Operators Definition and 1000 Threads

Hi, A general question.. In analytical mechanics, we take a given hamiltonian and re-write it in term of generalzed coordinates. In a way- we recode the hamiltonian to concern only the "essence" of the problem. However, it seems to me, that in QM we do the opposite- we look for operators that...

Homework Statement Determine [T]β for linear operator T and basis β T:((x1; x2]) = [2x1 + x2; x1 - x2] β = {[2; 1], [1; 0]} Homework Equations Now that would be MY question :rolleyes: The Attempt at a Solution Well the answer is [1, 1; 3, 0], but i have no idea what I'm even...

Homework Statement In my quantum class we learned that if two operators commute, we can always find a set of simultaneous eigenvectors for both operators. I'm having trouble proving this for the case of degenerate eigenvalues.Homework Equations Commutator: [A,B]=AB-BA Eigenvalue equation:A...

consider a particle in the box in one dimension with the length a. the hamiltonian is H. then the box's walls goes far away and the box length gets b. now the hamiltonian is H1. i like to know whether these two hamiltonians commute or not?

Hello, I was reading von neumann's mathematical foundations of quantum mechanics book and I am confused at a certain part regarding finding the resolutions of position and momentum operator. I will ask my first question (I will think more about the second one before posting :p) For anyone who...

I would just like some clarification and some assertion that I've got the right idea. Please correct everything I say! For any observable A over a finite-dimensional vector space with orthonormal basis kets \{|a_i\rangle\}_{i=1}^n we can write A = IAI = \left(\sum_{i=1}^n |a_i\rangle\langle...

Homework Statement Show that f(a†a)a† = a†f(a†a + 1) Where f is any analytic function and a and a† satisfy commutation relation [a, a†] = 1. The Attempt at a Solution I have used [a, a†] = aa†-a†a=1 to write the expression like f(a†a)a†= a†f(aa†) but I don't know what to do...

Homework Statement Find the hermitian conjugates, where A and B are operators. a.) AB-BA b.) AB+BA c.) i(AB+BA) d.) A^\dagger A Homework Equations (AB)^\dagger =B^\dagger A^\dagger The Attempt at a Solution Are they correct and can I simplify them more? a.)...

Just to check something: If A and B are operators and B|a> = 0, does this imply that <a|AB|a> = 0 ? Or can you not split up the operators like <a|A (B|a>) ? Thanks.

Hi! Could anyone please tell me the meaning of Tow operators are unitary equivalent. I tried Wiki but I did not get my goal.

Homework Statement The operators J(subscript x)-hat, J(subscript y)-hat and J(subscript z)-hat are Cartesian components of the angular momentum operator obeying the usual commutation relations ([J(subscript x)-hat, J(subscript y)-hat]=i h-bar J(subscript z) etc). Use these commutation...

How can I formally demonstrate this relations with hermitian operators?(A^{\dagger})^{\dagger}=A (AB)^{\dagger}=B^{\dagger}A^{\dagger} \langle x|A^{\dagger}y \rangle=\langle y|Ax \rangle ^* If \ A \ is \ hermitian \ and \ invertible, \ then \ A^{-1} \ is \ hermitian I've tried to prove them...

Homework Statement I need to prove that, <p'|\hat{x}p> = i\hbar\frac{d}{dp'}\delta{p-p'} i.e. find the position operator in the momentum basis p for p'... It's easy to prove that <x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'} (position operator in position basis for x') since I...

Hello everybody, long time reader, first time poster. I've searched the forums extensively (and what seems like 60% of the entire internet) for anything relevant and haven't found anything, please point me in the right direction if you've seen this before! Homework Statement Show that even...

Can anyone help with the 2 questions below: 1) Suppose f is a mathematical quantity that can take on the "states", or "values", f1, f2,...fi,...fn., where n can be finite or infinite. So, F is the Set { f1, f2,... fi,.. fn } or F = { f1, f2, fi,... fn } = { all...

Penrose says in “Cycles of Time” that rest mass isn't exactly a Casimir operator of the de Sitter group, so a very slow decay of rest mass isn't out of the question in our universe. If rest mass is strictly conserved, should it be a Casimir operator of the de Sitter group? Decay of rest...

