Operators Definition and 1000 Threads

Hi, physics undergraduate here. I don't know much about differential geometry yet, but I'm curious about this idea: Say I encounter a boundary value problem, and I'm not sure what coordinate system would be 'easiest' to solve the problem in. Is there some way to put the differential...

Hi all Homework Statement Show: (a^\dagger a)^2=a^\dagger a^\dagger a a +a^\dagger a wheres: a= \lambda x +i \gamma p a^\dagger= \lambda x -i \gamma p Homework Equations - The Attempt at a Solution Well, I haven't got much. I just tried to use the stuff given, put it into my...

What is the significance of the ladder operators eigenvalues as they act on the different magnetic quantum numbers, ml and ms to raise or lower their values? How do their eigenvalues relate to the actual magnetic transitions from one state to the next?

bra - ket?? Hi, maybe a stupid question, but i would like to know if, if We have a real number, but we are i a vector space, and the operator is hermitian, is |a> is equal to < a |*? i assume this, because if a is the vector (1,0) (spin up), and only real entries. im trying to make...

Hi, Homework Statement Let A and B be hermitian operators. Show that C=i[A,B] is hermitian aswell. Homework Equations - The Attempt at a Solution Well, I tried just to use the definition but I'm not sure if that's enough (my guess would be no lol)...

Hey, I have a question regarding the invariance of a 'mixed' Casimir operator under rotation, By 'mixed' Casimir operator I refer to: \vec{J}_1\cdot \vec{J}_2 Where J1 and J2 are two independent angular momenta. I want to show that this 'mixed' Casimir operator is invariant under...

Homework Statement If <h|Qh> = <Qh|h> for all functions h, show that <f|Qg> = <Qf|g> for all f and g. f,g, and h are functions of x Q is a hermitian operator Hints: First let h=f+g, then let h=f+ig Homework Equations <Q>=<Q>* Q(f+g)= Qf+Qg The Attempt at a Solution...

Greetings, Regarding operators, my understanding until today was that given the operators \hat{x} for x and \hat{p} for p, you could construct the operator corresponding to any classical quantity Q by expressing Q in terms of x and p, and then swapping x and p for \hat{x} and \hat{p}, ie., Q...

Hi there I am working through the problems in R.I.G. Hughes book the structure and interpretation of quantum mechanics and have hit a wall in the last part of the following question: Show that Sx and Sy do not commute, and evaluate SxSy-SySx. Express this difference in terms of Sz, and...

Determining the commutation relation of operators -- Einstein summation notation Homework Statement Determine the commutator [L_i, C_j] . Homework Equations L_i = \epsilon_{ijk}r_j p_k C_i = \epsilon_{ijk}A_j B_k [L_i, A_j] = i \hbar \epsilon_{ijk} A_k [L_i, B_j] = i \hbar...

If A and B are hermitia operators , then prove (A+B)^n is also hermitian. Justw ondering if this would suffice ? ∫ ψ^*(A+B) ∅ dt= ∫((A+B) ψ)^* ∅ dt assuming (A+B) is hermitian I can do that again ∫ ψ^*(A+B) ∅ dt= ∫((A+B) ψ)^* ∅ dt multiply them together ∫((A+B) ψ)^(2*) ∅^2...

Suppose that T: V →V is a linear operator and {v1, . . . , vn} is linearly dependent. Show that {T (v1), . . . , T (vn)} is linearly dependent. I'm pretty lost as to how to even go about doing this problem, but I'll take a crack at it. I'm not sure what the operator "T:V→V" means. It...

Hi all! If you are given an operator such that A|1> = √(1/3) |1> +√(2/3) |2>, how do we interpret it? I do know that 1/3 and 2/3 are probabilities but is this operator application on state one suggesting that this state in state 1 and 2 with probabilities 1/3 and /3 respectively? Thank you!

Homework Statement unitary operators on hilbert space Homework Equations is there a unitary operator on a (finite or infinite) Hilbert space so that cU(x)=y, for some constant (real or complex), where x and y are fixed non-zero elements in H ? The Attempt at a Solution I know the...

Hi there I've recently started studying quantum field theory and I'm trying to understand the field operators. One thing that bugs me is the difference between field operators and wave mechanics operators. For instance, let's take the kinetic energy operator in wave mechanics for a single...

