Operator Definition and 1000 Threads

Hi, I'm struggling to understand this concept. I think the term probably comes from functional analysis and I don't know any of the terms in that field so I'm having trouble understanding the meaning of what a compact linear operator is. I posted this in linear algebra because I'm reading...

Homework Statement Use ##D = \frac{d}{dx}##as a differential operator and the following $$(D - a)(D -b)[f(x)e^{\lambda x}] = e^{\lambda x} (D + \lambda -a)(D + \lambda -b)f(x)$$ to obtain $$(D^2 + D +1)[(Ax^2 + Bx + C)e^{ix}] = (iAx^2 + [iB + (4i + 2)A]x + 2A + (2i + 1)B + iC)e^{ix}$$ The...

I am trying to solve the following problem on an old Quantum Mechanics exam as an exercise. Homework Statement Homework Equations I know that the trace of an operator is the integral of its kernel. \begin{equation} Tr[K(x,y)] = \int K(x,x) dx \end{equation} The Attempt at a...

Can someone explain to me the mathematical intuition that motivates the embedding of quantum operators between the conjugate wave function and the (non-conjugated) wave function? That is, we write: \Psi^{*}\hat{H}\Psi, that is: \Psi^{*}(\hat{H}\Psi), so that \hat{H} operates on \Psi (not...

In class we recently learned that for a linear operator $$T: V \rightarrow V$$ and function $$g(t) = a_0 + a_1t + \dots + a_nt^n$$ one can define the operator $$g(T) = a_0I + a_1T + \dots + a_nT^n$$ (where $$I$$ is the identity transformation). We also recently learned about the exponential of...

For potential well problem for well with potential which is zero in the interval ##[0,a]## and infinite outside we get ##\psi_n(x)=\sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}##. If I want to get this result for well with potential which is zero in the interval ##[-\frac{a}{2},\frac{a}{2}]## and...

Homework Statement Calculate the action of the operator on the function f(x) Homework Equations Operator - exp(a*x^2*(d/dx)) The Attempt at a Solution

Greetings, My task is to prove that the angular momentum operator is hermitian. I started out as follows: \vec{L}=\vec{r}\times\vec{p} Where the above quantities are vector operators. Taking the hermitian conjugate yields \vec{L''}=\vec{p''}\times\vec{r''} Here I have used double...

Hello, now I'm reading Peskin Shroeder. I have a question about the Casimir operator on page 500 in Chapter 15. From the following eq, ## \ \ \ [T^b , T^a T^a ] = 0 \ \ \ \ \ \ \ (15.91) ## ## T^2(=T^a T^a) ## is an invariant of the algebra. Thus the author concludes that ##T^2## is...

e^{\alpha\frac{d}{dx}}=1+\alpha\frac{d}{dx}+\frac{\alpha^2}{2!}\frac{d^2}{dx^2}+...=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^n}{dx^n} Why this is translational operator? ##e^{\alpha\frac{d}{dx}}f(x)=f(x+\alpha)##

Homework Statement In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...

When I try to operate this command plot(plot::Inequality(x^2 + y^2 < 1,x = -1.5..1.5, y = -1.5..1.5)) It failed, and displayed Error: Unexpected MATLAB operator. How can I fix it?

Homework Statement Using <\hat{p}n> = ∫dxψ*(x)(\hat{p})nψ(x) and \hat{p} = -ihbar∂x and the definition of the Fourier transform show that <\hat{p}> = ∫dk|\tilde{ψ}(k)|2hbar*k 2. The attempt at a solution Let n = 1 and substitute the expression for the momentum operator. Transform the...

Hi can anyone tell me why in the fermionic number operator case: <0|N/V|0>= \sum_{\pm r}\int d^3 k a^{\dagger}(t,r)a(t,r) because if: N=a^{\dagger}(t,k)a(t,k) then after Fourier decomposition surely one gets: \int d^3 r d^3 r \frac{1}{(2Pi)^{3}} a^{\dagger}(t,r)a(t,rk) and when...

I was wondering if anyone knew if it is possible to construct an if else if with the ternary operator in C. I know that we can use it for if else, but what if you wanted multiple conditions for else if in your statement? printf("%d",(a>5)?1:(a<5)?0:10); //Just a silly example perhaps?