Homework Statement Consider a quantum system that acts on an N-dimensional space. We showed that any operator could be expressed as a polynomial of the form O=\sum_{m,n=1}^{\infty}o_{mn}U^m V^n where U and V are complementary unitary operators satisfying (U^N = V^N =1) Show that if O...

Homework Statement Please see attached pdf. Can anyone please advise on how to approach the questions or provide websites where I can read up on it? I am reading Griffiths but it doesn't seem to cover this much. And I don't know how to google on this topic because I can't type the characters...

In a right-handed cartesian coordinate system the divergence and curl operators are respectively: \nabla \cdot A= \frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z} \nabla \times \mathbf{A}= \begin{vmatrix} \widehat{x} & \widehat{y} &...

Hi, I have this question for a problem sheet: Use the unit operator to show that a Hermitian operator A can be written in terms of its orthonormal eigenstates ln> and real eigenvalues a as : A=(sum of) ln>a<nl and hence deduce by induction that A^k = (sum of) ln>a^k<nl I have no...

Homework Statement A bound quantum system has a complete set of orthonormal, no-degenerate energy eigenfunctions u(subscript n) with difference energy eigenvalues E(subscript n). The operator B-hat corresponds to some other observable and is such that: B u(subscript 1)=u(subscript 2) B...

So I'm doing some proofs on a Saturday night... working on proving that (AB+BA) is self-adjoint, that is (AB+BA)=(AB+BA)* (using a * instead dagger). What I want to know is if the following is true: (AB+BA)*=B*A*+A*B* ?

Hi. I'm just a bit stuck on this question: Write down two equations to represent the fact that a given wavefunction is simultaneously an eiigenfunction of two different hermitian operators. what conclusion can be drawn about these operators? Im not quite sure how to start it. Thanks!

Hello, I have an unitary operator f, and another binary linear operator g. I would like to find out a necessary and/or sufficient condition on f for the following to hold: f(g(a,b)) = g(f(a),f(b)) Is this always valid when f is linear?

Homework Statement The problem is number 11, the problem statement would be in the first picture in the spoiler. Basically, I'm trying to find if two operators commute. They're not supposed to, since they involve momentum and position, but my work has been suggesting otherwise, so I'm doing...

Hi there, This is my first post. In operator theory, what we mean by "The operator M_u (the multiplicative operator) acts isometrically from L^1 to L^1". Also, what is the anti-linear isometry. Thanks in advance.

Homework Statement Given a Hermitian operator A = \sum \left|a\right\rangle a \left\langle a\right| and B any operator (in general, not Hermitian) such that \left[A,B\right] = \lambdaB show that B\left|a\right\rangle = const. \left|a\right\rangle Homework Equations Basically those...

Homework Statement Let U and V be the complementary unitary operators for a system of N eingenstates as discussed in lecture. Recall that they both have eigenvalues x_n=e^{2\pi in/N} where n is an integer satisfying 0\leq n\leq N. The operators have forms U=\sum_{n}|n_u\rangle\langle n_u...

Homework Statement I have the following expressions for angular momentum components: L_1 = x_2\frac{\partial}{\partial x_3} - x_3\frac{\partial}{\partial x_2}, L_2 = x_3\frac{\partial}{\partial x_1} - x_1\frac{\partial}{\partial x_3}, L_3 = x_1\frac{\partial}{\partial x_2} -...

Hello! Im trying to do QFT on my own and its going fine.. except one confusion now. We have our operator fields corresponding to our observables, and our state which is a function of space and time. But doing the second quantization we get the creation and destruction operators which now...

Hi, We know that the Pauli matrices along with the identity form a basis of 2x2 matrices. Any 2x2 matrix can be expressed as a linear combination of these four matrices. I know of one proof where I take a_{0}\sigma_{0}+a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{3}\sigma_{3}=0 Here, \sigma_{0} is...

Homework Statement Consider two operators, A and B which satisfy: [A, B] = B ; B†B = 1 − A A. Determine the hermiticity properties of A and B. B. Using the fact that | a = 0 > is an eigenstate of A, construct the other eigenstates of A. C. Suppose the eigenstates of A form a complete...

