Operators Definition and 1000 Threads

a) V = U_1 ⊕ U_-1 where U_λ = {v in V | T(v) = λv} b) if V = M_nn(R) and T(A) = A^t then what are U_1 and U_-1 When V is a vector space over R, and T : V -> V is a linear operator for which T^2 = IV .

I know that the commutator of the position and momentum operators is ihbar. Can any other combination of two different operators produce this same result, or is it unique to position and momentum only?

An operator A defined by a matrix can be written as something like: A = Ʃi,jlei><ejl <eilAlej> How does this representation translate to a continuous basis, e.g. position basis, where operators are not matrices but rather differential operators etc. Can we still write for e.g. the kinetic...

It is said that each observable like position or momentum is represented by a Hermitian operator acting on the state space. And the Hamiltonian is the total energy of the system, kinetic and potential.. so it means the Hamiltonians encode or encompass the energy of all observables (like...

Hello Homework Statement For a free particle moving in one dimension, divide the following set of operators into subsets of commuting operators: [P,x, H, p] Homework Equations The Attempt at a Solution I don't get the statement itself What does the set represents for the...

Let $$(T_{n}) $$be a sequence in $${B(l_2}$$ given by $$T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). $$Show that $$T_{n}->T$$ given by $$T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). $$ I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.

For a set of energy eigenstates |n\rangle then we have the energy eigenvalue equation \hat{H}|n\rangle = E_{n}|n\rangle. We also have a commutator equation [\hat{H}, \hat{a^\dagger}] = \hbar\omega\hat{a}^{\dagger} From this we have \hat{a}^{\dagger}\hat{H}|n\rangle =...

Why do we multiply some operator A both on the left and on the right with, say, A and A^(-1) in order to perform some kind of conjugation? If it helps, the example I'm thinking of is the relationship between Schrodinger and Heisenberg operators in QFT. Thanks.

Let's say we have operator X that is Hermitian and we have operator P that is Hermitian. Is the following true: [X,P]=ihbar This is the commutator of X and P. This particular result is known as the canonical commutation relation. Expanding: [X,P]=XP-PX=ihbar This result indicates that...

Homework Statement This is quite a long problem, and I have most of it figured out, but I am getting stuck on the very last part of the problem. My problem is I do not understand how to find \left\langle\psi|a_1\right\rangle in the very last line. Is...

Looking through this matrix approach to the quantum harmonic oscillator, http://blogs.physics.unsw.edu.au/jcb/wp-content/uploads/2011/08/Oscillator.pdf especially the equations m \hat{ \ddot { x } } = \hat { \dot {p} } = \frac {i}{\hbar} [ \hat {H} , \hat {p} ] I'm getting the impression...

Homework Statement Calculate <i(\hat{a} - \hat{a^{t}})> Homework Equations |\psi > = e^{-\alpha ^{2}/2} \sum \frac{(\alpha e^{i\phi })^n}{\sqrt{n!}} |n> \hat{a}|n> = \sqrt{n}|n-1> I derived: \hat{a}|\psi> = (\alpha e^{i\phi})^{-1}|\psi> The Attempt at a Solution...

In Calculus, I am studying differentiation at the moment. The two equations is the basic Derivative function: (f(x+h)-f(x))/h and the alternative formula: (f(z)-f(x))/(z-x); and I can see how they both have their own purposes for finding the tangent line and such; but when will differentiation...

Homework Statement For the SHO, find these commutators to their simplest form: [a_{-}, a_{-}a_{+}] [a_{+},a_{-}a_{+}] [x,H] [p,H] Homework Equations The Attempt at a Solution I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first...

Say I have a 3x3 operator Q and I find its eigenvectors and eigenvalues. Now i know that those eigenvectors are the same as eigenfunctions so if i act on them with Q i will get the corresponding eigenvalue. What the question I am trying to solve asks is, Measure the quantity Q in state [b]...

Prove that if a continuous function e\left( x \right) on \mathbb{R} is eigenfunction of all shift operators, i.e. e\left( x+t \right) = \lambda_t e\left( x \right) for all x and t and some constants \lambda_t , then it is an exponential function, i.e. e\left( x \right)= Ce^{ax} for some...

Homework Statement [A,B] = C and operators A,B,C are all hermitian show that C=0 Homework Equations The Attempt at a Solution Since it is given that all operators are hermitian I know that A=A' B=B' and C=C' so i expanded it out to AB-BA=C A'B'-B'A'=C (BA)' - (AB)'=C...

Hi everyone, I was just working on some problems regarding the mathematical formalism of QM, and while trying to finish a proof, I realized that I am not sure if the following fact is always true: Suppose that we have two linear operators A and B acting over some vector space. Consider a...

