In the last post, definitions Part 1, I explained some of the terms that commonly come up in interpretation of QM and described their roles in that context. Here, I will define some other useful terms; these are more technical and less key to understanding most interpretation issues, but still handy in that context (and fun!) You can look up the equations that are involved; my concern here is with what is relevant to interpretations.
spin: This refers to a property of individual particles that behaves like an intrinsic angular momentum. When measured, it has a constant magnitude, and the component of it in the measured direction can only take on a few discrete values.
Spin-1/2 particles, such as an electron, have two possible eigenvalues of their measured spin component: + or - 1/2 hbar. When not measured, they are in a superposition of the eigenstates with the allowed values (or in an entangled state). Such a superposition is always an eigenstate of the spin measurement operator in some other direction, though an entangled state is not.
degenerate: While this term may refer to modern society, in the context of QM it means that there are more than one eigenfunctions of a particular operator with the same eigenvalue. Measurements based on that operator will not cause degenerate eigenfunctions to decohere - for example, if you measure energy and there are two eigenstates with the same energy, those two states will remain in a coherent superposition and the observer will not distinguish out a unique eigenstate.
bra and ket notation: Dirac invented this useful notation in which a function can be represented by a 'ket', or the second half of a bracket, written |label> where "label" is used to describe which function is being referred to. For example, if f(x) = sine(ax)exp(-bx^2), one could write |f_ab>.
A 'bra', or the first half of a bracket, represents the complex conjugate of the same function.
It is written < F_ab|
A 'bracket', such as < f | g >, represents the integral (sum over the configuration space) of the bra function (here, complex conjugate of f) multiplied by the ket function (here, g).
Often (though not always), bras and kets are normalized so that < f|f> = 1.
If you see two kets next to each other, such as |b>=|f>|g>, this means function b is the product of the function f that lives in the configuration space of one system and g which lives in another system: b(x,y) = f(x) g(y).
quantized: Some measurements have discrete possible outcomes, and such quantities are called quantized. For example, the energy levels of an electron in a hydrogen atom are quantized, but the energy of an electron that escapes the atom can take on a continuous set of values so it is not quantized.
It can also refer to obtaining a quantum mechanical model from a classical one.
'Quantum mechanics' originally referred to quantized quantities but is now used to describe the whole branch of physics which deals with related phenomena such as the wavefunction.
Plank's constant, h: This is a constant that appears in the Shrodinger equation. It sets the scale at which quantum phenomena have noticable direct effects. It has units of 'action', units which are those of (mass)(velocity)(position). More commonly encountered is hbar, which is h/2 pi. The plain h is more useful for full oscillating cycles, which are common enough with waves, while hbar is useful for instantaneous rates of change.
h = 6.63 x 10^-34 kg m^2 / s
geometric optics limit: While there are many issues involved in deriving the appearance of a classical world from the wavefunction model, if we grant the validity of the Born Rule for probabilities then an important part of the derivation of classical mechanics is simple after that:
When the wavelength of a wave is much smaller than the size of whatever openings it goes through, the spreading out of the wave become negligable. This is the same reason that light waves can be treated as coming out of a flashlight in straight lines, while sound waves much more noticably bend around corners. There are still small tails where a tiny portion of the wave's squared amplitude will spread off, which is why I invoked the Born Rule, since it lets us neglect that part.
Just as most of a light wave will move in a straight line, most of a quantum matter wave for a non-microscopic (macroscopic) object will follow the trajectory predicted by classical mechanics. The small bit that will not generally has a Born Rule probability so low that it is effectively impossible to ever measure.
Hilbert space: The state of a quantum system is given by the wavefunction, which is a function on configuration space. This can be thought of as representing a 'state vector' in an abstract space.
An ordinary vector in regular 3 dimensional space is a quantity which has both direction and magnitude - for example, velocity. It can be represented by x,y,z components: V = (vx, vy, vz). Thus, in a particular coordinate system, it is written as a function of a discrete index which can only take on 3 values; e.g. V(1) = vx, V(2) = vy, V(3) = vz.
A Hilbert space is a generalization of this to describe any function as a vector in some high-dimensional space.
Philosophically, thinking of the function as a vector implies that the particular coordinate system in which the components are spelled out is not very important. The physical nature of a velocity would be the same no matter what x,y,z coordinate system we are working in, but the components would look different.
quantum mechanical basis: This is analogous to choosing a coordinate system to write the vectors of the Hilbert space in. Changing a basis is analogous to rotating the directions of your coordinate system components.
Examples include 1) position basis, in which the wavefunction is a function of position as expected, 2) momentum basis, in which the wavefunction is a function of particle momentum, which is (mass)(velocity). It may not seem at first that the two pictures are equivalent, because in classical mechanics, knowing the momentum will not tell you the position. But in quantum mechanics, in which the wavefunction is a complex number valued function, knowing the wavefunction at every point in the momentum basis is enough to find it in the position basis and vice versa.
Just as there is rotational symmetry in ordinary space which makes it impossible to know if there is in nature any actual, fundamental coordinate system in which vectors really have three components, it is impossible to know what the "actual basis that nature uses" is in QM. Under the Copenhagen interpretation, the assumption was that there is no such thing - only things we can measure were considered real.
As will be seen, more mechanistic, literal views of mathematical models (such as the MWI is) seem to require some actual basis that nature would use.
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