Product Definition and 1000 Threads

I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am...

Under what conditions are the symbols \bigoplus and \times intechangangable?

I'm interested in what people know about the application of inner product structures (usually reserved for QM) to diff equations describing classical physics, in particular non- hermitician diff operator of the Fokker-Plank equation. Thanks.

The volume of a triangular prism is given by: v = ½ |a • b x c| Where b and c are two of the sides of the triangular face of the prism, and a is the length of the prism. The volume of a rectangular/parallelogram-based pyramid is given by: V = ⅓ |a • b x c| My question is, what are a, b...

Homework Statement Saw a calculation that put differentiation of power in terms of acceleration as follows: E=Fs dE/dt=Fv=P dP/dt=Fa=ma^2 It doesn't make sense to me because if power was changing, acceleration must change. Correct me if I'm wrong, but shouldn't the product rule be applied...

When we talk about the angle between two vectors while computing the cross product, which angle are we referring to exactly? According to most sources, the angle should be between 0 and π; yet according to my math book, "the angle is measured in an anticlockwise sense from a to b, if the vector...

Homework Statement If at some particular place and time the sun light is incident on the surface of the Earth along a direction defined by the unitary vector – vˆ , where vˆ =(4, 3, 5)/sqrt (50) and with a power density P, what is the total power captured by a solar panel of 1.4 m2 and with...

Hello, I'm having some difficulty with a conceptual question on a practice test I was using to study. I have the answer but not the solution unfortunately. 1. Homework Statement "For every differentiable function f = f(x,y,z) and differentiable 3-dimensional vector field F=F(x,y,z), the...

compute the product. $\left(\log_{2}\left({3}\right)\right)\cdot \left(\log_{3}\left({4}\right)\right)\cdot \left(\log_{4}\left({5}\right)\right)\cdots \left(\log_{126}\left({127}\right)\right)\cdot \left(\log_{127}\left({128}\right)\right)$ The answer to this is 7 I assume this can be done...

Homework Statement I have the operators ##D_{\beta}:V_{\beta}\rightarrow V_{\beta}## ##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}## ##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}## where each...

Hi, Assuming that A is a n x m random matrix and each of its entries are complex Gaussian with zero mean and unit-variance. Also, assume that b is a n x1 random vector and its entries are complex Gaussian with zero mean and variance=s. Then, what would be the variance of their product Ab? Any...

Hi, this is just a review exercise. Let M,N be n- and m- manifolds respectfully , so that the product manifold MxN is orientable. I want to show that both M,N are orientable. I could do some computations with product open sets of ##\mathbb R^n ## , or work with orientation double-covers...

Homework Statement I could prove a, trying b now. Homework Equations The definition of the cross prod.? The Attempt at a Solution https://www.dropbox.com/s/0sauaexkl4j2yko/proof_cross_prod.pdf?dl=0 I did not manage to get a scalar times v and a scalar times w. (No need to point this...

Homework Statement Given that \vec{V} and \vec{W} are vector operators, show that \vec{V}\times \vec{W} is also a vector operator. 2. The attempt at a solution The only way I know how to do this is by showing that the commutator with the angular momentum vector operator ( \vec{J}) is zero...

I'm asked to transform y(t) = x(t)*x(t) (where * is the convolution product) and x(t)= sinc(t)cos(2π10t) ( sinc(t)= sin(πt)/(πt) ).The attempt at a solution Clearly everything is simple if you know X(f), because y(t)=InverseFourier{ X(f)2 }. The problem is that I can't find X(f). By the way...

Homework Statement A damped harmonic oscillator is driven by a force of the form f(t)=h(t) t^2 Exp(-t), where h(t) is a Heaviside step function. The Oscillator satisfies the equation x''+2x'+4x=f(t). Use pencil-and-paper methods involving Fourier transforms and inverse transforms to find the...

Homework Statement Find vector product of C = A \times B of two vectors in oblique coord. system. Give explicit expressions of components of C in covariant and contravariant components (constructing reciprocal basis from direct basis will be useful). Homework Equations I am basically just...