Homework Statement There are two ways to write the momentum operator, p = (-i hbar d/dx) and p = (hbar / i)d/dx. How do you go from one to the other? Homework Equations The two I gave above. The Attempt at a Solution I tried to see if -ih = h/i by squaring both sides, but one came out...

Is something wrong in my assertions below? Suppose we have two quantum systems N and X. Let N is described by discrete observable \hat{n} (bounded s.a. operator with discrete infinite spectrum) with eigenvectors |n\rangle. Let X is described by continuous observable \hat{x} (unbounded s.a...

Hi all, Del = i ∂/∂x + j ∂/∂y + k ∂/∂z in x y z cordinate similarly I require to see the derivation of del in other coordinates too. Please give me a link for the derivation.

If you consider a bounded linear operator between two Hausdorff topological vector spaces, isn't the kernel *always* closed? I mean, if you assume singleton sets are closed, then the set \{0\} in the image is closed, so that means T^{-1}(\{0\}) is closed, right (since T is assumed continuous)? I...

How does finding the rotation operator for a spin 1/2 particle differ from finding that of a spin 1 particle?

Homework Statement Why does <0|\frac{1}{(2\pi)^3}∫ \hat{a}^{\dagger}(t,r) \hat{a}(t,r) d^{3} \textbf{k} |0> = \frac{1}{\pi^2}∫|β|^2 k^2 dk. Where \hat{a} and \hat{a}^{\dagger} and its conjugate are bogulobov transformations given by: \hat{a}(t,k) = \alpha(t)a(k) + β(t)b^{\dagger}(-k)...

Homework Statement Verify whether or not the operator L(u) = u_x + u_y + 1 is linear. Homework Equations An operator L is linear if for any functions u, v and any constants c, the property L(c_1 u + c_2 v) = c_1 L(u) + c_2 L(v) holds true. The Attempt at a Solution I feel...

Hey, I'm having trouble interpreting a question, as the solutions say something different... Anyways the question part d) below: So we want to determine the expectation value of the y-component of the electron spin on the eigenstate of Sx, now I would of thought this was given by...

Hey, I have a question on showing how the raising operator in QM raises a particular eigenstate by 1 unit, the question is showed below: I think I know how to do this but not sure if what I'm doing is sufficient: \hat{N}a^{\dagger}|n>=([\hat{N},a^{\dagger}]+a^{\dagger}\hat{N})|n>...

"Del" operator crossed with a scalar times a vector proof Homework Statement Prove the following identity (we use the summation convention notation) \bigtriangledown\times(\phi\vec{V})=(\phi \bigtriangledown)\times\vec{V}-\vec{V}\times(\bigtriangledown)\phi Homework Equations equation for...

I have the following situation: About the polarization of the photon, I introduce the basis: Horizontal polarization $|\leftrightarrow>=\binom{1}{0}$ Vertical polarization $|\updownarrow>=\binom{0}{1}$ The density matrix in this problem is: $$\rho =\frac{1}{2}\begin{pmatrix} 1+\xi...

Hello. I have some questions on operations. Suppose in the course of a derivation there is a mathematical statement of the form A+1=B+C then "+" is an operator acting on inputs "B" and "C". Question 1: Is the output of the operation "A" or the expression "B+C"? The reason I think the...

I have a problem where it's said that the operator Q is likely to be: Q=\sum^3_{i=1}[\frac{1}{2}B_i + I_{3,i}] I have to apply this to the proton wave function which is the same as you can see in equation (3.20) here...

Homework Statement How to prove that for any representation of the spin, the state e^{-i{\pi}J_x/\hbar}|j,m\rangle is proportional to |j,-m\rangle The exponential term is the rotation operator where J_x is the x-component of the total angular momentum operator, and |j,m\rangle is an...

Hey, I have the following question on Hermitian operators Initially I thought this expectation value would have to be zero as the eigenvectors are mutually orthogonal due to Hermitian Operator and so provided the eigenvectors are distinct then the expectation would be zero... Though...

The book I am going through says this : The below proposition is false for real inner product spaces. As an example, consider the operator T in R^2 that is a counter clockwise rotation of 90 degrees around the origin. Thus , T(x,y) = (-y,x). Obviously, Tv is orthogonal to v for every v in...

For the free particle the solution to the SE are eigenstates of the momentum. You get something like: ψ = Aexp(ik(x-vt)) + Bexp(-ik(x+vt)) , where k is a constant And my book then says that first term represents a wave traveling to the right and the second a wave traveling to the left. But I...

