Recent content by Master J

Apologies if I haven't followed the recent stream of replies, but I have noticed a few well-meaning responses that seem somewhat discouraging. The only thing that matters is whether or not you have a real passion, a real fascination with the subject. If you work hard at it, you can learn so...

I am just beginning graduate self-study of Special and General Relativity, so forgive me if my question seems niave. I have found the beautiful line " The essence of Special Relativity is that the laws of physics are Poincaré invariant" - Modern Mathematical Physics, Szekeres. The space...

Well, I'm still confused. Say I have an ODE who's solution family y(t) is unstable. That is, for increasing t, the solution curves diverge from each other. In this case, J = df(y, t)/dy < 0. So does this mean that ANY numerical method I use to solve this ODE will be unstable? With reference to...

I have been thinking about numerical methods for ODEs, and the whole notion of stability confuses me. Take Euler's method for solving an ODE: U_n+1 = U_n + h.A.U_n where U_n = U_n( t ), A is the Jacobian and h is step size. Rearrange: U_n+1 = ( 1 + hA ).U_n This method is...

Is there an explicit equation for the Stormer-Verlet numerical integration method for any problem? I usually only see it in a formulation that is specific to a given problem. Is there a general equation?

I've used many different power series representations of functions and seem to always take it for granted that functions which are "nice" and continuous have such a representation. But what is the criteria for a function to have a power series representation? I know of some that don't, but...

I'm having trouble seeing how an operator can be written in matrix representation. In Sakurai, for an operator X, we have: X = \sum \sum |a''> <a''| X |a'> <a'| since of course \sum |a> <a| is equal to one. Somehow, this all gets multiplied out and you get a square matrix with the...

Sorry, I should really learn LaTeX, but yes, that's what I meant ... I have from Wikipedia ( http://en.wikipedia.org/wiki/Bra-ket_notation ) that the complex conjugate of a bra is a ket, and vice versa. So, in the equation <B|X|A> = ( <A|X|B> )* , the * would some unnecessary? Again here, X is...

I understand that, I'm not confused. Perhaps we have digressed too far. My original question was, since accelerating electrons in a solid are responsible for reflection, emission and absorption of light, and since the lattice (the atoms, which themselves are charge distributions) oscillates...

In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17): <B|X|A> = (<A|X^|B>)* = <A|X^|B>* where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X. My question is, since a bra is the conjugate...

Thanks for the input guys! I was also thinking, considering an atomic mass is typically thousands of times that of a single electron, the frequencies of the vibrating lattice atoms must be far less. What kind of radiation would they emit, if say the electrons were emitting IR. Would it be...

I know that accelerating electrons in a solid are responsible for the emission, absorption and reflection of light. However, what role do the lattice vibrations play? These oscillating atoms are charge distributions and should emit radiation too, right? As far as I can remember, the...

Thanks for the input people, you have cleared up a lot! The notion of cosets is quite confusing, at least to me. They've made their appearance in a chapter on Vector Spaces and I haven't seen them before. Another minor detail I have come across is this (perhaps my set theory is lacking!)...

Perhaps I should have clarified...I meant that W would stand for any element of W. If the coset of the equivalence relation is u + W, this means as I understand it, that the equivalence relation for u only holds for u itself ( u - u = 0 \in W), and for any element of W. Is that correct?

In studying vector spaces, I came across the coset of a vector space. We have an equivalence relation defined as u = v \rightarrow u-v \in W where W is a subspace of V. the equivalence class that u belongs to is u + W. I can see why u must belong to this equivalence class ( the...