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The difference between a density matrix and a state vector is that the latter corresponds to having more complete information about the system. Let’s see how this plays out in a simple example.

A density matrix that is diagonal with entries of {1}/{2} means that we are talking of just a two dimensional state space (such as that of a single electron whose position we are ignoring), and that we have chosen a particular observable, such as spin component in the z direction, to determine the basis with respect to which you are representing the state.

In this context there are actually many different pure states for which the probability of having spin up or down in the z direction are equal. It could for example be certain to have a particular value in (say) the x direction, or the y, or any other direction perpendicular to the z axis. Each of these corresponds to a “ket” whose components (relative to the z eigenstates) are of the form {1}/{\sqrt{2}} multiplied by complex phase factors (and the different relative phases correspond to different directions of spin in the xy plane). On the other hand, representing the state by that density matrix means that we have prepared the electron in such a way that its spin has equal probability of being measured up and down in the z direction, but also that all ways of getting that result are equally likely so there is also equal probability of getting up or down in any other direction.

Source: (1002) Alan Cooper’s answer to Why can’t the density matrix with 1/2 in their diagonals be equal to a ket wave function with 1/2 in each entry. Do they not both describe how each 0 or 1 state has a 1/2 probability? – Quora

People often ask what it means for an electron to have spin 1/2.

Here is my attempt at an informal explanation.

It means that electrons (and most other elementary particles) are represented by wave functions or fields whose values are not given just by complex numbers (the “scalar” or “spin zero” case), but instead by complex vectors (of “internal” coordinates) which admit a finite dimensional representation of the rotation group. The action of a rotation on a state then corresponds to the usual change of position in space combined(*) with a reorientation of the “internal” coordinates.

It turns out (due to mathematics that I cannot usefully(*) insert here) that the possible results of measuring angular momentum corresponding to “internal” properties of a particle occur with increments of just half of those corresponding to measurements of the classical orbital angular momentum.

And the electron happens to be an example of the simplest kind of non-scalar field.

(*) – The crux of this can perhaps be inadequately explained by saying that the way the internal and external actions of the rotation group have to be related is such that one full rotation in the position space produces a sign reversal in the internal space and so to bring everything back to where it started actually requires two full rotations in the position space.

Source: (1001) Alan Cooper’s answer to Why is the spin of an electron equal to half? What does it mean by half? – Quora

Spacetime is like a movie reel, or better a stack of pictures on top of one another. Time is just a way of labelling the individual frames or pictures, and the word “move” just means to have different coordinates in space at different times ie to be at different places in different pictures. It doesn’t make sense to ask for the image of something in one picture to be in a different picture.

Source: (1001) Alan Cooper’s answer to If time is the 4th dimension, why can’t anything move in either direction like the other 3 axes? – Quora

For any motion with constant acceleration over a time interval, the product of acceleration times distance travelled over that interval is equal is equal to half the change in the square of the speed in the direction of acceleration. (This is just the junior high school version of the calculus identity \int{x’’ dx}=\int{x’’ x’ dt}=\int{x’ x’’ dt}=\int{x’ dx’}=\Delta(x’^2/2).)

So the quantity that is increased by applying force through a distance (to do “work” on a particle) is quadratic in its speed. But why is this important enough to give it a special name? That is because it allows us to define a quantity that is conserved throughout the evolution of any physical system.

As a result of Newton’s law of action and reaction (which is basically just a way of expressing conservation of momentum), in the motion of any system of particles the sum of mv^2/2 for all the particles plus the net work done against outside forces is a constant. We call this the “Energy” of the system and identify the part involving the speeds (that does not include work against the outside world) as the “kinetic” part of that energy – and the outside work (which includes an arbitrary constant depending on what we take as the starting point) is a called “potential” energy since it could in principle be returned to the system in future interactions.

Source: (1001) Alan Cooper’s answer to Why does kinetic energy increase quadratically, not linearly, with speed? – Quora

Spacetime with matter and gravity is not symmetrical. There is a particular singularity relative to which what we identify as our experience includes information (which we call memories) about events “closer” (not in spatial distance but in the spacetime metric) to the singularity (which we call the “past”), but not about those further away (which we call the “future”). Part of what is in our memory is experiences in which the scope of our memory was smaller. This gives us the feeling of becoming progressively further away from the singularity – ie of moving “forward” in time.

Source: (1001) Alan Cooper’s answer to Why do we experience the fourth dimension as constantly moving in one direction? – Quora

John Platts’s answer to Why does a mirror reverse things horizontally but not vertically? includes some nice illustrations and discussion but declares that the front-to-back reversal is not horizontal.

Interesting point. But I think we use the word “horizontal” to refer to any part of a line or plane that is perpendicular the line joining it to the Earth’s centre, regardless of whether or not our view of it actually appears parallel to the horizon. We often test for this property by looking at it from a position where it lines up (left-to-right) with the horizon but two people building a house will generally agree that a beam is horizontal even when they are not looking at it from such a position. (If that were not the case the word would be a lot less useful because the property of a beam being horizontal would depend on the observer and would be lost whenever she changes her orientation.)

So, in my understanding, horizontal lines aren’t necessarily parallel to the horizon in a perspective view, but (if the world was flat) their infinite extensions (would) all terminate on it (though due to the Earth’s curvature that actually happens a tiny bit above what we see as the visible horizon).

Naive students sometimes ask what would happen if electrons or other fermions were “forced” to violate the Pauli Exclusion Principle, but they never suggest any idea of what this would actually mean.

Whatever they are imagining probably involves thinking of those fermions as independent particles such that you can somehow keep track of which is which; but in quantum theory there is no way of doing that, and no-one has ever suggested an alternative theory that fits the facts and in which such separate identities make sense.

Source: (1000) Alan Cooper’s answer to What happens to fermions that are forced to violate the Pauli exclusion principle? – Quora

A Quora question asks: Can you deduce time is dilated based on blue shifted light? Or can light be blue shifted without the phenomenon of time dilation?

Here’s my answer:

Yes, and yes.

Taking the questions in reverse order, the non-relativistic Doppler effect (ie neglecting time dilation) is a blue shift for an approaching source and a red shift for one that is receding; and the non-relativistic calculation is reasonably accurate for speeds that are not too great. So, yes, light would be blue shifted without the phenomenon of time dilation. (But this is just a qualified “yes” because in fact no relative motion can occur without the phenomenon of time dilation, so I had to say “would be” rather than “can be”.)

With regard to the first question, the relativistic correction (attributable to time dilation) is basically to make a small red shift adjustment to the non-relativistic calculation (ie to make red shifts slightly stronger and blue shifts slightly weaker). So, by noting the difference between an observed blue shift and what would be predicted classically from the observed rate of approach, we can see evidence of the time dilation effect.

If the electron was orbiting they would be called “orbits”.

The word “orbital” therefore correctly indicates that they are not orbits (but suggests that they are related to what would be the allowed orbits if the electron was a particle with the only quantum effect being to confine it to discrete energy levels).

Source: (1000) Alan Cooper’s answer to Why are the atomic orbitals called ‘orbitals’ if an electron is not orbiting the nucleus? – Quora