Can you reference the results you are talking about? They do not seem correct as stated... for example you can certainly take a definite integral of a step function, which has a discontinuity.
Also, a couple of comments are in order regarding your initial post.
1) It doesn't make sense to ask whether a function is measurable with respect to a sigma algebra. What you should be asking is how to show that the function f is measurable with respect to the measure space (which...
You need to show the function is measurable with respect to the lebesgue measure. Thus given \alpha \in \mathbb{R} you must show that
\{x|f(x)< \alpha \}
is a lebesgue measurable set.
It is necessary for you to make some attempt at a solution. If you have not already done so, read a chapter in a classical mechanics book on central force motion. For example chp 8 in thornton and marion Classical Dynamics. The steps for solving such a problem will be outlined for you there.
The point is that they are exactly the same integrals. It doesn't matter if you use x or y or Ω or √ or ∏ to label the variable, it is simply the same integral. It is like asking if the solutions to the following equations will be the same or different:
λ+2=1
θ+2=1
it doesn't matter...
Not sure exactly what you mean by that. However, what you should do is set up the integrals needed to calculate <x^2>,<y^2> and <z^2>. Look for similarities between the integrals. Do they look the same or different? For example: consider the following integrals,
\int_0^5 (y+2)^2 dy
and...
The term 'expectation value' is an unfortunate one and really should be something more like 'ensemble average'. Imagine you know all of the accessible quantum states of a given system, and there are g of them. And you were to construct an ensemble of g systems one in each of the accessible...
I thought about it for a bit, read a bit on wikipedia and was going to summarize what I read, but i'll just give you the link.
http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics
I just think about it in the practical sense. A measurement is something you do in a lab with rulers and...
The difference is certainly half odd integral, but not necessarily 1/2 :D
And i believe the difference is that half integral spin particles permit spin multiplets with an even number of states, and integral spin particles permit spin multiplets with an odd number of states. Even and odd numbered...
Thanks so much for your reply! I don't really know anything about Lie algebra yet, but this definitely helps me understand a bit of what I need to learn in order to pursue the topic further. Could you possibly recommend a good book or books that I could buy that would cover these types of...
I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said "...
Homework Statement
Consider f(x) = x^3-5
and its splitting field K = Q(5^{1/3}, \omega)
where \omega = e^{2 \pi i/3}
Show that B = \{1, 5^{1/3}, 5^{2/3}, \omega, \omega 5^{1/3} , \omega 5^{2/3} \}
is a vector space basis for K over Q.
The Attempt at a Solution
I am just a bit...
Homework Statement
Show that if p is prime and e^{2 \pi i/p} is constructable
then p=2^k+1 for a positive integer k
Homework Equations
e^{i \theta} = Cos \theta + iSin \theta
The Attempt at a Solution
By definition, a complex number a+bi is constructible if a and b are...
Homework Statement
This is something ive been trying to prove for a bit today. My quantum mechanics book claims that the following two definitions about hermitian operators are completely equivalent
my operator here is Q (with a hat) and we have functions f,g
\langle f \mid \hat Q f \rangle...
In math you do not need to prove a definition. We simply notice that there is this combination of symbols that we often see, so we decide to give it a name: "dot product" or "cross product". From these definitions you can prove that the cross product is perpendicular to the original vectors for...
Actually now that I think of it. Why don't we just use
T=\frac{1}{2}I_{ij} \omega_i \omega_j
(summed over i and j) with the given tensor and have
\vec \omega \rightarrow \Omega (\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)
then we just have a sum of four terms for the kinetic energy
For me physics was much more difficult. But it depends completely on the professor. My physics professor (Kapp) wrote very difficult exams and expected a lot out of us. My calculus 1 professor did the exact opposite. I didn't even understand calculus until I took physics.
one problem that I see is that not every homomorphism is an automorphism. Consider the identity mapping. this is clearly a homomorphism but it is not bijective.
Also you said there are four homomorphisms from K to C. But what is C? is C just Aut(K/Q)?
Also I just want to clarify something: Is...
If the state of the particle is the nth eigenstate, then you can use that last formula. When it is in the ground state, it still holds with n=0. When you are given some initial wave function that is not one of the eigenstates, you need to calculate the c values with fourier's trick, and then...
If anybody has an explanation I would be happy to hear it. I have turned in the homework assignment already and decided it makes most sense to just calculate the c value for the bound state, and say
P(E \neq E_1 ) = 1 - |c_1|^2 = .17
What I am wondering about is how to express the state as a...
If the cube has its corner at the origin and is contained in the first octant, we want a rotation about the axis defined by the direction
( 1 , 1 , 0 )
correct? Just want to make sure we are on the same page with what the problem is asking..
you need to find a rotation matrix that takes the x direction unit vector to the direction of the diagonal of the face of the cube. Call the matrix R. Tensors transform according to the equation:
I'=RIR-1
then once you have this transformed tensor, you have
T=\frac{1}{2} I_i'_j \omega _i...