It is obvious to me how \hat {x} = x; \hspace{5 mm} \hat {p}_x = -i \hbar \frac {\partial} {\partial x} implies [ \hat {x} , \hat {p}_x ] = i \hbar and I can accept that these two formulations are mathematically equivalent, but I do not know how in general (or even in this specific...

Homework Statement Write down the 3×3 matrices that represent the operators \hat{L}_x, \hat{L}_y, and \hat{L}_z of angular momentum for a value of \ell=1 in a basis which has \hat{L}_z diagonal. The Attempt at a Solution Okay, so my basis states \left\{\left|\ell,m\right\rangle\right\}...

Hello, folks, A Sturm-Liouville operator is typically defined not on the whole space of C^2 functions, but rather on some subspace described by boundary conditions. My question is: are those subspaces closed (hence complete, hence Hilbert) in L^2? In case of an affirmative answer, how can...

The following is a problem statement. locally bounded (or locally (weakly) compact) differential operators of the Schwartz space of smooth functions on a sigma-compact manifold I realize this is very abstract. I expect the solution to be just as abstract. Thanks in advance.

Homework Statement The expectation value of the time derivative of an arbitrary quantum operator \hat{O} is given by the expression: d\langle\hat{O}\rangle/dt\equiv\langled\hat{O}/dt\rangle=\langle∂\hat{O}/∂t\rangle+i/hbar\langle[\hat{H},\hat{O}]\rangle Obtain an expression for...

Hello, I'm having trouble calculating this commutator, at the moment I've got: \left[a_{p},a_{q}^{\dagger}\right]=\left[\frac{i}{\sqrt{2\omega_{p}}}\Pi(p)+\sqrt{\frac{w_p}{2}}\Phi(p),\frac{-i}{\sqrt{2\omega_{p}}}\Pi(p)+\sqrt{\frac{w_p}{2}}\Phi(p)\right]=i\left[\Pi(p),\Phi(q)\right]=i\int...

The total energy of a particle in a harmonic oscillator is found to be 5/2 ~!. To change the energy, if i applied the lowering operator 4 times and then the raising operator 1 times successively. What will be the new total energy? i want the calculation please

Hello, I have really been banging my head the whole day and trying to figure this derivative out. I have a function of the following form: F = W * (I.J(t)) - (W * I).(W*J(t)) where I and J are two images. J depends on some transformation parameters t and W is a gaussian kernel with some fixed...

Is this correct in infinite dimensional Hilbert spaces? ## (\hat{A}_1 \otimes \hat{A}_2)^{-1}=\hat{A}^{-1}_1 \otimes \hat{A}^{-1}_2 ## ## (\hat{A}_1 \otimes \hat{A}_2)^{\dagger}=\hat{A}^{\dagger}_1 \otimes \hat{A}^{\dagger}_2 ## ## (\hat{A}_1 +\hat{A}_2) \otimes \hat{A}_3=(\hat{A}_1 \otimes...

Is there a treatment of "infinitesimal operators" that is rigorous from the epsilon-delta point of view? In looking for material on the infinitesimal transformations of Lie groups, I find many things online about infinitesimal operators. Most seem to be by people who take the idea of...

I want to consider a thought experiment: Suppose, at some point in the near future, the effects of irrelevant operators in the standard model are firmly confirmed by experiment. In other words, we see some effect, perhaps the muon g-2, which simply cannot be accounted for without including...

I'm studying Shankar's book (2nd edition), and I came across his equation (15.3.11) about spherical tensor operators: [J_\pm, T_k^q]=\pm \hbar\sqrt{(k\mp q)(k\pm q+1)}T_k^{q\pm 1} I tried to derive this using his hint from Ex 15.3.2, but the result I got doesn't have the overall \pm sign on the...

I have probably a silly question about correlation functions of composite operators. Why can't you just calculate a correlator with fields at different points x1, x2, x3, ... and then set a couple of the points equal at the end of the calculation to get the result? e.g., \langle 0...

Homework Statement Hi. I'm given a 3-dimensional subspace H that is made up of the states |1,-1\rangle, |1,0\rangle and |1,1\rangle with the states defined as |l,m\rangle and l=1 as you can see. The usual operator relations for L_{z} and L^{2} applies, and also: L_{+} = L_{x}+iL_{y} L_{-} =...

Can creation operators be used to find a matrix representation in a larger dimension? Is that maybe how I could find the 3D representation for SU(2) ?

