In defining the cotangent bundle, Spivak in his differential geometry book, says
Then later,
I don't understand what he means by this last comment. It seems to me that $y_*' \circ (x_*')^{-1}$ and $y_* \circ x_*^{-1}$ don't even have the same domain. How could they be equal?
Acuriousmind, I am trying to investigate about a small question in quantum mechanics. However given how in the past experienced told me I often present poorly and people have trouble figuring out what I want to do, I decided to first check whether my aim is sensible first before proceeding:
Background: We knew that a fermion wavefunction is antisymmetric. This means electrons obey pauli exclusion principle. A consequence of this is that the probability of finding two electrons of the same spin at the same position is 0 but there is nonzero probability of finding electrons of opposite spin at the same position. In the context of chemistry this means effectively this contributes an exchanging interaction between electrons in the molecular hamiltonian.
Since electron spin states can be a superposition of spin up and spin down. I want to understand how the magnitude of this exchange interaction changes with the relative angle of the spins of the two electrons, given two electrons at some fixed distance r apart
It's very hard to judge exactly how well or badly you've done in an exam. While it's tempting to beat yourself up about it the main thing is to put it behind you and draw a deep calming breath!
@ACuriousMind Uh, I don't quite understand that. We can have electron spin states like $\frac{1}{\sqrt{2}}(\lvert 0\rangle+\lvert 1\rangle)$ and $\lvert 1\rangle$. Since electron states are rays in hilbert space, and that we can plot these states on the bloch sphere, why is the notion of angle substended by the two rays not a well defined notion?
At most one component of the spin vector is ever without uncertainty due to the commutation relations and the unvertainty principle. You should not think of "spin of an electron" as being a well-defined vector beyond being able to say in which direction it is currently definite.
The Cambridge entrance didn't need to be that hard.
I don't think students are any brighter or duller now than they were in 1978, and I don't think there is any difference between students in the UK and India. The difference is in how hard the Indian students have to work for the exam, and I'm not sure that's a good thing.
I don't think getting crushed by interesting, but primarily ad-hoc super-hard technical problems is the point of education: but since most of the students want to get in engineering, JEE is invading the standard education system too.
@ACuriousMind Hmm, so that means there isn't a continuum of cases between parallel spins (e.g. both spin up, and the component under consideration is the z component) and antiparallel spins (one spin up and one spin down), that is, there is no such thing as "one electron is spin up and the other electron is spin left (superposition of up and down by equal amounts)"?
@Secret Oh, there is such a thing. But that doesn't mean that the notion of "angle between spins" is well-defined. When you say "spin up", you mean that you have an eigenstate of the z-spin component, and "spin left" means that you have an eigenstate of the y-spin component.
You can't really talk about the "angle" between the spins of a spin-up and spin-left state, because the "spin vector" is never definite in all components.
Hmm I see. In that case I am a bit lost at how to understand the exchange interaction of two electrons beyond the parallel spin and antiparallel spin cases because I thought there is a continuum of cases that connect between the parallel spin to the antiparallel spin cases
@ACuriousMind yeah, but since "angles between" the z and y eigenstates of spins are not well defiend due to the uncertainty in the two other components of the "spin vector", I don't know what is the parameter that describes this continuum
It's just a shame you're not allowed to split the code up into different files. Putting the board and GFX structs in their own file would make the code very elegant.
You could even use the "angle". My entire point was that that's not an "angle between spins". Stop always inferring more into what I say than I actually do.
Instead of having one variable for the board and then an array of size UNDO_DEPTH for the history we could have just one array of size UNDO_DEPTH+1 where [0] is always the current board
user228700
I didn't have much in the morning so I've had a whole lot for lunch just now; Rice, a bag of potato chips and two nutella sandwiches :-P
@Kaumudi.H I had beefburgers yesterday, then some hot cross buns, then some custard slices then ice cream (rum and raisin). I fancy something with chips/french fries today.
@BernardoMeurer Yes, I'd just use a separate array. I don't see any advantage to a single array with live board in it. Either way you'll be copying the entire board each move.
user228700
@JohnRennie 24th April so yes, straight back...sort of :-)
Use an array, but keep the live board at the end of the array. Then when you do the move/collapse you don't need intermediate storage because you use the next free board for the storage.
And the undo is then just a matter of stepping the current board index backwards.
And use the array as a ring buffer. So you're allowed infinite moves but you only keep the last UNDO_DEPTH boards.
Or I guess if you were very functional programming oriented, you could just store the composition of move functions, and then unwind that to go to previous states?
Rather than storing the "entire state" at each point that is
@alarge There isn't a way to undo a move, so an undo would mean starting at the first board and replaying all the moves up to some point. You could do that but it's a pain.
@BernardoMeurer Every n moves prune the list, where n is some number that's appropriate to the memory available. Pruning the list on every move seems excessive.
@JohnRennie Well, I could've implemented things in stupid ways, used my professor's sample code, and chosen not to do any of the extra functionality :P
The game shouldn't needs to know details of how the board does the memory management, so you need to keep the cell access functions in the board struct.
The only things that move to the Game struct are the Move function (because that's game logic) and the history management.
A magnetic needle of magnetic moment $6.7 \times 10^{-2} A m^2$ and moment of inertia $7.5 \times 10^-6 kg m^2$ is performing simple harmonic oscillations in a magnetic field of $0.01T$. Time taken for 10 complete oscillations is:
why the commutation relation between momentum and position operators should be [Q,P]=ih? is it necessary for QM to be compatible with classical mechanics? If so, is there any classical limit for the anti-commutation relation for fermions?
K. I think molar ones are written in capital $C_p, C_v$ and the standard, non-molar folks written as $c_p, c_v$. So that's an intentional confuzzlement for sure.
If $(M,\mathcal{F})$ is codimension 1 foliation on a compact manifold s.t. there's a compact leaf $L$ with $H^1(L; \Bbb R) = 0$ then $M$ fibers over the circle.
Aka strong topological restrictions on manifolds admitting foliations with certain leaves
(This doesn't hold C^0, so gives examples of C^0 foliations which cannot be promoted to a C^1 foliation too!)
In defining the cotangent bundle, Spivak in his differential geometry book, says
Then later,
I don't understand what he means by this last comment. It seems to me that $y_*' \circ (x_*')^{-1}$ and $y_* \circ x_*^{-1}$ don't even have the same domain. How could they be equal?
