Operator Definition and 1000 Threads

Hi. In a question I needed to figure out whether -\frac{i\hbar}{m} \hat{p} is hermitian or not. Since the constant doesn't matter this is similar to whether i \hat{p} is hermitian or not. I thought that since \hat{p} is hermitian, then i times it would not be, since it would not...

Homework Statement Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations The Attempt at a Solution...

hi everyone, im currently trying to teach myself the D operator technique, as opposed to the 'guess method' which i don't really like. I stumbled upon this on yahoo answers: (D^2 + 1)y = 4 cos x - sin x Find the complementary function by solving the auxiliary equation: m² + 1 = 0...

Hi there, I learn from the text that the exponential of an operator could be expanded with a series such that e^{\hat{A}} = \sum_{n=0}^{\infty} \frac{\hat{A}^n}{n!} So if the eigenvalue of the operator \hat{A} is given as a_i e^{\hat{A}}|\psi\rangle will be a matrix with diagonal elements...

Hey guys, this is my first time working on a genetic algorithm. It seems to me that the algorithm is primarily defined by how you choose to define your crossover operator and fitness function. Let's say the crossover takes two parents and produces one child. Is it necessary/good/bad/etc that the...

i am completely lost as to how to go from p^ \frac{1}{m}∇p to \frac{m}{m+1} ∇p^\frac{m+1}{m}

Homework Statement I want to show that tr\left(\hat{\rho}_{mixed}\right)=1 tr\left(\hat{\rho}_{mixed}^{2}\right)<1 when \hat{\rho}_{mixed}=\frac{1}{2\pi}\int_{0}^{2\pi}d \alpha \hat{\rho}(\psi) Homework Equations tr\left(\psi\right)= \sum_{n}\langle n|\psi|n\rangle...

What does the expectation and deviation of an operator mean?? The way I understood it was every observable has a operator to it and the expectation of the observable uses the operator to calculate the deviation ... for ex :: <p>=integral( (si)* momentum operator (si) ) dx ... so what does...

Hi, A fuller view can be found here: (i) What is this operator? (ii) What does this operator mean if Sp is a 5x5 matrix and (Z'Z)-1 is a 3x3 matrix?Thanks a lot.

One can easily prove that \nabla \cdot f is invariant under a rotation of the reference frame, however I would like to prove that the divergence operator itself is invariant (same principle, different approach). In other words I want to prove that \mathbf \nabla = \mathbf e_x...

Hello, I am going through the book Introduction to QM by D.Griffiths. In the third chapter the book says the eigen functions of the momentum operator do not belong to the Hilbert space ... But the only condition that a vector belongs to the Hilbert vector space is that the integral...

Could someone please give a definition what the symbol of an integral operator is? Does it have asymptotic expansions as symbols of differential operators? I know about symbols of differential operators but, shame on me, heard nothing about the symbol of an integral operator. Any help in...

There is a paper written by R.T. Seeley in the Proceedings of Symposia Pure Mathematics that I've seen cited by several papers I've been reading, but I can't find it anywhere. The citation given is R. SEELEY, Complex Powers of an Elliptic Operator, “Singular Integrals (Proc. Symp. Pure Math...

Homework Statement Find spectrum and eigenvalues of operator from L^2(-1,1) to L^2(-1,1) T(f)(t) = ∫(t+s)^2f(s)ds The integral is taken over [-1,1] 2. The attempt at a solution I have already proven that this operator is self-adjoint and compact. However, I have now idea how to find...

I have general function sum_(k=1, to n) f(k) = F(n) ex.) sum_(k=1, to n) k = F(n) solution: F(n) = (1+n)n/2 But If I have inverse problem?: ex.) sum_(k=1, to n) f(k) = n^2 how to get f(k) ? Generaly, If I have F(n), how to get f(k) for sum_(k=1, to n) f(k) = F(n) Is...

The position operator in coordinate representation is: Xab=aδ(a-b) this is diagonal as expected The momentum operator turns out to look like Pab=-ih∂aδ(a-b) Now, this is not supposed to be diagonal because it does not commute with X. However it looks pretty diagonal to me. What am I...

How is the following operator linear??! Homework Statement Is the following a linear vector function? F(r) = r - ix Homework Equations A function is linear if: F(r1 + r2) = F(r1) + F(r2) AND F(ar) = aF(r) The Attempt at a Solution F(r1 + r2) = (r1 + r2) - ix F(r1) +...

Hello everyone, I have enquired about inversion in a sphere here in past: https://www.physicsforums.com/showthread.php?t=440759 Although that time I could not come back to the discussion (apologies to Jason), later I went through some of the properties mentioned by him...

