Operators Definition and 1000 Threads

Let H be a Hilbert space and let S be the set of linear operators on H. Is there a proper subset of S that is dense in S?

Studying old exam papers from my college I came across the following: Given linear operators A,\,B: V\rightarrow V, show that: \textrm{rk}AB\le \textrm{rk}A My solution: Since all v \in \textrm{Ker}B are also in \textrm{Ker}AB (viz ABv=A(Bv)=A(0)=0) and potentially there are w \in...

Homework Statement in quantum mechanics, if we have a wave function, and an operator, we can know the eigenvalue from the eigen equation:\hat{F}\phi=f\phi. but how we obtain the mathematical form of operator \hat{F}? Homework Equations \hat{x} \rightarrow x ? \hat{p} \rightarrow -ih...

Homework Statement Homework Equations Stated in the question. The Attempt at a Solution It is a eigenfunction of L_z as it has no dependence on Z? Not sure if I can just state this, I do need to actually prove it but I can't get the calculations to work. I managed a similar...

Homework Statement \phi_1 and \phi_2 are normalized eigenfunctions of observable A which are degenerate, and hence not necessarily orthogonal, if <\phi_1 | \phi_2> = c and c is real, find linear combos of \phi_1 and \phi_2 which are normalized and orthogonal to: a) \phi_1; b) \phi_1+\phi_2...

I'm looking for a good website for understanding Quantum Mechanics (i.e. Time Independent Schrodinger Eq'n, Harmonic Oscillators, Rigid Rotors, etc) The operator is linear if the following is satisfied: A[c*f(x)+d*g(x)]=c*A[f(x)]+d*A[fg(x)], where A = an operator of any kind I'm having...

hi. i'm reading "quantum mechanics in hilbert space" and a don't get a basic point for bounded operators. def. 1 a set S in a normed space N is bounded if there is a constant C such that \left\| f \right\| \leq C ~~~~~ \forall f \in S def. 2 a transformation is called bounded if it maps...

Hermitian operator--prove product of operators is Hermitian if they commute Homework Statement If A and B are Hermitian operators, prove that their product AB is Hermitian if and only if A and B commute. Homework Equations 1. A is Hermitian if, for any well-behaved functions f and g...

Hello! I have a task to do where I do not know where to start or where to find more information. At first, this is just the problem statement: Velocity operator \mathbf{\hat{v}} is defined by the following equations: \frac{d}{dt} \mathbf{\bar{r}} = \left< \psi | \mathbf{\hat{v}} |...

I am having a problem with a couple of problems involving commutating operators. Homework Statement 1. How do i find the commutation operators of x and ∂/∂x 2. If the angular momenta about 3 rotational axes in a central potential commute then how many quantum numbers we would get? And why...

Hello! I found on this webpage: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/costate.pdf page 1, on the bottom that e^{\phi^* a } f(a^{\dagger} , a ) = f(a^{\dagger} + \phi^*, a) e^{\phi^* a } I have tried to prove this, writing both as taylor series, but the problem is to...

Hello. I am having trouble realizing the following relation holds in Lagrangian Mechanics. It is used frequently in the derivation of the Euler-Lagrange equation but it is never elaborated on fully. I have looked at Goldstein, Hand and Finch, Landau, and Wikipedia and I still can't reason...

Homework Statement Find the commutator \left[\hat{p_{x}},\hat{p_{y}}\right] Homework Equations \hat{p_{x}}=\frac{\hbar}{i}\frac{\partial}{\partial x} \hat{p_{y}}=\frac{\hbar}{i}\frac{\partial}{\partial y} The Attempt at a Solution [\hat{p}_{x}...

I'm confused about these two forms of the raising/lowering operators for the harmonic oscillator. When each one is used? a_+\psi_n=i\sqrt{(n+1)\hbar\omega} \psi_{n+1} a_-\psi_n=-i\sqrt{n\hbar\omega} \psi_{n-1} a_+|\psi_n\rangle=\sqrt{n+1} |\psi_{n+1}\rangle...

Hi, I came across a line (http://www.springerlink.com/content/t523l30514754578/) about how the trace of a linear operator is not, in general, independent of the choice of orthonormal basis. The link states that such an operator may have a trace that converges for one basis but not another...

