planck formula
Background radiation (2.7K) of the universe
1.6 DeBroglie hypothesis and Devisson-Germer experiment
Why Bohr’s orbitals of atoms are quantized?
(L = mvrn = n= )
Photon case:
p = E/c = hν/c = hλ
ω = 2πν
λ = h/p
k = 2π/ λ
p = =k where = = h/(2π)
deBroglie Hypothesis:
Any particle moving with a momentum p would have a wave of matter with its
wavelength as
λ = h/p
For an electron in an atom, the stationary orbits in the Bohr model has an integral number
of wavelengths.
2πrn = nλ
This experiment demonstrated the wave nature of the electron, confirming the
earlier hypothesis of deBroglie. Putting wave-particle duality on a firm
experimental footing, it represented a major step forward in the development of
quantum mechanics.
Low energy electron diffraction:
d sinθ = n λ
Wave-particle duality: matter exhibits properties of both waves and particles.
1.7 Heisenberg Uncertainty Principle
The more precisely the position is
determined, the less precisely the
momentum is known in this instant,
and vice versa.
--Heisenberg, uncertainty paper, 1927
One cannot measure values (with arbitrary precision)
of certain conjugate quantities, which are pairs of
observables of a single elementary particle. These
pairs include the position and momentum
sinθ ~ θ
λ = h/p ~ h/px
∆y ~ a
∆py/px ~ θ
a sin θ = λ
∆y ∆py/px ~ h/px
∆y ∆py ~ h
1.8 References and homework
R.l. Liboff, Introductory Quantum Mechanism