Product Definition and 1000 Threads

Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example: The...

So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...

Hi everyone. Yesterday I had an exam, and I spent half the exam trying to solve this question. Show that ##\left\langle\Psi\left(\vec{r}\right)\right|\hat{p_{y}^{2}}\left|\phi\left(\vec{r}\right)\right\rangle =\left\langle...

Let $A=\left[\begin{array}{c}1 & 2 & -3 \\ 2 & 0 & -1 \end{array}\right] \textit { and } B=\left[\begin{array}{c}3&2 \\ 1 & -1 \\ 0 & 2 \end{array}\right]$ Find $(AB)^T$$AB=\left[ \begin{array}{cc}(1\cdot 3)+(2\cdot1)+(-3\cdot0) & (1\cdot2)+(2\cdot-1)+(-3\cdot2) \\ (2\cdot3)+(0\cdot1)+...

Homework Statement Force F⃗ =−11j^N is exerted on a particle at r⃗ = (8i^+5j^)m. What is the torque on the particle about the origin? Express your answer using two significant figures. Enter coordinates numerically separated by commas.Homework EquationsThe Attempt at a Solution F: 0i, -11j...

Hi. I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get ΣnΣm xn/(n!) (-x)m/(m!) without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ? Thanks

From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...

Homework Statement [/B] Hopefully this is in the correct section I looked around for others but this seemed like the right one. Find the scalar, vector, and parametric equations of the plane that passes through the points P(1,0,4), Q(3,1,-6), and R(-2,3,5). Homework EquationsThe Attempt at a...

Homework Statement Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2 $$ where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$ Homework Equations I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this...

Homework Statement Which one of the following products will not be obtained in the given Wurtz reaction? The Attempt at a Solution Well, (a) will definitely be the major product. We know that this reaction may proceed by a free radical or an ionic mechanism. But what's that got to do with the...

Homework Statement Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that Exp(A+B) = Exp(A)Exp(B)Exp(-1/2 [A,B]) Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of T(s)...

Homework Statement Prove: If A and B each have at least two elements, then not every element of P(A×B) has the form A1 ×B1 for some A1 ∈ P(A)and B1 ∈ P(B). Homework EquationsThe Attempt at a Solution Suppose A = {1, 2}, B = {3, 4}. AXB = {(1,3), (1,4), (2,3), (2,4)} P(A) = {{1}, {2}, {1,2}...

Homework Statement EDIT: ##U(n)## is the set of relatively prime numbers less than ##n## and ##U_k(n) = \lbrace x \epsilon U(n) : x \equiv 1 (\operatorname{mod} k) \rbrace##. I'm trying to finish the proof of this statement(s): Suppose ##s## and ##t## are relatively prime. Then ##U(st) \approx...

Hey, I am currently reading over the linear algebra section of the "introduction to quantum mechanics" by Griffiths, in the Inner product he notes: "The inner product of two vector can be written very neatly in terms of their components: <a|B>=a1* B1 + a2* B ... " He also took upon the...

I was studying Group Theory on my own from a mathematics journal and got confused at some point where it defines Cartesian products, from binary one, say (A × B), to n-tuples one, say (A_1 × A_2 × ... × A_n). What confuses me when I tried to read it is that the definition made for infinite...

Homework Statement A switching network has 4 inputs and a single output (Z) as shown in the figure below. The output Z is 1 iff the binary number represented by ABCD ( A is the MSB) is an even number greater than 5. Find : a) The standard POS of Z (abbreviated form). b) The standard SOP of...

Homework Statement Prove that the quaternion group ##Q_8## is not isomorphic to a semi-direct product ##H\rtimes_\rho K## for non-trivial groups ##H## and ##K##. Homework EquationsThe Attempt at a Solution My idea is to look at the subgroups of ##Q_8## and to show that the intersection of any...

Homework Statement [/B] The scalar triple product computes the magnitude of the moment of a force vector about a specified line. It is M = ( r×F ) ⋅n , where is the position vector from the line to the point of application of the force and is a unit vector in the direction of the line. Prompt...

<Moderator's note: Moved from a technical forum and thus no template.> Here is a problem and solution given in my book. this is using a symbol in cartesian product . I checked other books for cartesian product examples but there was no such symbol being used. Here is my solution without...

For example, does this always hold true? M_ab = v_a × w_b If not, where does it break down?

I’m having trouble understanding the relationship between how work is both a dot product and integral. I know that work equals F • D and also the integral of F(x): the area under the curve of F and D. However, let’s say that I have a force vector <3,4> and a displacement vector of <3,0>. The...

I'm having a little trouble with this : We have ##(\alpha\vec{a})\cdot b = \alpha(\vec{a}\cdot\vec{b})## but shouldn't it be ##|\alpha|(\vec{a}\cdot\vec{b})## instead since ##||\alpha \vec{a}||=|\alpha|.||\vec{a}||## ? ##(\alpha\vec{a})\cdot b = ||\alpha\vec{a}||.||\vec{b}||.\cos\theta =...

Hello, we defined the wedge-product as follows Alt is the Alternator and the argument of Alt is the Tensor poduct of one k-form and a l-form (in this order w and eta). Suppose we have the wedge product of a 0-form (a smooth function) and a l-form , so the following may result: $$\frac{1}{l!}...

Homework Statement This isn't really a homework problem, as the entire solution is laid out in the text. My question is in regards to a possible typo, which I have highlighted in blue in the given picture. Usually I don't like to second guess the text, but this one has been absolutely plagued...

