Identity Definition and 1000 Threads

If we can identify ##|c_n|^2## as the probability of having an energy ##E_n##, then that equation is just the bog standard one for expectation. But the book has not proved this yet, so I assumed it wants a derivation from the start. I tried $$ \begin{align*} \Psi(x,t) = \sum_n c_n...

Prove that $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$.

(Sinx-2cosx)/ (cotx - sinx) Substitute tan instead of cot (Tanx(sinx-2cosx)/(1-sinx) What do I do from here I don't think what I did there is correct That's why I didn't expand the tan to sin/cos

This is quite literally a showerthought; a differential equation is a statement that holds for all ##x## within a specified domain, e.g. ##f''(x) + 5f'(x) + 6f(x) = 0##. So why is it called a differential equation, and not a differential identity? Perhaps because it only holds for a specific set...

Hi, everyone. I'm trying to get the next identity It is in the format <j1, j2; m1, m2 |j, m>. I hope you can help me

Prove $\tan 3x=\tan \left(\dfrac{\pi}{3}-x\right) \tan x \tan \left(\dfrac{\pi}{3}+x\right)$ geometrically.

Hi, K₁cos(θt+φ)=K₁cos(θt)cos(φ)-K₁sin(θt)sin(φ)=K₁K₂cos(θt)-K₁K₃sin(θt) Let's assume φ=30° , K₁=5 5cos(θt+30°) = 5cos(θt)cos(30°)-5sin(θt)sin(30°) = (5)0.866cos(θt)-(5)0.5sin(θt) = 4.33cos(θt)-2.5sin(θt) If only the final result, 4.33cos(θt)-2.5sin(θt), is given, how do I find the original...

Consider the process e^-\rightarrow e^-\gamma depicted in the following Feynman diagram. The spin-averaged amplitude with linearly polarised photons is \overline{|M|^2}=8\pi\alpha\left(-g^{\mu\nu}+\epsilon^\mu_+\epsilon^\nu_-+\epsilon^\mu_-\epsilon^\nu_+\right)\left(p_\mu p^\prime_\nu+p_\nu...

I know there is an identity involving the Laplacian that is like ##\nabla^2 \vec A = \nabla^2 A## where ##\vec A## is a vector and ##A## is its magnitude, but can't remember the correct form. Does anyone knows it?

I'm trying to come up with a proof of the operator identity typically used in the Mori projector operator formalism for Generalized Langevin Equations, e^{tL} = e^{t(1-P)L}+\int_{0}^{t}dse^{(t-s)L}PLe^{s(1-P)L}, where L is the Liouville operator and P is a projection operator that projects...

Hi All, I try to prove the following commutator operator Identity used in Harmonic Oscillator of Quantum Mechanics. In the process, I do not know how to proceed forward. I need help to complete my proof. Many Thanks.

any tip to start?

Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...

I have an identity $$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2})$$ which should give us $$3(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m}$$ But I have to derive it using the Einstein summation notation. How can I approach this problem to simplify things ? Should I do something like...

I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with: δjj δkl - δjl δkj = δkl - δlk Clearly I did not take the right approach in this proof and have no clue as to how to proceed.

Hello, If I have a quadratic form ##q## on a ##\mathbb{R}## vectorial space ##E##, its associated bilinear symmetric form ##b## can be deduce by the following formula : ##b(., .) = \frac{q(. + .) - q(.) - q(.)}{2}##. So that, an homogeneous polynomial of degree 2 can be associated to a blinear...

I'll write down my calculations, and I would like if someone can point me to my mistakes. $$\partial_t \int T^{00}(x^i x_i)^2 d^3 x = -\int T^{0k}_{,k}(x^i x_i)^2 d^3 x = \Dcancelto[0]{-\int (T^{0k}(x_ix^i)^2)_{,k}d^3 x} +\int (T^{0k}(x_i x^i)^2_{,k})d^3 x$$ After that: $$\partial_t \int...

I spent a good amount of time thinking about it and in the end I gave up and asked to a friend of mine. He said it's a 1-line-proof: just "integrate by parts" and that's it. I'm not sure you can do that, so instead I tried using the identity: to express the first term on the right-hand side...