Hi all, I have quite basic questions about the general properties of operators in quantum field theory. When quantizing the free scalar field, for instance, you promote the classical fields to operators and impose suitable commutation relations (canonical quantization). In momentum space the...

Homework Statement If A has eigenvalues 0 and 1, corresponding to the eigenvectors (1,2) and (2, -1), how can one tell in advance that A is self-adjoint and real. Homework Equations e=m^2 The Attempt at a Solution I can show that A is real: it has real orthogonal eigenvectors and...

I can't seem to find information regarding this anywhere. I understand why when the ladder operators act upon an energy eigenstate of energy E it produces another eigenstate of energy E \mp\hbar \omega. What I don't understand is why the following is true: \ a \left| \psi _n \right\rangle...

Two quantum mechanics operators are infinitesimal translation and time evulotion operators.Is there an infinitesimal translation time evolution operator similar to relativistic mechanics?

Homework Statement The title presents my problem. I know in principle how to find eigenvalues and eigenfunctions of the Hamiltonian if it depends only on orbital operators or in spin operators. On the other hand I have no clue how to solve it if there are both types of operators. The...

Homework Statement The title presents my problem. I know in principle how to find eigenvalues and eigenfunctions of the Hamiltonian if it depends only on orbital operators or in spin operators. On the other hand I have no clue how to solve it if there are both types of operators. The...

I'm working through a proof that every linear operator, A, can be represented by a matrix, A_{ij}. So far I've got which is fine. Then it says that A(\textbf{e}_{i}) is a vector, given by: A(e_{i}) = \sum_{j}A_{j}(p_{i})e_{j} = \sum_{j}A_{ji}e_{j}. The fact that its a vector is fine...

Why can we say that: <x'|e^{i\hat{x}}|x>=e^{ix'}\delta(x'-x) where where \hat{x} is an operator? I mean if \hat{x}|x>=x|x> we may write <x'|\hat{x}|x>=x<x'|x>=x\delta(x'-x) but in the expression at the top, we have an exponential operator (something I've never come across...

Homework Statement This has been driving me CRAZY: Show that \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle and \langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle Homework Equations Raising/lowering eigenvalue equations: a |n...

Right so I've had an argument with a lecturer regarding the following: Suppose you consider P_4 (polynomials of degree at most 4): A(t)=a_0+a_1t+a_2t^2+a_3t^3+a_4t^4 Now if we consider the subspace of these polynomials such that a_0=0,\ a_1=0,\ a_2=0}, I propose that the dimension of of this...

Hi all, I've been looking over some results from functional analysis, and have a question. It seems that often times in functional analysis, when we want to show something is true, it often suffices to show it holds for the unit ball. That is, if X is a Banach space, then define b_1(X) =...

I am about to start a graduate program in signal processing. A lot of the literature that I've been recently browsing, uses the concept of operators on functions - such as a differential, or Fourier transform operator. I really like this "framework" (for lack of understanding) but have never...

Homework Statement I'm just not sure how to change the operators in summation, can anyone help? Let s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k what is s_{2n}?Homework Equations s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k The Attempt at a Solution s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k or...

Hi, Will give a 30 minute talk in class about (group) and Unitary Operators. Could anybody suggest a suitable soursde suitable for a presentation (keeping the class interested) in first year Grad. level . Thank you

My lecturer has written A | \alpha_n> = a_n |\alpha_n> => A = \sum_n a_n | \alpha_n>< \alpha_n | and B | \alpha_k> = b_k |\alpha_k> => B = \sum_k b_k | \alpha_k>< \alpha_k | Where A is a hermitian operator. I understand he's used the properties of the unitary projector operator here, but is...

Forgive me if I am putting this in the wrong place, but this is my first post here. The question that I have is directed to the more experienced researchers than I am, I guess. In the Hamiltonian formulation of QFTs we write everything in terms of the ladder operators, right? So in practice...

im the very beginner of learning c programe so i would like to ask the follow questions and hope someone can give me some idea:: i know: 3%2 = 1 8%3 = 2 12%4 = 0 but I am not sure about the followings (or my concept is wrong or right): 12%100 = 12 1%2 = 1 can i say when the...

I am confused about operators and eigenstates. What does "an operator has two normalized eigenstates" mean ? Is there a way I can make a physical interpretation ? How are measurements made with these ?

