Product Definition and 1000 Threads

Hello, Throughout my undergrad I have gotten maybe too comfortable with using Dirac notation without much second thought, and I am feeling that now in grad school I am seeing some holes in my knowledge. The specific context where I am encountering this issue currently is in scattering theory...

That may sound really silly, and that may be due to my lack of understanding of the operations itself, but: if ##|\vec{a}\times\vec{b}|=|\vec{a}|\cdot|\vec{b}|sin\theta##, being ##\theta## the angle between the two vectors, how could ##\vec{b}\times\vec{a}## be different? Wouldn't it be just the...

Hello All: i have a question regarding the steps after your team finish developing a software , my relative and her team finished developing a software but they don't know how to sell it , it is their first software , they start from scratch algorithm then code ,...etc now after one year...

Writing both ##\vec{U}## and ##\vec{B}## with magnitude in all the three spatial coordinates: $$ \vec{U}\times \vec{B}= (U_{x}\cdot \widehat{i}+U_{y}\cdot \widehat{j}+U_{z}\cdot \widehat{k})\times (B_{x}\cdot \widehat{i}+B_{y}\cdot \widehat{j}+B_{z}\cdot \widehat{k})$$ From this point on, I...

Can I just interpret that some of energy of P + Q (maybe KE) is converted into mass of X + Y? Another question: is it possible that mass of products = mass of reactants? Thanks

hi guys i was thinking about the inner product we choose in quantum mechanics to map the elements inside the hilbert space to real number which is given by : $$\int^{∞}_{-∞}\psi^{*}\psi\;dV$$ or in some cases we might introduce a weight function dependent on the wave functions i have , it seems...

Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator...

As per the image, I am supposed to select all the valid statements. Apparently I'm only partially correct, and so I took another look at the statements. I believe the third statement is wrong, since c * (a_m*a_{m+1}*a_{m+2}*...*a_n) =/= (c*a_m)(c*a_{m+1})(c*a_{m+2})*...*(c*a_n) Thus there...

I am beginning this new general physics course and I have encountered a question involved with what I assume to be cross products, a topic that I have very little experience with. I am not looking for a direct answer to the problem but advice on what steps should be taken in order to learn how...

I don't really know how to begin. I've done alkylations by having two of the same compounds react with each other e.g. two aldehydes but never started out with dimethyl malonate. I was thinking I need 1,4 dibromobutane to form the cyclopentane ring but apart from that I'm clueless

I'm learing about antennas in a course, and we are using Jin's Electromagnetic text. This isn't a homework problem, I'm just trying to understand what I'm supposed to do in this situation. This part of the text discusses how to evaluate a radiation pattern. One of the steps to evaluate the...

I will post the answer here, part of which I do not follow. I do not follow the outer-product part. I know that I should multiply two terms together if they are in the same space. However, in this problem, I do not know how to determin which term belongs to which space. It seems, sometimes...

A coordinate system with the coordinates s and t in R^2 is defined by the coordinate transformations: s = y/y_0 and t=y/y_0 - tan(x/x_0) , where x_0 and y_0 are constants. a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system is well defined. Express the...

Show that $3^{2008}+4^{2009}$ can be written as a product of two positive integers each of which is larger than $2009^{182}$.

Mentor note: Moved from a technical section, so is missing the homework template. Hi, I'm always not sure how to prove something in math and I'm wondering if this is enough. ##\vec r \cdot (\vec u + \vec v) ## ##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s## ##\vec r \cdot (\vec u +...

Well, the question is in the title.

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...

I thought this was kind of a cool proof of the product rule. Let ##F(x)## and ##G(x)## be cumulative distribution functions for independent random variables ##A## and ##B## respectively with probability density functions ##f(x)=F'(x)##, ##g(x)=G'(x)##. Consider the random variable...

When i read this in the book "A VECTOR APPROACH TO OSCILLATIONS" i was a little shocked, because first it make quotients of vectors, and after this he defines this planar product, i searched this in google: i found nothing. Anyway, this operations make sense if we imagine the vectors...

Product of two Hermitian matrix ##A## and ##B## is Hermitian matrix only if matrices commute ##[A,B]=0##. If that is not a case matrix ##C=AB## could have complex eigenvalues. If A=\sum_k \lambda_k|k \rangle \langle k| B=\sum_l \lambda_l|l \rangle \langle l| AB=\sum_{k,l}\lambda_k\lambda_l|k...

Seems to me the answer is a specific vector: The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!) How to write this in vector language?

Hello, Within Griffith's text - chap 12 section 12.2 page 423 - this is a brief summary of Bell's Theorem and description of Bell's 1964 work. There is a table on page 423 showing the spin of the electron and positron (from pi meson decay) - these would be in the singlet state, one would be...

Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we...

for the sum, ##\frac {1}{∝^3}##+##\frac {1}{β^3}##=##\frac {β^3+∝^3}{∝^3β^3}## =##\frac {(∝+β)[(∝+β)^2-3∝β]}{∝^3β^3}## =##\frac {-b}{a}##...

Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...

As part of the final stage of a problem, there is some algebraic manipulation to be done (from the solution manual): But I'm getting lost somewhere: Also a bit of general advice needed: This is part of a self-study Calculus course, and I often have difficulty with bigger algebraic...

I'm reading Fundamentals of Astordynamics by Bate, Mueller, White and having trouble with this passage (pg15): "2. Since in general a⋅a' = a a'..." I don't think that this is the case. For instance in uniform circular motion r⋅r' = 0. Would appreciate if anyone has some insight into this.