What does it mean for a creation or annihilation operator to act on the state <x|p>. For example: a_p e^{ip \cdot x}

Suppose T is an injective linear operator densely defined on a Hilbert space \mathcal H. Does it follow that \mathcal R(T) is dense in \mathcal H? It seems right, but I can't make the proof work... There is a theorem that speaks to this issue in Kreyszig, and also in the notes provided by my...

I have been posting on here pretty frequently; please forgive me. I have an exam coming up in functional analysis in a little over a week, and my professor is (conveniently) out of town. We proved in our class notes that if T:X\to X is a compact operator defined on a Banach space X, \lambda...

Hi Here's the problem I am trying to do. a) Is the state \psi (\theta ,\phi)=e^{-3\imath \;\phi} \cos \theta an eigenfunction of \hat{A_{\phi}}=\partial / \partial \phi or of \hat{B_{\theta}}=\partial / \partial \theta ? b) Are \hat{A_{\phi}} \;\mbox{and} \;\hat{B_{\theta}}...

Is it true that if T: X\to Y is a compact linear operator, X and Y are normed spaces, and N is a subspace, then T|_N (the restriction of T to N) is compact? It seems like it would work, since if B is a bounded subset of N, it's also a bounded subset of X and hence its image is precompact in Y...

Homework Statement Prove or disprove: if U is in the vector space of complex n x n matrices, then U is unitary if and only if U= e^iA, where A is some self adjoint matrix in same vector space, all of whose eigenvalues lie in the interval [0,2pi) Homework Equations A is self adjoint; A*...

Is it easy to show that T: X \to Y is a compact linear operator -- i.e., that the closure of the image under T of every bounded set in X is compact in Y -- if and only if the image of the closed unit ball \overline B = \{x\in X: \|x\|\leq 1\} has compact closure in Y? One direction is (of...

couple of questions a) the operators not commuting would also be true of position and momentum operators in classical mechanics (x d/dx -d/dx x) f(x) so the non-commutation does not inherently constitute a proof for the uncertainty principle, or do you just not care about the uncertainty at...

For spin 1/2 particles, I know how to write the representations of the symmetry operators for instance T=i\sigma^{y}K (time reversal operator) C_{3}=exp(i(\pi/3)\sigma^{z}) (three fold rotation symmetry) etc. My question is how do we generalize this to, let's say, a basis of four...

some, like momentum appear to be, but are all of them?

In the free case,we decomposite the free Hamiltonian into the creation and annihilation operators, i just wonder why this ad hoc method can not be used to the interaction theory?

I'm just revising some Quantum Mech and I have two questions. I know that if two operators commute say for instance [\hat{A},\hat{B}] = [\hat{B},\hat{A}] = 0 Then the observables that the operators extract from the wavefunction can be measured exactly (without losing information about the...

What is the general strategy in solving vector equations involving grad and the scalar product? In particular, I want to express \Lambda in terms from \mathbf U \cdot \nabla\Lambda = \Phi but it looks impossible, unless there is some vector identity I can use. Thanks in advance.

Given some state \left|\psi\right\rangle, and two operators \hat{A} and \hat{B}, how do you prove that if \langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle then \hat{A} = \hat{B} ?

In Greiner & Muller's 'Quantum Mechanics: Symmetries' (section 3.5) they explain that where a system possesses a symmetry, the corresponding Hamiltonian must be 'built up' from the Casimir operators of the corresponding symmetry group. Does anyone know of a reference where this is gone into...

Apparently - that is, if I'm to believe Kolmogorov - we have the following for a bounded linear operator A between two normed spaces: \sup_{\| x \| \leq 1} \|Ax\| = \sup_{\|x\| = 1} \|Ax\| But why?

Hi, I understand that we use i\hbar\partial/\partial t and -i\hbar\nabla for the energy and momentum operator but I would like to know how this identification is made. I can see that it works for a wave of the form e^{i(kx-\omega t)} and using the relation E=\hbar\omega and the relation...

Homework Statement The quantum simple harmonic operator is described by the Hamiltonian: \hat{H} = -\frac{h^{2}}{2m}\frac{d^{2}}{dx^{2}} + \frac{1}{2}m\omega^{2}x^{2} Show how this hamiltonian can be written in terms of the raising and lowering operators: \widehat{a}_{+} =...