Homework Statement The problem is to show that, \hat{a_{+}}|\alpha>=A_{\alpha}|\alpha+1> using \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha>It's not hard to manipulate \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha> into the form...

If A is an operator, is it correct/allowed to say: Ae^{iA} = e^{iA}A Thanks

im working through a proof and am stuck on the last line. i can't understand why \nabla_a \nabla_b \omega_c - \nabla_b \nabla_a \omega_c + \nabla_b \nabla_c \omega_a - \nabla_c \nabla_b \omega_a + \nabla_c \nabla_a \omega_b - \nabla_a \nabla_c \omega_b=0? any advice?

Can somebody please explain the following? Given the measurements of 2 different physical properties are represented by two different operators, why is it possible to know exactly and simultaneously the values for both of the measured quantities only if the operators commute?

I forgot the exact terminology for these types of operators but here goes. take for example the operators x, and p. the commuter equals i(h bar), and the eigenvectors are Fourier transforms of each other. my question is, how do you go about proving at least one of the properties listed...

Homework Statement Consider the vector space of square-integrable functions \psi(x,y,z) of (real space) position {x,y,z} where \psi vanishes at infinity in all directions. Define the inner product for this space to be <\phi|\psi> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}...

Prove that a normal operator with real eigenvalues is self-adjoint Seems like a simple proof, but I can't seem to get it. My attempt: We know that a normal operator can be diagonalized, and has a complete orthonormal set of eigenvectors. Let A be normal. Then A= UDU* for some...

Hey, My brain seems to have shut down. Let's say I'm working in the space H_a \otimes H_b and I have an operator A and B on each of these spaces respectively. Furthermore, consider a general state in the tensor space |\psi \rangle = | x_1 \rangle |y_1\rangle + |x_2 \rangle |y_2 \rangle...

I am working on a homework problem from quantum mechanics. In order to solve the problem I need to derive the raising and lowering operators. In order to to this I did the following: S+operator = <1,i | S+operator | 1,j > where i = 1, 0, -1 with i = 1 corresponding to row one etc...

Homework Statement I need to find the momentum space function for the ground state of hydrogen (l=m=0, Z=n=1) Homework Equations \phi(\vec{p}) = \frac{1}{(2\pi\hbar)^{3/2}}\int e^{-i(\vec{p}\cdot\vec{r})/\hbar}\psi(\vec{r})d^3\vec{r}...

as an example, say I am operating on r^2 with 2/r*d/dr : do I differentiate r^2 first, then times by 2/r or times by 2/r and then differentiate. Confused as they give different answers

If A and B are hermitian operators, then AB is hermitian only if the commutator=0. basically i need to prove that, but i don't really know where to start ofther than the general <f|AB|g> = <g|AB|f>* obv physics math is not my strong point. thanks :)

I have the following two equations #1 d(A(t))/dt=A(t)B where A is some matrix that depends on parameter t, and B is another matrix, d is the differential this can be simplified to by multiplying both sides by the left inverse of A(t), A^-1(dA(t))=B*t which allows me to solve...

Homework Statement (from "Advanced Quantum Mechanics", by Franz Schwabl) Show, by verifying the relation \[n(\bold{x})|\phi\rangle = \delta(\bold{x}-\bold{x'})|\phi\rangle\], that the state \[|\phi\rangle = \psi^\dagger(\bold{x'})|0\rangle\] (\[|0\rangle =\]vacuum state) describes a...

Partial differential equations represented as "operators" Homework Statement Partial differential equations (PDEs) can be represented in the form Lu=f(x,y) where L is an operator. Example: Input: u(x,y) Operator: L=∂xy + cos(x) + (∂y)2 => Output: Lu = uxy+cos(x) u + (uy)2 Homework...

If the Hamiltonian is given by H(x,p)=p^2+p then is it Hermitian? I'm guessing it's not, because quantum-mechanically this leads to: H=-h^2 \frac{d^2}{dx^2}-ih\frac{d}{dx} and this operator is not Hermitian (indeed, for the Sturm-Liouville operator O=p(x)\frac{d^2}{dx^2}+k(x)\frac{d}{dx}+q(x)...

Seen a couple of pieces on gauge theories of finance equating arbitrage opportunities to curvature, for this to work the curvature must therefore not be constant, instead it would be some sort of a function of capital flows