Find the limit as x approaches 0 of x2/(sin2x(9x)) I thought I could break it up into: limit as x approaches 0 ((x)(x))/((sinx)(sinx)(9x)). So that I could get: limx→0x/sinx ⋅ limx→0x/sinx ⋅ limx→01/9x. I would then get 1 ⋅ 1 ⋅ 1/0. Meaning it would not exist. However the solution is 1/81...

I am reading a lot of stuff on advanced algebra and running into these questions. Thank you

This is Jackson's 3rd edition 1.12 problem. So, for both ## \phi ## and ## \phi' ##, I started from Green's second identity: ## \int_V ( \phi \nabla^2 \phi' - \phi' \nabla^2 \phi )dV = \int_S ( \phi \frac {\partial \phi'} {\partial n} - \phi' \frac {\partial \phi} {\partial n} ) dS ## And...

Summary: could you explain why this equality is a quadratic form identity? i read this equality (4.26) here w depends on two variables. it is written that if B is bounded (L2) then it is a quadratic form identity on S. what does it mean? is it related to the two variables? next the author...

I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.

Attempt: $$\nabla \times ( a\nabla b) = \epsilon_{ijk}\frac{\partial}{\partial x_j}(a\frac{\partial b}{\partial x_k})\hat e_i$$ $$ = \epsilon_{ijk}\big(\frac{\partial a}{\partial x_j}\frac{\partial b}{\partial x_k}+a\frac{\partial b}{\partial x_j\partial x_k}\big)\hat e_i$$ $$= \nabla a \times...

Hi I've been reading Peskin & Schroeder lately and I have some confusions over the ward identity. So I think I understand how the identity works at a practical level but not exactly where it comes from. To illustrate my questions (which are difficult to state generally), I will make use the...

I am reading the A First Book of Quantum Field Theory. I have reached the chapter of renormalization, where the authors describe how the infinities of the self-energy diagrams can be corrected. They have also discussed later how the infrared and ultraviolet divergences are corrected. Just before...

From the properties of tensor product, ##H^{\otimes n} \cdot H^{\otimes n} =\left ( H_1 \cdot H_1 \right ) \otimes \left ( H_2 \cdot H_2 \right ) \otimes \cdots \otimes \left ( H_n \cdot H_n \right ) =I \otimes I \otimes \cdots \otimes I =I## where ##H_i## acts on the ##i^{th}## qubit. But I...

Hi all, I've come across an interesting matrix identity in my work. I'll define the NxN matrix as S_{ij} = 2^{-(2N - i - j + 1)} \frac{(2N - i - j)!}{(N-i)!(N-j)!}. I find numerically that \sum_{i,j=1}^N S^{-1}_{ij} = 2N, (the sum is over the elements of the matrix inverse). In fact, I...

Why are we allowed to differentiate both sides of something like ##y=x^2## but not something like ##x=x^2## I believe the answer might be that the first equation is an identity that is true for all values while the second equation is an equation and is only true for some values. Although...

prove this identity (sinx+cosx)/(secx+cscx)= sinxcosx if you could list out the steps it would be appreciated

So, I saw the answer but I couldn't understand it. But I think it can be solved by tan(a)+tab(b)+tan(c)=tan(a)*tan(b)*tan(c) (where a+b+c=Pi) , but I don't know how to transfer it to its inverse. The answer:

If a>1, a cannot = 1, x>0, show that Loga(1/x) = log1/x(a). (COULD NOT SOLVE)

Prove the binomial identity: $$\sum_{j=0}^{n}(-1)^j{n \choose j}=0$$ - in two different ways

Hello, In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows: $$-\frac{1}{4}F_{\mu \nu}^{2}=\frac{1}{2}A_{\mu}\square A_{\mu}-\frac{1}{2}A_{\mu}\partial_{\mu}\partial_{\nu}A_{\nu}$$ where: $$F_{\mu\nu}=\partial_{\mu}...

