Key definitions for QM: Part 3

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In this post some additional QM terms will be defined. These often come up in applications of QM and might come into play for interpretation issues.

Hamiltonian: In classical mechanics, the Hamiltonian is a function of the configuration and velocities that equals the energy of the system, giving the energy as a sum of that of the various types of energy in the system. In QM, it is a corresponding linear operator on the wavefunction. The Hamiltonian appears in the Shrodinger equation. Its eigenstates have definite values for energy. A wavefunction that is an energy eigenstate will not undergo change as time passes except as a standing wave, undergoing phase rotations.

commute: Let A and B be operators. They commute if A B psi = B A psi for any function psi. This is written as AB = BA or [A,B]=0.

If two operators don't commute, then measurements associated with one of them will change the probabilities for values of measurable quantities associated with the other, and they can not be measured simultaneously.

Position does not commute with momentum (which is mass times velocity). Spin measurements in different directions also don't commute.

(Heisenberg's) uncertainty principle: There is a minimum uncertainty for the product of measurable quantities that don't commute. This follows from the math (and the Born Rule). Most famously, the product of (spread in position) (spread in momentum) >= hbar/2.

This can be understood roughly as follows: Momentum is related to the wavelength of sinusoidol patterns in the wavefunction - those are its eigenstates (actually there is an imaginary component as well - the wavefunction is complex-number-valued). If the wavefunction is concentrated near a point (small uncertainty in position), then it must be built up out of a superposition of a wide range of sinusoidal functions. If on the other hand it is in a nearly sinusoidal pattern, then it must be spread out over a large range of positions.

(The Born Rule comes into play because we assume the usual relation between probability and the square of the wavefunction.)

There is a similar uncertainty principle that relates uncertainty in energy and time.

Wikipedia's article has a more in-depth explanation.

boson: Particles with "integer spin" have spin component eigenvalues that are integer multiples of hbar. Such particles are bosons, which means that they have a tendency to occupy the same states as identical particles of the same type; technically, their wavefunctions are symmetric with respect to exchanging the particles. Photons (particles of light) are bosons, which lets them reinforce each other and produce the classical-seeming behavior of electromagnetic fields.

fermion: Particles with half-integer spin are fermions, which means they cannot occupy the same state as identical particles of the same type; technically, their wavefunctions are anti-symmetric with repect to exchanging the particles.
The connection between spin values and boson/fermion behavior is a consequence of relativistic quantum field theory (QFT). Actually in QFT there are no particles, just quantized excitations of the fields. It is not surprising that treating excitations of the fields as though they were particles (as is done in the nonrelativistic approximation, used very often in QM) would require some special treatment of the so-called particles with respect to the symmetry of exchanging them.

Pauli exclusion principle: This is the principle that, as mentioned above, no two fermions of the same type can occupy the same state. Electrons are fermions, so this is very important in atomic physics and chemistry. Atoms have various shells of electrons which can be though of as built up by adding one electron at a time. When an inner shell is fully occupied, another electron can't occupy one of those states, so it will end up in the next shell out. (If placed in an even higher shell out, it will fall to the innermost shell it can, emitting a photon to carry away the extra energy.)

Shrodinger Picture: This is the usual formalism in which the wavefunction varies with time while measurable quantities are associated with fixed linear operators. There is a global time.

Heisenberg Picture: This is a formalism in which the wavefunction is static but linear operators vary with time, giving the same Born Rule probabilities for measured outcomes. In relativistic quantum field theory, this picture has fewer problems of being mathematically well defined (with infinite renormalization, or re-scaling of certain quantities) than the Shrodinger picture, and is also the only local formulation of QM. (Local meaning that things defined at points in ordinary space only interact with their neighbors.)

However, this formalism is harder to work with. It is also believed that infinite renormalization will not be necessary for a fundamental model that includes quantum gravity.

The Heisenberg picture has the strange feature that interactions carry labels with them of what has been interacted with, and these proliferate as more and more systems interact. Basically, at each point, a field operator encodes information about the correlations of the field at that point with the set of field configurations over all space. At each point in space the field operator must be capable of carrying an unlimited amount of information about all of space and updating it as time passes. Although there are no shortage of infinities in most models of physics, this would seem surprisingly inefficient for the fundamental working of nature. Of course, nature has surprised us before.

For more detailed and technical information about locality and label proliferation in the Heisenberg picture see
Locality in the Everett Interpretation of Heisenberg-Picture Quantum Mechanics
Locality in the Everett Interpretation of Quantum Field Theory

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