Operators Definition and 1000 Threads

Is it possible to write series ##\ln x=\sum_na_nx^n##. I am asking you this because this is Taylor series around zero and ##\ln 0## is not defined. And if ##A## is matrix is it possible to write ##\ln A=\sum_na_nA^n##. Thanks for the answer!

If a system is made up by two subsystems, for example, the atom and the photon. and let's assume the state of the atoms is described by |\phi\rangle, while the state of the photons can be described by |n\rangle, The Kronecker product of the |\phi\rangle and |n\rangle can be used to describe the...

Hi, I am currently going over this and got me thinking about a scanario where you have A -> BC Where A is S = 0, L=0, B is S = 1 L=0, C is S=1 L =0 (I'll use S = intrinsic spin, L = angular momentum, J = Total Angular momentum, |L-S|=< J =< L+S) Maybe such a decay doesn't exist, but I'm just...

The definition of tensor operator that I have is the following: 'A tensor operator is an operator that transforms under an irreducible representation of a group ##G##. Let ##\rho(g)## be a representation on the vector space under consideration then ##T_{m_c}^{c}## is a tensor operator in the...

Homework Statement a) The operators ##a## and ##a^{\dagger}## satisfy the commutation relation ##[a,a^{\dagger}] = 1##. Find the normalization of the state ##|\psi \rangle = C (a^{\dagger} )^2 |0\rangle##, where the vacuum state ##|0\rangle## is such that ##a|0\rangle = 0## b)A one...

Hello, I am currently studying ladder operator for a simple harmonic operator as a method for generating the energy values. This seem like a simple algebra question I am asking so I do apologize but I just can't figure it out. Here are my operator definitions...

I would appreciate if someone could set me straight here. I understand if I have an arbitrary operator, I can express it in matrix component notation as follows: Oi,j = <vi|O|vj> Is it possible to get a representation of the operator O back from this component form. I'm more interested...

Hi, So, I am working through section 5.2 of Sakurai's book which is "Time Independent Perturbation Theory: The Degenerate Case", and I see a few equations I'm having some trouble reconciling with probably because of notation. These are equations 5.2.3, 5.2.4, 5.2.5 and 5.2.7. First, we...

Homework Statement Homework Equations The Attempt at a Solution I have tried inserting the first wavefunction into Lz which gets me 0 for the eigenvalue for the first wavefunction. Is this correct? For the second wavefunction, I inserted it into Lz and this gets me -i*hbar*xAe^-r/a which...

Trying to get my head around this one. Given that you can have a proton and an electron in a hydrogen atom for example, and they can create a singlet or triplet configuration, with spin 1 and spin 0 respectively. The total spin operator can be derived as: S^2 = (Se + Sp)^2 = Se^2 + Sp^2 +...

So I understand the commutation laws etc, but one thing I can't get my head around is the fact that L^2 commutes with Lx,y,z but L does not. I mean if you found L^2 couldn't you just take the square root of it and hence know the total angular momentum. It seems completely ridiculous that you...

In this video at around 9:00 , Carl Bender demonstrates a method of solving y''+a(x)y'+b(x)y=0. He first rewrites it in terms of differential operators D2+a(x)D+b(x))y(x)=0, then factors it (D+A(x))(D+B(x))y=0 then multiplies it out to determine B(x). I thought we would get...

After read this stretch https://en.wikipedia.org/wiki/Closed_and_exact_forms#Vector_field_analogies, my doubts increased exponentially... 1. A scalar field correspond always to a 0-form? 1.1. The laplacian of 0-form is a 2-form? 1.2. But the laplacian of sclar field is another scalar field...

Hi guys, This is a problem which is bothering me right now. The angular momentum operators (Lx, Ly, Lz), when expressed in spatial rotations consists of derivatives in \theta and \phi. This would suggest that there are, at any point in space, only two linearly independent operators (since there...

If the gradient of f is equal to differential of f wrt s: \vec{\nabla}f=\frac{df}{d\vec{s}} so, what is the curl of f and the gradient of f in terms of fractional differentiation?

