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Lecturer
Available information
- course policies
- announcements & suggested reading
- problem sets/solutions, practice exams
- student grades
Problem sets
- posted by Thursday
- due following Thursday afternoon
- late homework not accepted (solutions are published online)
- one lowest homework score will be dropped when calculating grades
Grading
- Exam 1 20%
- Exam 2 20%
- Final 40%
- Problem sets 20%
Collaboration on problem sets encouraged, but everybody has to submit own solution.
Textbooks
- Gasiorowicz: required
- French & Taylor: strongly recommended
- Feynman, Lectures on Physics: selected chapters
Texts as reference & reading as preparation. Lectures are basis; notes will be posted.
Learning goals for 8.
- boundary between classical and quantum physics
- understand crucial experiments that paved way for development of quantum mechan ics
- understand & interiorize probability amplitude and interference concepts that are at the heart of QM
- single-particle quantum mechanics for external degrees of freedom; Schr¨odinger equation
- internal degrees of freedom; e.g., spin: 8.05; many body quantum physics: 8.06 and beyond
- some formal structure of QM (operators, expectation values, commutators, Dirac notation) further development: 8.
- understand interface between mathematical structure (Schr¨odinger equation as partial differential equation) and physical interpretation, measurement, uncertainty, correla tions, and entanglement
- study important QM systems: harmonic oscillator, hydrogen atom
- At the end of this course you should be able to:
- solve simple QM single-particle problems in one and three dimensions (scat tering, tunneling, bound states)
- give a physical interpretation of mathematical entities (operators, wavefunc tion, state representation in different bases, Fourier transform, Heisenberg un certainty relation)
- appreciate & understand the all-importance of interference effect (addition of probability amplitudes) in QM
- 8.04: only non-relativistic QM
Problems with/failures of classical mechanics (CM)
- CM fails at microscopic level
- CM cannot explain, e.g.,
- stability of individual atoms
conducting walls the modes satisfy λn = (^2) nL^ , n ≥ 1 integer. There are infinitely many short- wavelength modes inside the container. If each contains average energy kB T , then the energy stored inside the container must be infinite.
QM. The mode frequency νn = (^) λ^ c n sets a natural energy scale (photon energy) E (^) n = hνn,
(h = 6. 6 × 10 −^34 J s is Planck’s constant), modes whose natural energy scale· E (^) n is much larger that kB T are not thermally populated, they remain empty and carry no thermal energy. High-energy modes with E (^) n � k (^) B T are “frozen out”. They do not carry thermal energy. →Energy inside box remains finite, spectrum and energy per mode agree with experi ments. (Planck formula: 8.044).
Heat capacity of diatomic gas
Monatomic gas of N atoms has heat capacity (energy stored at temperature T ) given by C (^) V = 23 Nk (^) B, in agreement with measurements. There are three translational degrees of freedom per atom, each degree of freedom stores kinetic energy 12 k (^) B T. For a gas of N diatomic molecules, we expect CV = 72 Nk (^) B, 2N atoms with translational degrees of freedom, or 3 center-of-mass translational degrees of freedom, 2 rotational de grees of freedom, 2 vibrational degrees of freedom (one kinetic energy, one potential en ergy). However, observation at room temperature is CV = 25 Nk (^) B.
(a) One vibrational degree of freedom. (b) Two rotational degrees of freedom.
Figure II: Degrees of freedom of a diatomic molecule.
Explanation. Vibrational mode with frequency ν has natural energy scale E = hν � kB T , is “frozen out,” does not contribute to heat capacity at room temperature. At high temperature k (^) B T � hν : C (^) V → 27 Nk (^) B.
What about electronic degrees of freedom inside atom?
Also frozen out. E (^) n ∼ 1 eV� k (^) B T = 401 eV at room temperature.