Hello everyone, I'm going through some lecture notes and there are some things I don't understand about the whole derivation of the angular momentum multiplet. It's said that the skew-symmetric 3x3 matrices J_i are the infinitesimal generators of the rotation group SO(3). Later, however...

Homework Statement How to calculate? ## \sum _{i,j} \langle 0|\prod_n \hat{C}_n \hat{C}^+_i\hat{C}_j \prod_n \hat{C}^+_n|0 \rangle ##Homework Equations ##\hat{C}^+, \hat{C}## are fermionic operators. ##\{\hat{C}_i,\hat{C}^+_j\}=\delta_{i,j}##The Attempt at a Solution I have a question. What is...

Is the following a theorem? yes or no If A and B are non-commuting Hermitian operators (or matrices), there does not exist Hermitian operators C and D such that AB-BA = CD. (Or, as special case, ...there does not exist a Hermitian operator C s.t. C= AB-BA) Thanks

Homework Statement Find the adjoints for x2d/dx and d2/dx2 Homework Equations I know that (x2)_dagger=x2 and that (d/dx)_dagger=-(d/dx). The Attempt at a Solution I solved x2d/dx by doing the following: (x2)_dagger * (d/dx)_dagger= (x2) * (-d/dx) thus the answer should be...

Hi, I'm confused by a sentence in a set of lecture notes I have on quantum mechanics. In it, it is assumed there is some representation \pi of SO(3) on a Hilbert space. This representation is assumed to be irreducible and unitary. It is then said that the operators J_i, which are said to...

Homework Statement Let ## \hat{A} = x ## and ## \hat{B} = \dfrac{\partial}{\partial x} ## be operators Let ## \hat{C} ## be defined ## \hat{C} = c ## with c some complex number. A commutator of two operators ## \hat{A} ## and ## \hat{B} ## is written ## [ \hat{A}, \hat{B} ] ## and is...

I'm, slowly, working through "Quantum Physics" by Stephen Gasiorowicz, second edition. On page 49, he gives the equation i\hbar\frac{\partial ψ(x, t)}{\partial t} = -\frac{\hbar^{2}}{2m}\frac{\partial ^{2}ψ(x, t)}{\partial x^2} He then makes the identification (h/i)(\partial/\partial x) =...

Homework Statement Calculate the expectation value for a harmonic oscillator in the ground state when operated on by the operator: $$AAAA\dagger A\dagger - AA\dagger A A\dagger + A\dagger A A A\dagger)$$ Homework Equations $$AA\dagger - A\dagger A = 1$$ I also know that an unequal number of...

A little behind in this subject, but I understand raising & lowering operators to almost be factors of the Hamiltonian operator. raising - Ahat dagger = 1/√2 (x/a - a*(δ/δx)) lowering - Ahat = 1/√2 (x/a + a*(δ/δx)) I also have the Hamiltonian as; Hhat = (Ahat dagger * Ahat + 1/2)...

Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem: \begin{bmatrix} \partial_x^2 + \mu + u(x) & u(x)^2 \\ \bar{u(x)}^2 & \partial_x^2 + \mu +...

Homework Statement Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could...

Can somebody please explain bilinear maps, tensor products and p-summing operators in an easy-to-understand way. As though explaining to an undergraduate student who just knows basic linear algebra and basic functional analysis. And please give some nice examples to make the explanations more...

Hello, we known that for each linear operator \phi:\mathbb{R}^n\rightarrow \mathbb{R}^n there exists an adjoint operator \overline{\phi} such that: <\phi(\mathbf{x}),\mathbf{y}>=<\mathbf{x},\overline{\phi}(\mathbf{y})> for all x,y in ℝn, and where <\cdot,\cdot> is the inner product. My...

So I recently learned that you can derive all four of the propositional logic operators (~, V, &, →) from Nand alone. As I have understood it, so long as you have negation, and one of the other operators, you can derive the rest. Like P → Q can be defined as ~P V Q. However, I learned that...

I'm stuck on a question in atkins molecular quantum mechanics 4e (self test 1.9). If (Af)* = -Af, show that <A> = 0 for any real function f. I think you are expected to use the completeness relation sum,s { |s><s| = 1. I'm sure the answer is simple but I'm stumped.

Hi, I have an algorithm that I have to test, and it gives me certain variables at different stages of time. I also have a "result" (I guess you can call it that), that these variables are supposed to amount to, in some mathematical fashion, at those equal points in time. This gives me a...

My understanding is that in general, operators -- corresponding to observables -- act on a state (itself a member of an infinite-dimensional Hilbert space), and the eigenvalue is the value in that state (at least, if it's a pure state). To get an expectation value, you take the dot product of...