Homework Statement \hat{A} is an anti-unitary operator, and it is known that \hat{A^2}= -\hat{I}, show that |\Psi> is orthogonal to \hat{A}|\Psi> Homework Equations I know that \hat{A} can be represented by a unitary operator, \hat{U}, and the complex conjugation operator, \hat{K}...

Homework Statement This problem is about the momentum squared operator. First, I state how I saw the derivation for the momentum operator. Then I state how I attempt to (and fail to) derive the momentum squared operator using the same methods. Homework Equations <p> = ∫ ψ*(ħ/i ∂/∂x)ψ dx The...

Homework Statement calculate \frac{d}{dt}e^{\hat{A}t} where \hat{A} \neq \hat{A}(t) in other words operator A doesn't depend explicitly on t. Homework Equations The Attempt at a Solution \frac{d}{dt}e^{\hat{A}t} = (\frac{d}{dt}(\hat{A})t + \hat{A})e^{\hat{A}t} =...

I recently got interested in some aspects of quantum optics and have a basic question. There is an uncertainty relation concerning the phase and the particle number of a mode. How is this phase observable defined? Where can I read about such basic concepts without dwelling deeply into higher...

[b]1. (D+1)(D-x)(2e^x+cosx) [b]2. None [b]3. (D+1)(D-x)(2e^x+cos(x))=D(2e^x-sinx-2xe^x-xcosx)+1(2e^x-sinx-2xe^x-xcosx)=-4e^x+2e^x-xcosx-2cosx+xsinx-sinx The correct answer is 2e^x(xsinx-3sinx+2xcosx)

Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...

If we have a hermitian operator Q and we know it's matrix representation [Q], does that mean that [Q2] = [Q]2? For example, I'm pretty sure that's the case for p2 for a harmonic oscillator. We have p=ic(a+-a-) and so p2=c2(a+-a-)(-a++a-)*=c2(a+-a-)(a+-a-)=p p Which tells us that [p2]=[p]2...

Let us take a function defined by y=cx To differentiate that, we use the operator d/dx \frac{d}{dx} y = \frac{d}{dx} cx^{-1} By the chain rule/implicit differentiation on the left and normal differentiation on the right we get, \frac{dy}{dx} = -1c x^{-2} What confuses me is the proper...

Homework Statement Let x=(x1,x2,x3), Ax:=(x2-x3,x1,x1+x3), Bx:=(x2,2x3,x1) Find: (3B+2A2)x. Homework Equations The Attempt at a Solution Warning: I have no idea what I'm doing! (3B+2A2)x = 3Bx+2A2x 3Bx = (3x2,6x3,3x1) Now to find 2A2x. Considering that an index has a...

Hi there, Let X be a Hilbert (Banach) space, and spanned by a set S, say. Let A be linear bounded operator on X into itself. Suppose that the operator is well known on S, that is Aa_i=b_i for all a_i\in S. First, is this operator unique on X? if yes, can we find Aa, for general element...

Homework Statement The hamiltonian for a given interaction is H=-\frac{\hbar \omega}{2} \hat{\sigma_y} where \sigma_y = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right) the pauli Y matrix Homework EquationsThe Attempt at a Solution So from the time dependant schrodinger...

In QFT of a real Klein-Gordon-Field, the field operator \phi(x) is an observable. Mathematically, this is the case because it is a sum (over all k) of a and a^\dagger and this yields a Hermitian operator. Physically, I can understand this because this equation would describe, for example, a...

Homework Statement L : R^n → R is defined L(x1 , . . . , xn ) = sum (xj) from j=1 to n. The problem statement asks me to find an estimation for the operation norm of L, where on R the norm ll . llp, 1 ≤ p ≤ ∞, is used and on R the absolute value.The Attempt at a Solution from, ll Lv...

Hi, I am wondering what mistake i might have made in the following. (v, Au) = [(v, Au)*]* = [(A†v, u)*]* = (u, A†v)* = ((A†)†u, v)* = (v, (A†)†u), where v and u are arbitrary vectors in a Hilbert space. That proves that A = (A†)†, doesn't it? My tutor says some of the steps are wrong...

The letters next to p and L should be subscripts. [Lz, px] = [xpy − ypx, px] = [xpy, px] − [ypx, px] = py[x, px] −0 = i(hbar)
py 1.In this calculation, why is [x, px] not 0 even they commute? 2.Why is py put on the left instead of the right in the second last step? i thought it should be...

