Operator Definition and 1000 Threads

A basic question, not a homework problem. Say I have the expression: 5x = 10 Can I apply the derivative operator, d/dx, to both sides? d/dx(5x)=d/dx(10) would imply 5=0. I thought you can apply operators to both sides of an equation. Why can't you not do it in this case?

Consider two self-adjoint operators A and B with commutator [A,B]=C such that [A,C]=0. Now I consider an operator which is a function of A and is defined by the series ## F(A)=\sum_n a_n A^n ## and try to calculate its commutator with B: ## [F(A),B]=[\sum_n a_n A^n,B]= \\ \sum_n a_n...

Homework Statement I think there may be something wrong with a problem I'm doing for homework. The problem is: Solve the IVP with the differential operator method: [D^2 + 5D + 6D], y(0) = 2, y'(0) = \beta > 0 a) Determine the coordinates (t_m,y_m) of the maximum point of the solution as a...

I was studying quantum states in quantum field theory and I came across the formula for defining eigenstates: |n> = [(a†)n / sqrt(n!)] * |0> However, my book did not actually define ground state |0> (meaning the book did not give some function or numbers or anything like that to define what...

Homework Statement Find the eigenvector of the annhilation operator a. Homework Equations a|n\rangle = \sqrt{n}|{n-1}\rangle The Attempt at a Solution Try to show this for an arbitrary wavefunction: |V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle a|V\rangle = a\sum_{n=1}^\infty c_{n}|n\rangle...

Suppose y is a positive vector. Let p and x be two positive matrices with N rows, where ##p_j## and ##x_j## denotes the j:th row in these matrices, so that j = 1,…,N. Does the following hold: \inf_{k=1,...,N} [\sup_{l=1,...,N} [p_k(y-x_k)]] = \inf_{k=1,...,N} [p_k(y-x_k)] where ##p_k(y-x_k)##...

Let (X,d) be a metric space. An operator $T:X\to X$ is said to be quasi nonexpansive if T has at least one fixed point in X and, for each fixed point p, we have $d\left(Tx,p\right)\le d\left(x,p\right)$ (1) And also we give a...

Hi, I wish to get position operator in momentum space using Fourier transformation, if I simply start from here, $$ <x>=\int_{-\infty}^{\infty} dx \Psi^* x \Psi $$ I could do the same with the momentum operator, because I had a derivative acting on |psi there, but in this case, How may I get...

Consider a general Sturm-Liouiville problem ##[p(x) y'(x) ]' + [q(x) + \lambda \omega (x) ] y(x) = 0 ## ##\quad \Leftrightarrow \quad \hat{L} y(x) = \lambda \omega(x) y(x) \quad \text{with} \quad \hat{L} y(x) = -[p(x) y'(x)]' - q(x) y(x) ## where ##p(x), p'(x), q(x), \omega(x)## are...

Hi all, I have a severe confusion about the time-ordering operator. It is the best thing ever, I think, since it simplifies many proofs, due to the fact that operators commute (or anti-commute, but let's take bosonic operators for simplicity) under the time-ordering. However, sometimes I...

Hello, This is probably a very easy questions about the one-dimension momentum operator derivation. So you take the d<x>/dt to find the "velocity" of the expectation value. At one point in the derivation early on, you bring in the d/dt into the integral of the expectation value. The book I'm...

What is ternary operator in c++ and its use?

I've been given the question "What is ∇exp(ip⋅r/ħ) ?" I recognise that this is the del operator acting on a wave function but using the dot product of momentum and position in the wave function is new to me. The dot product is always scalar so I was wondering if it would be correct in writing...

Hi guys, I'm trying to understand why does the amplitude of the electric field operator in a cavity is fixed at \left ( \displaystyle\frac{\hbar\omega}{\epsilon_{0}V} \right )^\frac{1}{2} Every book I read says it is a normalization factor... but, normalizing an operator?, what is the meaning...

Homework Statement py and pz are components of the momentum . Do they compatible operators? Homework Equations compatible operators equation The Attempt at a Solution i think I have to use commutation

Homework Statement Find the action of the operator ## e^{\vec{a} . \vec{\nabla}} \big( f(\theta,\phi) . g(r) \big) ## where \nabla is the gradient operator given in spherical coordinates, f and g are respectively scalar functions of the angular part ## ( \theta, \phi) ## and the radial part ##...