I was looking for a hint on a problem in my professor's notes (class is over and I was just auditing). I want to show that if T:V→V is a linear operator on finite dimensional inner product space, then if T is diagonalizable (not necessarily orth-diagonalizable), so is the adjoint operator of...

http://upload.wikimedia.org/math/2/b/2/2b2fe1336915a03e04930c11b27f4585.png The above link shows the material derivative. Which is the derivative that follows a volume of fluid throughout its movement through a fluid. How is this derived from a chain rule? Is the v in that equation the...

Homework Statement Homework Equations The Attempt at a Solution I don't know what's wrong with my work. I can't obtain the eigenvector provided in the model answer. [SIZE="5"]My work [SIZE="5"] Model Answer

Homework Statement I'm trying to find the divergence of a vector field (a fluid flow vector), but the vector takes the form u = u(x,y,z,t) The Attempt at a Solution I only really know how to take the divergence of a time-independent vector, so I'm guessing I just take the partial...

eigenvalues of a compact positive definite operator! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥... be the eigenvalues of A in decreasing order. Can...

I have bumped into a term a^\dagger \hat{O}_S | \psi \rangle I would really like to operate on the slater determinant \psi directly, but I fear I cannot. Is there any easy manipulation I can perform?

hi,my friends.I have a perturbation operator problem. v=ezE; why this formula is right?how to deduce it?a is bohr radius. thank you!

Hi I am reading a paper (http://arxiv.org/abs/0901.3105), where they after eq. (3) mention something I can't understand. First of all, (3) comes from the master equation of a collection of N atoms in a cavity. They say that (page 2, right after (3)): The last term describes the coupling of...

In the QM Hamiltonian, I keep seeing h-bar/2m instead of p/2m for the kinetic energy term. H-bar is not equivalent to momentum. What am I missing here?

Hey, I recently had an exam where the quantum state were on the form |\psi\rangle = \frac{1}{\sqrt{2}} ( |+\rangle + i |-\rangle ) Here I formed the density operator for the pure state \rho(t) = |\psi\rangle \langle \psi| = \frac{1}{2} ( |+\rangle + i |-\rangle )( \langle +| - i \langle...

So, I'm studying Second Quantization for fermions and came across this equation. I was just wondering why there is a summation needed? And why do we do it with (i≠p).? Please can someone explain this to me? Reply and help is much appreciated.

Homework Statement Show that the expectation operator E() is a linear operator, or, implying: E(a\bar{x}+b\bar{y})=aE(\bar{x})+bE(\bar{y}) Homework Equations E(\bar{x})=\int_{-\infty}^{+\infty}xf_{\bar{x}}(x)dx With f_{\bar{x}} the probability density function of random variable x...

I was wondering about this: The identity operator writes a vector in the basis that is used to express the identity operator: 1 = Ʃlei><eil But if you are to apply it to a vector in a given basis A should the lei> then be expressed in terms of their own basis or in terms of A?

We all know the greek letter delta is the mathematical symbol that represents "change in." I though about a new form of delta: Δn. Where n2 = the # of terms when you expand the delta operator. For example: the usual Δx = x2 - x1 But now: Δ2x = (X4-X3) - (X2-X1). We can see that for Δ2...

What are the eigenfunctions of the spin operators? I know the spin operators are given by Pauli matricies (https://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin), and I know what the eigenvalues are (and the eigenvectors), but I have no idea what the eigenfunctions of the...

when we have a certain state ψ(t) and it is acted on by an operator A of eigenstates a, b, c and eigen vectors la>, lb>, lc> does it mean that after measuring A ( if the result was 'a'), the state lψ(t)> becomes in state la>?

Homework Statement Consider the set of functions {f(x)} of the real variable x defined on the interval -\infty< x < \infty that go to zero faster than 1/x for x\rightarrow ±\infty , i.e., \lim_{n\rightarrow ±\infty} {xf(x)}=0 For unit weight function, determine which of the...

I'm trying to understand the construction of the T(ε) operator and why it is equal to I-iεG/hbar. The textbook I'm using (Shankar) talks defines the translation operator with the phase factor: T(ε)\left|x\right\rangle=e^{i \epsilon g(x)/\hbar}\left|x+\epsilon\right\rangle and...