Homework Statement For the ground state of a particle moving freely in a one-dimensional box 0≤x≤L with rigid reflecting end points, the uncertainity product (Δx)(Δp) is (A) h/2 (B) h√2 (C) >h/2 (D) h/√3Homework Equations The uncertainity principle says that - (Δx)(Δp) ≥ ħ/2 Ground state energy...

Homework Statement Prove that for vectors A,B and C, A.(B+C)=A.B+A.C and prove the property that for two vectors, A and B the dot product is equal to A^ie_i . B^je_j = e_i.e_jA^iB^j Homework Equations Only use the definition where for two vectors a and b the (length of a)(length of b)cost...

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition of...

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition...

Use the UPC scheme to determine the check digit for the number $07312400508$. here is the example from the book ok so going with dot product with 07312400508 \begin{align*}\displaystyle &\quad (0731 2 4 0 0 5 0 8)\cdot(3,1,3,1,3,1,3,1,3)\\ &= 0\cdot3+7\cdot1 +3\cdot3+1\cdot1...

Homework Statement I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space. Homework EquationsThe Attempt at a Solution The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...

Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...

Let's take two orthogonal curves in polar coordinates of the form ##\langle r,\theta \rangle##, say ##\langle r,0\rangle## and ##\langle r,\pi/2\rangle##. Cleary both lines are orthogonal, but the dot product is not zero. This must be since I do not have these vectors in the form ##\langle...

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Homework Statement Let ##x,y \in \mathbb{R^n}## not null vectors. If for all ##z \in \mathbb{R^n}## that is orthogonal to ##x## we have that ##z## is also orthogonal to ##y##, prove that ##x## and ##y## are multiple of each other. Homework Equations We can use that fact that ##<x ...

Greetings! This is probably the most simplest of questions but I wanted to ask that which particle carries most of the energy of an electron-proton collision (as in p + e− → n + νe), the neutron or the neutrino? What happens if we use a higher energy proton or electron? For example we use...

The product of 2 and a number is the same as 12 subtract the number. Find the number. If you could please help me with this, thank you.

Homework Statement Show that an element has order 2 in ##S_n## if and only if its cycle decomposition is a product of commuting 2-cycles. Homework EquationsThe Attempt at a Solution Suppose that ##\sigma \in S_n## and ##| \sigma | =2##. Let the cycle decomposition of ##\sigma## be the...

Homework Statement a magnetic moment of U = 1 (i) + 2 (k) , surrounded by a magnetic uniform field of B= 3 (i) + 4 (j) - 1 (k) find the potential energy in mJ ? Homework Equations dot product of 2 vectors ( ui*bi)+(uj*bj)+(uk*bk) = or finding the module of both vector and doing AB...

Hi there. It is obvious that if you have two differentiable functions ##f(x)## and ##g(x)##, then the product ##h(x)=f(x)g(x)## is also smooth, from the chain rule. But if now these functions are multivariate, and I have that ##h(x,y)=f(x)g(y)##, that is ##f(x,y)=f(x)## for all y, and similarly...

This may be bit weird but really do we get any product after the reaction of 118 elements of the periodic table.We know masses of reactant = masses of product.But no so in nuclear fission and fusion.Anyhow finally what might be that product?and what is going to be its state.Please explain me...

Wolfram says that an example of an inner product space is the vector space of real functions whose domain is an closed interval [a,b] with inner product ##\langle f, g\rangle = \int_a^b f(x) g(x) dx##. But ##1/x## is a real function, and ##\langle 1/x, 1/x\rangle## does not converge... So how is...

Homework Statement A cruise ship that can hold up to 62 people provides excursions to groups of 42 or more. If the group contains 42 people, each person pays $70. The cost per person for all members of the party is reduced by $1 for each person in excess of 42. Find the size of the group that...

Homework Statement Among all the pairs of numbers with a sum of 236, find the product whose pair is maximum. Homework EquationsThe Attempt at a Solution I got the correct solution to this, but I feel it was a weak way to do it and I'd like to know how to do it more efficiently. For example...

So I'm reading through some stuff trying to learn tensors, and I wanted to stop myself to grasp this concept before I go further. Along the way it was mentioned in one of the sources I'm looking at that other than the dot product and cross product, you could multiply vectors in another way, and...

The Attempt at a Solution Is there some hindrance caused by the π bond in (c)? How do I approach the problem anyway?

Homework Statement n(n+1)(n+2)(n+3) cannot be a square Homework Equations Uniqueness of prime factors for a given number The Attempt at a Solution I'm not sure but I think I've proved a stronger case for how product of consecutive numbers cannot be squares. I don't know whether it is right...

Hello, I have the following product, and I am looking for a summation equivalent \prod_{k=1}^K\left(1-\frac{1}{x_k+1}\right) Is this doable? I tried to use partial fraction but got nowhere! Thanks in advance

Hi. If I have a vector v , say for velocity for example then v.v = v2 and I differentiate wrt t v.v I get 2v.dv/dt but if I differentiate v2 I get 2v dv/dt but v.dv.dt is not the same as v dv/dt so what am I doing wrong ? Thanks

i am learning physics for fun and also to make taking the actual courses easier. i have a physics textbook because i took physics for my associate's degree but ended up withdrawing because i took too many courses. anyway, i like "connected" knowledge in math though i know it is a bit of a...

Homework Statement Is the following conclusion correct? Assume there's an equation with vectors on both sides. Taking the dot product of this equation with vectors on both sides loses information, but information will not lose when taking dot products with higher rank tensors on both sides...