If you have some Hamiltonian represented by a 2x2 matrix ## H = \left( \begin{array}{cc} 0 & \Delta \\ \Delta & 0 \end{array} \right) ## And you want to use the time evolution operator ## U = \exp ( - \frac{i}{\hbar} H t ) ## it says that ## U = \exp (- \frac{i \Delta}{\hbar} t) ## Why...

Hey guys, I'm doing a third year course called 'Foundations of Quantum Mechanics' and there's this thing in my notes I don't quite get. I was hoping to get your help on this, if you don't mind. It's about Hermitian conjugate operators. The sentences go (v, Au) = (A†v|u) <v|A|u> = <v|(A|u>)...

Given an interior operator on the power set of a set X, i.e. a map \phi such that, for all subsets A,B of X, (IO 1)\enspace \phi X = X; (IO 2)\enspace \phi A \subseteq A; (IO 3)\enspace \phi^2A = \phi A; (IO 4)\enspace \phi(A \cap B) = \phi A \cap \phi B, I'm trying to show that the set...

Given a vector v\in H: R2->R3 which is a function from R2 to R3, and an operator G: H->H on the space H in which v lives, what is the most commonly used symbol to refer to this mapping, i.e., G(v)=G\otimes v or something else?

The four vector J_μ is the source of the Electromagnetic potential four vector A_μ. If I wanted a source term for the Dirac equation (not that I think there is such a thing) could it be (must it be) a spinor source term (what ever that means)? Related question, consider the two Feynman diagrams...

Hello, I have a problem where I must get value of a given binary bit in decimal number: f(decimal_number, bit_position) = bit_value For example, getting bit values in decimal 14 (binary 1110): f(14, 0) = 0 f(14, 1) = 1 f(14, 2) = 1 f(14, 3) = 1 In general it's easy: (14 &...

[Closed] Question on the form of a vertex operator in a proof Ok, never mind - I decided to find the solution in a different way.. This is a little too specialized anyway. (Is there a way to delete the thread?) Hi, I am reading paper [1] and I found that formula (33), \psi(xy)\psi^*(y)=\frac...

Homework Statement A particle of mass m moves in the potential energy V(x)= \frac{1}{2} mω2x2 . The ground-state wave function is \psi0(x)=(\frac{a}{π})1/4e-ax2/2 and the first excited-state wave function is \psi1(x)=(\frac{4a^3}{π})1/4e-ax2/2 where a = mω/\hbar What is the average value of...

Hello, Sorry if the question sounds silly, but can a continuous matrix be seen as a differential operator? First of all, let me state that I have no idea what a continuous matrix would formally mean, but I would suppose there is such an abstract notion, somewhere? Secondly, let me tell...

Reading through my electrodynamics textbook, I frequently get confused with the use of the del (nabla) operator. There is a whole list of vector identities with the del operator, but in some specific cases I cannot figure out what how the operation is exactly defined. Most of the problems...

Homework Statement Let T: \ell^{2} \rightarrow \ell be defined by T(x)=x_{1},\frac{1}{2}x_{2},\frac{1}{3}x_{3},\frac{1}{4}x_{4},...} Show that the range of T is not closed The Attempt at a Solution I figure that I need to find some sequence of x_{n} \rightarrow x such that...

Homework Statement The operator T maps from L^p(-2,2)\rightarrow L^p(-2,2) is defined (Tf)(x) = f(x) x Show that the operator maps from L^p(-2,2) into the same. Homework Equations p is a natural from 1 to infinity. Holders inequality Substitution integrals The Attempt at a Solution I look at...

If I consider the problem of for example the hydrogen atom. I.e. a central force problem with an effective potential V(r) that depends only of r, the distance between the positively charged nucleous and the negatively charged electron. In the Schrödinger's equation, one considers the...

Here's the definition. Let T be a linear operator on a finite-dimensional inner product space V. If \|\vec{T(x)}\| = \|\vec{x}\| \\ for all x in V, we call T a unitary operator. The question is asking about for all x in some orthornormal basis for V. Isn't that the same as for all x in V?

What does it mean for Sturm-Liouville operator to be symmetric w.r.t an inner product? I was reading in a book that it is symmetric but that was about a certain integral being zero and inner products had not even been mentioned.

Say you have some operator A with an incomplete set of eigenstates, but the state of the system is such that it happens to be expressible as a sum (possibly infinite, or integral, whatever) of the eigenstates of A, and let's say the eigenvalues are real and whatever is necessary...we may assume...

I have a doubt making a little of thinking of basic notions of QM and I think that the answer should be very simple but I can't make up my mind. So, here I go: I usually hear that when we want to go from to Quantum Mechanics to Classical Mechanics, one have to make h go to 0 and then the magic...