[FONT=Times New Roman]Homework Statement From Linear Algebra with applications 7th Edition by Keith Nicholson. Chapter 9.2 Example 2. Let T: R3 → R3 be defined by T(a,b,c) = (2a-b,b+c,c-3a). If B0 denotes the standard basis of R3 and B = {(1,1,0),(1,0,1),(0,1,0)}, find an invertible matrix P...

Hi everyone, I am currently preparing myself for my Bachelor thesis in local quantum field theory. I was encouraged by my advisor to read the books of M. Reed and Simon because of my lag of functional analysis experience but I have quite often problems understand the “obvious” conclusions. For...

During a course of QFT my teacher said that in this theory is not possible to use the operator X for the position in order to construct with the momentum P and the spin S a set of irreducible operators that charachterize particles, and that we need a different point of wiev: the irreducible...

As fields ##\phi ## are ill defined at precise time and position i read that fields have to be smeared. So we have test functions f in bounded regions in space time. We have a Hilbert space and ##\phi (f) ## is an operator which acts on H. Maybe we can retrieved the usual wave function when it...

Hamiltonian operator commutes with the linear momentum operator and for a particle in the box its wavefunction is Nsin(nπx/a) , N is the normalization constant But I found this wavefuntion is not a eigenfuntion for the momentum operator, why? Isn't the two operators commut with each other?

Homework Statement Let T:V→V be a linear operator on a vector space V over C: (a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection (b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...

Sorry if the answer is very simple, but I just had a question regarding error propagation when using a modulo operator in intermediate steps. For example, I have ## \theta = arctan(\frac {A}{B}) ## and then I do ## \theta ## % ##2\pi## (modulo ##2\pi##). This gives me an answer between ##...

Hey JO. The Hamiltonian is: H= \frac{p_{x}^{2}+p_{y}^{2}}{2m} In quantum Mechanics: \hat{H}=-\frac{\hbar^{2}}{2m}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial x^{2}}) In polar coordinates: \hat{H}=-\frac{\hbar^{2}}{2m}( \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}...

Hi all, Quick quantum question. I understand the total angular momentum operation \hat{L}^2 \psi _{nlm} = \hbar\ell(\ell + 1) \psi _{nlm} which means the total angular momentum is L = \sqrt{\hbar\ell(\ell + 1)} But how about applying this to an arbitrary superposition of eigenstates such as...

The experiment is described at p26 of http://arxiv.org/abs/quant-ph/0402001 In this experiment we see that we sum measurement results and not measure the sum. Is then the quantum measurement operator not : $$S=A\otimes B\otimes\mathbb{1_{64}}-\mathbb{1_{4}}\otimes A\otimes...

For particle in the box wave function, it is the eigenfunction of kinetic energy operator but not the eigenfunction of momentum operator. So, do these two operators commute? (or it has nothing to do with commutator stuff?) How about for free particle? For free particle, the wave function is...

For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?

In a multi-particle system, the total state is defined by the tensor product of the individual states. Why is it the case that operators, say position of 2 particles, is of the form X⊗I + I⊗Y and not X⊗Y where I are the identities for the respective spaces and X and Y are the position operators...

Hello, I'm sorry if this question sounds silly, but in QM the Momentum Operator is ##{\hat{p}}=-i{\hbar}{\nabla}## . In Relativistic QM in Flat Space, this operator can be written ##{\hat{P}_{\mu}}=-i{\hbar}{\partial}_{\mu}## . Would it be correct, then, to say that in curved spacetime the...

Hello. I need help with the following: Suppose a basis set of states $\varphi_i$. Calculate the effect of the operator $\widehat{R} \equiv \Pi_i \left( \widehat{Q} -q_i \right)$ on an arbitrary state $\Psi$, assuming that the equation $\widehat{Q} \varphi_i = q_i \varphi_i$ is satisfied. I'm...

Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu

How to prove that eigenvalue of lowering operator is zero?

Homework Statement Using J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle Derive that : \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)] Homework Equations J_- = J_x - iJ_y J_+ = J_x + iJ_y The...