Homework Statement Determine the matrix representation of the rotation operator R(\phi k) using the states |+z> and |-z> as a basis. Using your matrix representation verify that R^{\dagger}R=1 The Attempt at a Solution Do I need to write R| \psi> in terms of a matrix. If I...

Homework Statement Let U and A be two linear maps related by U=e^iA. Show that U is unitary if A is self-adjoint. Give a counterexample to show that U can be unitary if A is not self-adjoint. Homework Equations Self-adjoint: A*=A The Attempt at a Solution OK, so I had no problem...

Homework Statement Consider the matrix elements of \hat{x} in momentum space. That is, evaluate \langle p | \hat{x} | \psi(t) \rangle in terms of the momentum space wave equation \langle p | \psi(t) \rangle . Homework Equations \langle x | p \rangle = \frac{1}{\sqrt{2 \pi \hbar}}...

I can't seem to reconcile a part of the vector model of spin and some of its operators. to quote wiki just above the "Bohr model" section: http://en.wikipedia.org/wiki/Vector_model_of_the_atom#Mathematical_background_of_angular_momenta "2.The magnitude of the vectors must be...

Homework Statement Show that < l,m | Lx2 - Ly2 | l,m > = 0 Homework Equations L2 = Lx2 + Ly2 + Lz2 [ Lx, Ly ] = i h Lz [ L, Lz ] = i h Lx [ Lz, Lx ] = i h Ly The Attempt at a Solution I tried substituting different commutation values in place of Lx and Ly, but I'm...

Homework Statement Consider the following operators acting in the linear space of functions Ψ(x) defined on the interval (∞,∞) (a) Shift Ta: TaΨ(x)=Ψ(x+a), a is a constant (b) Reflection (inversion) I: IΨ(x)=Ψ(x) (c) Scaling Mc: McΨ(x)= √c Ψ(cx), c is a constant (d) Complex conjugation K...

Hi. I'm reading David Tong's notes (Causality at page 36: http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf ) on QFT and I'm currently trying to understand the causality requirement that [O_1(x), O_2(y)] = 0 \ \ \forall \ \ (x-y)^2 < 0. For two operators O1 and O2. He then states that this...

Hi! I was studying Shankar's Principles of Quantum Mechanics, but I was stuck to understand a relation concerned with operators. Homework Statement I've learned that in order to get the equation of motion, I should apply the Schrodinger's equation to the given Hamiltonian operator. In...

This has already been adressed here: https://www.physicsforums.com/showthread.php?t=173896 , but I still didn't get the answer. The Harmonic Oscillator is fully described (according to my favourite QM book) by the HO Hamiltonian, and the commutation relations between the position and momentum...

I'm trying to work out equation 2.45 in these notes: http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf Anyway \phi(\vec{x}) and \pi(\vec{x}) are given in equations 2.18 and 2.19 When I multiply out \pi(\vec{x}) \vec{\div} \phi(\vec{x}) I get four terms. After normal ordering I can combine two...

Homework Statement Evaluate the commutator \left[\frac{d}{dx},x\right] Homework Equations \left[A,B\right]=AB-BA The Attempt at a Solution \left[\frac{d}{dx},x\right]=1-x\frac{d}{dx} I don't know how to figure out x\frac{d}{dx}.

Homework Statement Homework Equations The Attempt at a Solution I solved part a) correctly, I believe, giving me ψ = e^{-(√(km)/\hbar)x^{2}} and a normalization constant A = ((π\hbar)/(km))^{-1/4} I'm having difficulty with part b. I'm not exactly sure how I create a...

I'm self-studying Griffith's Intro to Quantum Mechanics, and on page 100 he makes the claim that the eigenfunctions of operators with continuous spectra are not normalizable. I can't see why this is necessarily true. Hopefully I am not missing something basic. Thanks in advance.

My QM book introduces operators like the momentum operator \hat{\mathbf{p}} which act on their eigenstates to produce new states like \hat{\mathbf{p}}|\mathbf{p}\rangle = \mathbf{p}|\mathbf{p}\rangle. But how can we interpret a state like that? How is multiplication between a Euclidean vector...

Hello! First post, over here, hopefully it'll pique some interest, and awesome answers, but any and all answers are welcome! So, point is, we were given a guest lecture over how to 'jokingly' find answers to specific problems using methods that appear to not be justified; now, after the...

significance of commutation of operators? how do u show that the position and momentum operators do not commute?

why is only one component of angular momentum is quantised, and what determines which component is quantised?