I've tried with this work in attachment. i&m not sure of my answer is correct.

I don't have a clue as to how to go about proving (or verifying) the equation above. It would be very hard to take individual values of i,j and k and p,q and r for each side and evaluate ##3^6## times! More than that, I'd like a proof more than a verification. Any help would be welcome.

I don't really know where to start. Trying to use cosine rule but failed because no information about angle. Thanks

Hello! I was wondering if it is possible to express the gravitational energy as a product of the gravitational field by a moment, as we do with the magnetic and electric energy? Would this require the existence of bodies with negative mass? How could we relate this to the existence or total...

Summary:: Inner Product Spaces, Orthogonality. Hi there, This my first thread on this forum :) I encountered the above problem in Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGraw-Hill) Chapter 7 - Inner Product Spaces, Orthogonality. Using some particular values for u and v, I...

Today I was reading about geometric algebra and a kind of vector product that combines the dot and cross/wedge products together and it got me thinking about the meaning of "product". My math background is from an engineering perspective and I've always just accepted the dot and cross products...

If ##\tilde{U}_0 \cdot \tilde{A} = 0## in one frame then I would imagine it is also zero in another frame because from my understanding is that dot products are invariant under boosts. So let's boost to the rest frame of O. In that frame ##\tilde{U}_{0T} = \left( c, 0,0,0 \right)## and as...

I would like to extend the convergence of the Euler product over primes, and I tried to do so in the exact manor it was done for the Dirichlet series, namely, given a completely multiplicative sequence ##a( {kj} ) =a(k) \cdot a(j)\text{ and }a(1)=1##, the Dirichlet series ##\xi (s) :=...

I know that taking the scalar product of the harmonic (Laplacian) friction term with ##\underline u## is $$\underline u \cdot [\nabla \cdot(A\nabla \underline u)] = \nabla \cdot (\underline u A \nabla \underline u) - A (\nabla \underline u )^2 $$ where ##\underline u = (u,v)## and ##A## is a...

Suppose you have a complex-valued function of a complex variable (namely, ##z=x+iy, \, \, x,y\in \mathbb{R}##) defined as the assumed convergent infinite product $$F(z)=\prod_{k=1}^{\infty}f_{k}(z)$$ Further suppose ##F(x+iy)=u(x,y)+i v(x,y)##, where u and v are real-valued functions. How to...

How to prove that direct product of two rep of Lorentz group ##(m,n)⊗(a,b)=(m⊗a,n⊗b)## ? Let ##J\in {{J_1,J_2,J_3}}## Then we have : ##[(m,n)⊗(a,b)](J)=(m,n)(J)I_{(a,b)}+I_{(m,n)}⊗(a,b)(J)=## ##=I_m⊗J_n⊗I_a⊗I_b+J_m⊗I_n⊗I_a⊗I_b+I_m⊗I_n⊗J_a⊗I_b+I_m⊗I_n⊗I_a⊗J_b## and...

Hi everyone, I was attempting the following past paper question below: I have found a value for the coefficient c and I think I have calculated the inner product of <x|x>. I've attached my workings below. But I'm not sure what to do next to answer the last part of the question which asks...

I've always had a fascination with infinite products. I like them, I do. To stimulate our ensuing conversation, I here post Knopp's two-way series-to-product (and vice-versa) "doorway" out of his book, Theory and Applications of Infinite Series pg. 226: Maybe that'll break the ice... please...

I have numerous points of confusion: what does it mean that the matrices are within the exponential? How do I go about doing the matrix multiplication to prove the given form of CZ matches the common form, the 4x4 matrix? Update: using the fact that exp(At)=∑ ((t^n)/n!)*A^n, where A is a...

I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...

##\prod_{j \neq i}^{6} (\lambda_{i}-\lambda_{j})##

Hi, This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...

ok I don't don't know de jure on this so ... is it just plug and play?? find factors of -48 $-1(48)=-48$ $-2(24)=-48$ $-3(16)=-48$ $-4(12)=-48$ $-6(8)=-48$ check sums for positive number $-1+48=47$ $-2+24=22$ $-3+16=13$ $-4+12=8$ $-6+8=2$it looks like c. 5

(scroll to bottom for problem statement) Hello, I am wondering if someone could break down the problem statement in simpler terms (not so math-y). I am struggling with understanding what is being asked. I will try to break it down to the best of my ability Problem statement:Consider the inner...

$$<p_1 p_2|p_A p_B> = \sqrt{2E_1 2E_2 2E_A 2E_B}<0|a_1 a_2 a_{A}^{\dagger} a_{B}^{\dagger} |0>$$ $$=2E_A2E_B(2\pi)^6(\delta^{(3)}(p_A-p_1)\delta{(3)}(p_B-p_2) + \delta^{(3)}(p_A-p_2)\delta^{(3)}(p_B-p_1))$$ The identity above seemed easy, until I tried to prove it. I figured I could work this...

Hi all; I have a very basic understanding of sequences and series and recently encountered a sequence which really has me confused: $$(\frac{1}{5}+(\frac{1}{5}+(\frac{1}{5}+(...)^2 )^2)^2)^2$$ What type of sequence would you call this? I couldn't even google it because I couldn't work out how to...

I have a simple question...the control qubit is A and the the target is B. The cnot is applied on |1A> <0A|⊗|0B0C>. ... How does it work. Thanks in advance.

Summary:: I'm always unable to find the products of a chemical reaction. No Matter how much concept I study I can't ever get products from reasoning. Here is the question: What is the major product of this reaction? . OPTIONS : MY ATTEMPT : First of all I can see that in...