Can anyone help me with the Proof of Parseval Identity for Fourier Sine/Cosine transform : 2/π [integration 0 to ∞] Fs(s)•Gs(s) ds = [integration 0 to ∞] f(x)•g(x) dx I've successfully proved the Parseval Identity for Complex Fourier Transform, but I'm unable to figure out from where does the...

$\begin{align*} \frac{\cot {x}-1}{\cot{x}+1}&=\frac{1-\sin 2x}{\cos 2x}\\ \frac{\cos {x}-\sin x}{\cos{x}+\sin x} \frac{\cos x-\sin x}{\cos x-\sin x}&= \\ \frac{\cos^2x-2\sin x\cos x+\cos^2 x}{\cos^2 x-\sin^2 x} \end{align*}$ so far..

Problem: Let $\phi(x), x \in \Bbb{R}$ be a bounded measurable function such that $\phi(x) = 0$ for $|x| \geq 1$ and $\int \phi = 1$. For $\epsilon > 0$, let $\phi_{\epsilon}(x) = \frac{1}{\epsilon}\phi \frac{x}{\epsilon}$. ($\phi_{\epsilon}$ is called an approximation to the identity.) If $f \in...

I'm having a little trouble proving the following identity that is used in the derivation of the Baker-Campbell-Hausdorff Formula: $$[e^{tT},S] = -t[S,T]e^{tT}$$ It is assumed that [S,T] commutes with S and T, these being linear operators. I tried opening both sides and comparing terms to no...

fig 1 Given: eiΘ= cosΘ + i sinΘ (radians) eπi=-1 Deduced e2πi=(-1)2 e2πi=1 e(2/3)πi=11/3 e(2/3)iπ=1 e(2/3)iπ=cos(2i/3)+i sin(2i/3) e(2/3)iπ=-1/2+i(3/2) -1/2+i(31/2/2)=1 where n is greater than or equal to 1 or n=a/b where a is greater than or equal to 1 and b is odd 1n=1 ∴...

Show that(A-B)-C=A-(BUC)

Homework Statement Without using vector identities, show that ##\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0##. Homework Equations The definitions and elementary properties of the dot and cross products in terms of Levi-Civita symbols. The "standard" calculus III identities for the divergence...

what is the need to show that r belongs to S U {0} in proof (https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity#Proof) r is zero afterall, whether it lies in S U {0} or not doesn't affect.

I'm going through Peter Smith's book on Godel's Theorems. He mentions a simple formal theory ("Baby Arithmetic") whose logic needs to prove every instance of 'tau = tau'. Does every 'standard deductive apparatus' include the common identity axioms (e.g. 'x = x')?. The axioms of "Baby...

I think I have found a majot error in Neuenschwander's book on Noether's Theorem, but I'd like some confirmation from someone familiar with the book or with the Rund_Trautman identity for fields. As far as I can see the extension of the R-T identity for fields seems to be Neuenschwander's work...

Homework Statement Homework Equations The Attempt at a Solution [/B] Is my answer correct ?

Could someone clarify the notion of a ring with identity element $1=0$? Apparently it's just the zero ring, but then why do we always talk about a ring with identity $1 \ne 0$? It's like having to talk about prime $p \ne 1$; instead we don't define $1$ as a prime but also because we would lose...

Hello I have the following quantum circuit identity for converting a controlled U gate (4x4 matrix) into a series of CNOT gates and single qubit gates $$ U= AXA^{\dagger}X$$ where A is a unitary matrix. Here is a picture of the mentioned identity. Can someone help me understand conceptually...

I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc. But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc. I am trying to understand some of the subtleties of moving to 3D. In particular, for any...

Homework Statement Prove that $$\bf{ a \times ( b \times c ) = \phi [ b(a \bullet c) - c(a \bullet b) ]} $$ for some constant phi Homework EquationsThe Attempt at a Solution So I have used the unit vectors i, j, and k and found out that phi = 1. With the main part of the proof, we are not...

Homework Statement [/B] Prove the following identity: \vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A}) Homework Equations Kronecker's...