Operators Definition and 1000 Threads

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}_{+} =...

Problem: ----------- I’m trying to understand how to generally find Eigen functions/values (either analytically or numerically) for Hamiltonian with creation/annihilation operators in many-body problems. Procedures: -------------- 1. I setup a simple case of finite-potential well...

Hi folks, I was wondering if the two Casimir operators of the SU(3) color gauge group were of any physical significance, or corresponded to any familiar physical properties. For example, I know that in the Poincare group the two Casimirs correspond to mass and spin: is there a similarly...

Hi all, I'm working on Taylor's text on scattering (a reference from Peskin and Schroeder). They define the Moller operators \Omega which are isometric, satisfying \Omega^{\dagger}\Omega=1 This is not necessarily the same as unitary in an infinite dimensional space, the difference being...

The differential of a function may me interpreted a the the dual of its gradient. What is the interpretation of the Dolbeault operators?

I recently thought of this, please excuse me if it is way off the mark! If I act on a state with a hermitian operator, am I able to find the psi(p) (momentum), where I had psi(x) (position) before (and wise versa)? Or does the operator do what it appears to do, and that is find the derivative...

Homework Statement I have been given the following problem - the expectation value of px4 in the ground state of a harmonic oscillator can be expressed as <px4> = h4/4a4 {integral(-infin to +infin w0*(x) (AAA+A+ + AA+AA+ + A+AAA+) w0 dx} I think I know how to proceed on other...

Homework Statement Let V be a n-dimensional real vector space and L: V --> V be a linear operator. Then, A.) L can always be diagonalized B.) L can be diagonalized only if L has n distinct eigenvalues C.) L can be diagonalized if all the n eigenvalues of L are real D.) Knowing the...

A Hermitian matrix is a square matrix that is equal to it's conjugate transpose. Now let's say I have a Hermitian operator and a function f: [ H.f ] The stuff in the square is the complex conjugate as the functions are in general complex. If I do not consider the matrix representation of...

Homework Statement http://img818.imageshack.us/f/screenshot20110423at733.png/ http://img856.imageshack.us/f/screenshot20110423at733.png/ If it'll help you guys help me understand this, here are the solutions: http://img828.imageshack.us/f/screenshot20110423at752.png/ Homework Equations...

I don't understand the following step: using \hat{}a*\hat{}a = (\hat{}H/\hbarw ) -1/2 <n|\hat{}a*\hat{}a|n> = n<n|n>. my first thoughts were to use a|n> = sqrt n | n-1> but I don't think that's relevant if you sub in a*a and separate it into two expressions I don't see what good that would do

Homework Statement I'm given the expression for the operator Pr Y Pr Y= -ih(bar) (1/r) d/dr (r Y) I want to find Pr^2 Y so I have dotted Pr with Pr I expect to get: -h(bar)^2 (1/r) d/dr [ Y(Y + r dY/dr)] but my notes have omitted the first Y in the above bracket and I...

Hi, I've found the expectation value of Sz, which is hbar/2 (|\psiup|2 - |\psidown|2) by using the formula: <Si> = <\psi|Si\psi> where i can bex, y or z and \psi is the 'spinor' vector. I tried to find Sx using the same formula, however, I could only get as far as: hbar/2 ((\psiup)*\psidown...

a linear operator T: X -> Y is bounded if there exists M>0 such that: ll Tv llY \leq M*ll v llX for all v in X conversely, if i know this inequality is true, is it always true that T: X ->Y and is linear?

Homework Statement Show that y2 is non-linear. Homework Equations ^O (ay1 +by2) = a ^O(y1) + b ^O(y2). The Attempt at a Solution No idea!

Hi, I was wondering: What is the physical reason for only choosing linear operators to represent observables?

I'm trying to understand how to manipulate equations with del operators. If I have a equation like : div( A + B ) = div(E) and assume A,B,E are twice differential vectors do div cancel? can I say E = A + B? If I write is like this div( A + B - E ) = 0 div( A + B - (A + B)) = 0...

I have a rather fundamental question which I guess I've never noticed before: Firstly, in QM, why do we define the expectation values of operators as integral of that operator sandwiched between the complex conjugate and normal wavefunction. Why must it be "sandwiched" like this? From...

I'm am trying to derive the relations: a|n\rangle=\sqrt{n}|n-1\rangle a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle using just the facts that [a,a+]=1 and N|n>=|n> where N=a^{\dagger}a (which implies \langle n|N|n\rangle=n\geq 0). This is what I've done so far: [a,a^{\dagger}]=1 \Rightarrow...

in the experimental side of QM, I know i can use a slit to measure the q. but what about p or E? and how to conciliate the measure with the theory?: after the measurement of a slit i'll measure q and the system will collapse in a autovector of q, like |q> but it will evolve like a...

Homework Statement If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? Homework Equations The Attempt at a Solution My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must...

If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must be either upper triangular or lower triangular. This is what I found when I tried...