Evaluate $\displaystyle \lim_{{n}\to{\infty}} \prod_{k=3}^{n}\left(1-\tan^4\dfrac{\pi}{2^k}\right)$.

I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...

Homework Statement IF G, H and G+H are invertible matrices and have the same dimensions Prove that G(G^-1 + H^-1)H(G+H)^-1 = I 3. Attempt G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1 = (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1 =2GH^-1+...

Hello, I am confused a little bit on why for a reaction with a given activation energy, one should run at high temperatures if the activation energy of the desirable product, D, is greater than the undesirable product, U. To illustrate this, I have an example with two reactions ##A + B...

Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3]. My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then, w-2x+3z=0 and 2x+y+3z=0 So, she solves the system and get the following: Subspace= {...

Hello! (Wave) Sentence: If $A,B$ are sets, there is the (unique) set, of which the elements are exactly the following: $\langle a,b\rangle: a \in A \wedge b \in B$. Proof: Remark: $\langle a,b\rangle:=\{ \{a\},\{a,b\}\}$ If $a \in A$, then $\{ a \} \subset A \rightarrow \{ a \} \in...

Hello everyone, I was wondering if the following claim is true: Let ##G_1## and ##G_2## be finite cyclic groups with generators ##g_1## and ##g_2##, respectively. The group formed by the direct product ##G_1 \times G_2## is cyclic and its generator is ##(g_1,g_2)##. I am not certain that it...

Homework Statement Hey guys, So here's the problem I'm faced with. I have to show that (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) , by acting with the quabla (\Box) operator on the following...

I have recently delved into linear algebra and multi-linear algebra. I came to learn about the concepts of linear and bi-linear maps along with bases and changes of basis, linear independence, what a subspace is and more. I then decided to move on to tensor products, when I ran into a problem...

In T. S. Blyth's book on Module Theory, the author uses a large 'times' symbol (similar to a capital X) for the Cartesian Product as seen in the text below (taken from Blyth page 58) Can someone help me with the Latex code for such a symbol?Peter

The problem: Suppose G is Abelian with two representations as the internal direct product of subgroups: G=HxK1, G=HxK2. Assume K1 is a subset of K2 and show K1=K2. My attempted solution: I took the element (e_H, k_2), where e_H is the identity element of H and k_2 is an arbitrary element in K2...

Homework Statement in the photo, the 23Na form 24Na Am i right? if i am right, the mass of 23Na should decraese , and mass of 24Na should increase. but why the solution provided is the mass of 24Na decreses and increases at the time of delta t ?? i can't understand Homework EquationsThe...

Hello! (Wave) There is the following sentence in my notes: Let $A$ be a set. We define the set $I_A=\{ <a,a>, a \in A \}$. $$A \times A=\{ <a_1,a_2>: a_1 \in A \wedge a_2 \in A \}$$ Then $I_A$ is a relation, but does not come from a cartesian product of sets. Could you explain me the last...

Homework Statement [/B] Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal. Show that the product <x, y> A = x · Ay is an inner product for R3. Homework Equations None The Attempt at a Solution I've shown that x · Dy is...

Homework Statement Find angle between vectors if \cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...

Hi, Let us suppose we have three real matrices A, B, C and let \circ denote the Hadamard product, while AB is the conventional matrix product. Is this relation true for all A, B, C matrices: C \circ (AB) = A( C\circ B)? I looked at it more thoroughly and I realized that this assumption is...

If I want to take the wedge product of $$\alpha = a_i\theta^i $$ and $$\beta = b_j\theta^j$$ I get after applying antisymmetrization,$$ \alpha \Lambda \beta = \frac{1}{2}(a_ib_j - a_jb_i)\theta^i\theta^j$$ My question is it seems to me that antisymmetrization technique doesn't apply to the...