Homework Statement Given that \hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r}) , show that \hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) Homework Equations Above The Attempt at a Solution I tried \hat{p}\hat{p} =...

Hi everyone, I was hoping that someone might be able to tell me if what I'm doing is legit. Firstly, I start by saying that a unitary transform can be made between two sets of operators (this is defined in this specific way); b_{n} = \Sigma_{m} U_{mn}a_{m} (1) Now this is the bit I'm...

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an orthonormal sequence consisting of the eigenvectors of $|T|=(T^*T)^{0.5}$, (${\lambda}_{n}$)...

Most of us are familiar with the fact that in Loop gravity the area and volume observables have discrete spectrum. The discrete spectrum of the area operator, leading to a smallest positive measurable area, has lots of mathematical consequences that have been derived in the theory. It helps...

Hello everybody. I am new here, and also new to quantum mechanics. This is the question to which I can't answer neither in mathematical nor physical way. a,b → observables (like position and momentum) A,B → corresponding operators. "It is possible for particles to be in a state of...

I have a doubt on the second quantization formalism. Suppose that we have two spin-1/2 fermions which can have just two possible quantum number, 1 and 2. Consider the wave function: $$ \psi(r_1,r_2)=\frac{1}{\sqrt{2}}\left(\psi_1(r_1)\psi_2(r_2)-\psi_1(r_2)\psi_2(r_1)\right). $$ The second...

I can't figure this one out given that the coordinate operator is continuous, it's hard to imagine "matrix elements". But presumably since the coordinates of the system (1d free particle) are always real valued, would this make the coordinate operator Hermitian?

Most part of the fundamental quantum mechanics rely upon finding some operators \hat{X} that commutes with hamiltonian and is able to simultaneously diagonalize \hat{X} and hamiltonian. Actually what do you mean by diagonalize simultaneously?? Is there any relation with diagonalize the...

I have two possible bases (a,b) and (a',b'). If I also have the transformation matrix P, such that P(a,b)=(a',b'), am I correct in assuming that I can change an operator A, from the (a,b) basis to the (a',b') basis by applying A' = P_transposed * A * P ?

Homework Statement Hi,guys. I have a hard time understanding algebra and tricks of operators. So i have few examples: 1)[\hat{p}2x,xn] 2)[\hat{l}z,x],where \hat{l}z=x\hat{p}y-y\hat{p}x Homework Equations The Attempt at a Solution 1)[\hat{p}2x,xn]= [\hat{p}x \hat{p}x,xn]=...

Homework Statement I am trying to calculate the expectation value of an operator in the Lipkin model of nuclear physics. The background isn't important because my problem in really just a math problem. Homework Equations The anticommutation relation \begin{align*} a_{p\sigma}...

Hello all, I hope this is the write sub-forum for this question. I have been looking at the Laplacian of a 2-D vector field. It is explained nicely by this Wikipedia article here. My question is more regarding how these operators work together. So, in the case of the Laplacian, it tells me...

Hey Guys, I regard two operators \Psi , \Phi , that don't commute. Does the product-rule, looks like that? $$\nabla (\Phi \Psi) = \Psi (\nabla \Phi) +\Phi (\nabla \Psi) $$ THX

OK I'm not sure if this should go in the math or quantum forum, but as I'm learning these in introductory QM I post the questions here. Please move the thread if the section is inappropriate. Anyway, some questions: * What is an inner product space? * What is a hilbert space? * What are...

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

If we have two fermion operators with a known anti-commutator AB+BA, what do we do if we find ourselves with AB-BA in an equation? Does this automatically vanish for fermions? if not, is there anything we can say about in general?

Hello! I'm trying to understand QFT for the moment and have a question regarding the basic. So we have a vectorspace (Hilbertspace) of our states. The operator \phi(x) measures the amplitude at point x, whereas the operator \pi(x) measures the momentum density.. The ladder operator...

Suppose I have B: X\to Y and A: Y\to Z, where X,Y,Z are Banach spaces and B\in \mathcal L(X,Y) and A\in \mathcal L(Y,Z); that is, both of these operators are bounded. Does it follow that AB \in \mathcal L(X,Z) and \| AB \|_{\mathcal L(X,Z)} \leq \|A\|_{\mathcal L(Y,Z)} \|B\|_{\mathcal L(X,Y)}...