If H is a Hermitian operator, then its eigenvalues are real. Is the converse true?

In Quantum Mechanics, we have linear operators which can act on a ket to produce a new ket. However, we also allow the same operators to act on a bra vector to produce a new bra vector. That is, if \langle\phi| is a bra and A is an operator, the action of A on \langle\phi| is to produce a new...

I'm reading about selection rules, and the book is talking about how if you have a parity switching operator in between two wave vectors of opposite (definite) parity, the result is 0. For example, we have \left\langle2,0,0 \right|\hat{X}\left|2,0,0\right\rangle = 0 because...

As far as I understand, the momentum operator is: \hat{p} = -i \hbar \frac{\partial}{\partial \hat{q}} Where I'm not sure at this point if it's mathematically correct to talk about a derivative wrt. the position operator - but the point is, as far as I understand, that this equality is true...

Hello! I´m trying to read Georgi's book on Lie algebras in particle physics but am confused about the start of chapter 4. Georgi writes that "A tensor operator is a set of operators that transforms under commutation with the generators of some Lie algebra like an irreducible representation of...

Hi there, As you know, we can represent a Linear vector operator as a matrix product, i.e., if T(u) = v, there is a matrix A that u = A.v. What about a linear operator of matrices. I have a T(X) = b where X belongs to R^n_1Xn_2 and b belongs to R^p. What is a suitable representation of...

Which is the role of hermitian and unitary operators in quantum mechanics and which operator is neither hermitian nor unitary

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite. Any references, comments...

Spin 1/2--Raising and Lowering operators question Hi, Quick question regarding raising and lowering operators. Sakurai (on pg 23 of Modern QM), gives the spin 1/2 raising and lowering operators S_{+}=\hbar \left|+\right\rangle \left\langle-\right| and S_{-}=\hbar \left|-\right\rangle...

Hi, There are, for example, lists of spherical tensor operators for l=\text{integer} steps, e.g. l=0,1,2,.... T_{k}^{q}(J)\rightarrow T_{0}^{0}=1, \quad T_{1}^{\pm 1}=\mp \sqrt{\frac{1}{2}}J_{\pm},\quad T_{1}^0=J_z and this continues forever. I was wondering if there are operators...

I'm having trouble seeing how an operator can be written in matrix representation. In Sakurai, for an operator X, we have: X = \sum \sum |a''> <a''| X |a'> <a'| since of course \sum |a> <a| is equal to one. Somehow, this all gets multiplied out and you get a square matrix with the...

I'm confused about index calculation in eq. 8.25, Mandl QFT textbook. Can anyone give me a detailed explanation showing the equality below? X=\frac{1}{2}A_{\delta \alpha}^+(\bf{p'})\Gamma_{\alpha \beta}(\bf{p'})A_{\beta \gamma}^+(\bf{p})\widetilde{\Gamma}_{\gamma\delta}...

I have only heard about the use of ladder operators in connection with the harmonic oscillator and spin states. However, I would expect them to be useful in other systems as well. For example, can we find ladder operators for the discrete states of the hydrogen atom, or any other system with...

Hello. I have this problem at hand: Homework Statement A quantum mechanical system has a hamilton operator \hat{H} and another, time independent operator \hat{A}_{0}. Construct a time dependent operator \hat{A}(t) so that: <ψ(t)|\hat{A}_{0}|ψ(t)> = <ψ(0)|\hat{A}(t)|ψ(0)> for all states...

Hello, I'm trying to figure out what hypothesis I need to swap the Re{} (or Im) operator and the integral sign, but I can't find anything on the matter. I guess either it's a trivial question or a rare one. Can someone help me? Thanks in advance.

So I'm starting to learn how to program and I just learned how these operators work (difference between pre and post, etc.). However, I have a question, what real application can these operators have in actual software development? I mean, if for instance I have a variable with an X value...

I am interested in knowing that in QM what vector/tensor operators are? In fact how do they differ from scalar operators?

I have read in different places something like the following: Hermitian operators have real eigenvalues Hermitian operators/their eigenvalues are the observables in Quantum Mechanics e.g energy I am not sure what this means physically. Let us say I have a Hermitian operator operating on a...