Homework Statement Considera 1.0 C charge moving with a velocity of v = -2.0i + 2.0j - 1.0k in a magnetic field of B = -4.0i + 1.0j – 3.0k. What force is this charge experiencing? What is the angle between the velocity and magnetic field vectors? Homework Equations F = q(E + v x B)...

Hi! (Wave) If $A,B$ are sets, the set $\{ <a,b>=\{ a \in A \wedge b \in B \}$ is called Cartesian product of $A,B$ and is symbolized $A \times B$. If $A,B,C$ sets, then we define the Cartesian product of $A,B,C$ as: $$A \times B \times C:=(A \times B) \times C$$ But.. is it: $(A \times B)...

Hi all, Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it : 1)Basically, understanding how/why the...

Homework Statement Use the LC symbol to calculate the following: $$\nabla \times \frac{\vec{m} \times \hat{r}}{r^2}$$ Where ##\vec{m}## is just a vector, and ##\hat{r}## is the unit radial vector and ##r## is the length of the radial vector. Homework Equations On the Levi Civita symbol...

Is this just a coincidence that cross product can be found from determinant of 3*3 matrix? what is the differences between wedge product and cross product?Thanks.

A fluid motion has velocity $$\underline{u}=\sin{(at)}\hat{\imath}+\hat{\jmath} \times r +\cos{(at)}\hat{k}$$ I need to know what is $$\hat{\jmath} \times r$$ to find Vorticity and other things.

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? As I'm not being able to precisely phrase my doubt, consider this example: Hilbert space of a two dimensional particle is the direct product of...

Some words before the question. For two smooth manifolds M and P It is true that T(M\times P)\simeq TM\times TP If I have local coordinates \lambda on M and q on P then (\lambda, q) are local coordinates on M\times P (right?). This means that in these local coordinates the tanget vectors are...

Homework Statement So a kaon moving at some speed in the +x direction spontaneously decays into one pion and one anti-pion. The anti-pion moves away with velocity of 0.8c, and the pion moves away with velocity of 0.9c. Mass of kaon = 498 MeV/c^2 Mass of pion/anti-pion = 140 MeV/c^2...

Dear friends, I have been told that if ##\{a_n\}_{n\in\mathbb{N}}##, ##\{a_{-n}\}_{n\in\mathbb{N}^+}##, ##\{b_n\}_{n\in\mathbb{N}}## and ##\{b_{-n}\}_{n\in\mathbb{N}^+}## are absolutely summable complex sequences -maybe even if only one i between ##\{a_n\}_{n\in\mathbb{Z}}## and...

Is there an intuitive reason or proof demonstrating that in general dimensions, there is no direct analogue of the binary cross product that yields specifically a vector? I came across Wedge Product as the only alternative, but am just learning linear algebra and don't quite comprehend yet...

HEY GUYS! (Wave) ok so i have this question i did. and now I am reviewing for the test and i looked at how i did it and i did in the most complicated way ever. i don't FULLY understand chegg's method. so i hope someone can provide me with the SIMPLEST method possible. thank u! (Blush) (p.s...

Hi, how can I compute the general solution of a system of linear equations? Non-square systems for example. I have the book Linear Algebra via exterior products, but it is the worst book in the history of math books, I think I'll burn it somehow, whatever. I can calculate the solution with the...

Hello! I have a problem in my calculus based physics class regarding vectors. The problem says: Vectors A and B have a scalar product -6.00 and their vector product has magnitude 9.00 what is the angle between these two vectors? Here is how I approached it: -6=|A||B|cos (theta) 9=|A||B|sin...

I'm asked to prove the following using Levi-Civita/index notation: (\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = [\mathbf{a,\ b, \ d}] \mathbf c - [\mathbf{a,\ b, \ c}] \mathbf d \ I'm able to prove it using triple product identities, but I'm completely stuck...

So I've been working on physics homework and we have some vector/dot product questions. This is really long, but the questions I have really are rudimentary at best. I have seven total questions. You're given two vectors that only have an x and y component, A, and B, and the positive Z axis is...