Homework Statement Using Dirac Notation prove for the Hermitian operator B acting on a state vector |ψ>, which represents a bound particle in a 1-d potential well - that the expectation value is <C^2> = <Cψ|Cψ>. Include each step in your reasoning. Finally use the result to show the...

Hi All, I just found this site and this is my first post here. I am working on getting my masters in polymer chemistry and started taking a class this semester which is pretty much all calculus and linear algebra and I just have a hard time with these subjects. I got a homework problem that I...

1. Evaluate the following (i.e. get rid of the operators)...

The attached picture shows a representation of a general operator, which I found quite weird. The matrix elements are calculated in the position basis as far as I can tell, but I am not sure how. Do they do something like? <klTlk'> = ∫ dx dx' <klx><xlTlx'><x'lk'> In that case what happens to...

For bosons we define states as eg. ln> = l1 0 1 ... > where the numbers denote how many particles belong to the j'th orbital. And similarly for fermions. We then define creation and anihillation operators which raise and lower the number of particles in the j'th orbital: c_j...

How do I compute the commutator [a,e^{-iHt}], knowing that [H,a]=-Ea? I tried by Taylor expanding the exponential, but I get -iEta to first order, which seems wrong.

Other than Bernouilli, Euler, and Lagrange, who else discovered an irrational number in which transcendental operators have been developed to simplify physics and geometry?

As we know, all operators representing observables are Hermitian. In my undersatanding, this statement means that all operators representing observables are Hermitian if the system can be described by a wavefunction or a vector in L2. For example, the momentum operator p is Herminitian...

Let T be a (possibly unbounded) self-adjoint operator on a Hilbert space \mathscr H with domain D(T), and let \lambda \in \rho(T). Then we know that (T-\lambda I)^{-1} exists as a bounded operator from \mathscr H to D(T). Question: do we also know that (T-\lambda I)^{-1} is self-adjoint? Can...

When solving for the spherical harmonic equations of the orbital angular momentum this textbook I'm reading.. Does this mean that there must be a max value of Lz which is denoted by |ll>? Normally the ket would look like |lm>, and since m is maxed at m=l then |ll> is the ket consisting of the...

Homework Statement Hi we have Lorentz operators J^{\mu\nu} = i(x^{\mu}\partial^{\nu} - x^{\nu}\partial^{\mu}) and these have [J^{\mu\nu}, J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} + \eta^{\mu\sigma}J^{\nu\rho} - \eta^{\mu\rho}J^{\nu\sigma} - \eta^{\nu\sigma}J^{\mu\rho}) Now define...

What do we mean by the two operators are commutative or non commutative? I wanted to understand the physical significance of the commutative property of the operators. We are doing the introduction to quantum mechanics and there are many things that are really confusing. Any help will be...

Are functionals a special case of operators (as written on Wiki)? Operators are mappings between two vector spaces, whilst a functional is a map from a vector space (the space of functions, say) to a field [or from a module to a ring, I guess]. Now, the field is NOT NECESSARILY a vector...

Homework Statement Assume A and B are normal linear operators [A,A^{t}]=0 (where A^t is the adjoint) show that det AB = detAdetB Homework Equations The Attempt at a Solution Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets...

Greetings, Just checking if I'm getting this ... please correct me if I'm wrong. The value of the wavefunction is 'probability amplitude' in discrete case and 'probability amplitude density' in continuous case. The former is a dimensionless complex number and the latter is the same...

Homework Statement If a particle has spin 1/2 and is in a state with orbital angular momentum L, there are two basis states with total z-component of angular momentum m*hbar l L,s,Lz,sz > which can be expressed in terms of the individual states ( l L,s,Lz,sz > = l L,Lz > l s,sz > ) as l...

The fundamental idea of these operators is that we can use them to add particles to our system to a specific eigenstate. Now my book has examples of these operators of which the harmonic oscillator ladder operators are used. But thinking about it, this example does not make sense for me. The...