Homework Statement let A be a lowering operator. Homework Equations Show that A is a derivative respects to raising operator, A†, A=d/dA† The Attempt at a Solution I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can...

Hi . I've just encountered something called the displacement operator which is the exponential of a parameter multiplied by a vector but I thought the argument of an exponential had to be a scalar. Is this not true ?

I have a linear integral operator (related to integral equations) $(Ky)(x)=\int_{a}^{b} \,k(x,s) y(s) ds$ and another one $(Ly)(x)=\int_{a}^{b} \,l(x,s) y(s) ds$ both are continuous Before I proceed can I write: $Ky=\int_{a}^{b} \,k(.,s) y(s) ds$ ? (I saw...

Homework Statement Suppose A^ and B^ are linear quantum operators representing two observables A and B of a physical system. What must be true of the commutator [A^,B^] so that the system can have definite values of A and B simultaneously? Homework Equations I will use the bra-ket notation for...

Hello, could you please give me an insight on how to get through this proof involving operators? Proof: Given an eigenvalue-eigenvector equation, suppose that the vectorstate depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t...

a In the formula above, on the left hand side, ρ(0) is a system's density operator in its initial state. a is the annihilator operator of the system, and a+ is the create operator of the system. ρss is the system's density operator in its steady state. But I don't understand why this formula...

A vector operator ##\mathbf{V}## is defined as one satisfying the following property: ## [V_i,J_j] = i\hbar \epsilon_{ijk}V_k## where ##\mathbf{J}## is an angular momentum operator. My question is what is the role of ##\mathbf{J}##, does it have to be the total angular momentum from all...

I have just done a question and then looked at the solution which I don't get. The question gives a wavefunction as u = x - iy. It then asks if this function is an eigenfunction of the kinetic energy operator in 3-D. Applying this operator to u gives zero. I took this to mean that u is an...

Homework Statement I am working through a time independent perturbation problem and I am calculating the first order correction to the energy, and I am stuck operating the perturbation : v = i b (Exp[i g x]-Exp[-i g x]) on the ground state |0>. Homework Equations <0| v |0> = 1st order...

Suppose I have a system of fermions in the ground state ##\Psi_0##. If I operate on this state with the number operator, I get \langle \Psi_0 | c_k^{\dagger} c_k | \Psi_0 \rangle = \frac{1}{e^{(\epsilon_k - \mu)\beta} + 1} which is, of course, the fermi distribution. What if I operate with...

Can someone please explain to me why the expression ##[-\Box + U''(\Phi(r))]## is called the fluctuation operator? I was also wondering how to derive the following for the ##l^{th}## partial wave of the above operator: ##-\frac{d^2}{dr^2}-\frac{3}{r}\frac{d}{dr} + \frac{l(l+2)}{r}+...

Hello, I am going through this and I am totally confused. Where do they come up with ##\mid 1 \hspace {0.02 in} 0 \rangle = \frac {1}{\sqrt{2}}(\uparrow \downarrow + \downarrow \uparrow)##? They just use the lowering operator, but I'm wondering if the switch in order from equation 4.177 and the...

Hi there, The question about the helicity operator ## h= S . \bf{p} ## ( where S is 2 by 2 matrix, with ##\sigma^i ## on the diagonal ), that as mentioned in a reference as [arXiv:1006.1718], it commutes with the Dirac Hamiltonian ## H = \gamma^0 ( \gamma^i p^i + m ) ## equ. (3.3), due to...

Hello, For the spin angular momentum operator, the eigenvalue problem can be formed into matrix form. I will use ##S_{z}## as my example $$S_{z} | \uparrow \rangle = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac {\hbar}{2} \begin{pmatrix} 1 \\ 0...

Homework Statement Derive the infinitesimal rotation operator around the z-axis. Homework Equations My book gives this equation (which I follow) with epsilon the infinitesimal rotation angle: $$ \hat{R}(\epsilon) \psi(r,\theta, \phi) = \psi(r,\theta, \phi - \epsilon) $$ but I just don't get...

Homework Statement I have quite a straightforward question on the taylor expansion however I will try to provide as much context to the problem as possible: ##T(a)## is unitary such that ##T(-a) = T(a)^{-1} = T(a)^{\dagger}## and operates on states in the position basis as ##T(a)|x\rangle =...