Operators Definition and 1000 Threads

The fraction of members of a set of numbers \{k_i\}_{i=1}^N that are equal to a specific number k can be written as \frac 1 N\sum_{i=1}^N\delta_{kk_i} Now consider an ensemble of N identical systems. Because of the above, the operator f_k^{(N)} defined by...

Let V be the vector space of all real integrable functions on [0,1] with inner product <f,g>=\int_0^1 f(t)g(t)dt Three linear operators defined on this space are A=d/dt and B=t and C=1 so that Af=df/dt and Bf=tf and Cf=f I need to find the eigenvalues of these operators: For A...

Homework Statement None of the operators Lz, Ly and Lx commute. Show that [Lx,Ly] = i(h-cross)Lz Homework Equations Where Lx is defined as Lx=-ih ( y delta/delta x - z delta/delta y) Where Ly is defined as Ly=-ih ( z delta/delta x - x delta/delta z) Where Lz is defined as Lz=-ih (...

I'm not exactly sure if this belongs in introductory or advanced physics help. Homework Statement In my book, the author was explaining the proof of the Uncertainty relation between po position and momentum. It simply stated that [x,p]= ih(h is reduced) But when I tried to verify it I got...

I'm reading a paper on NMR, and the authors keep referring to the operators I_x, I_y, I_z . What are these operators? I keep finding them mentioned in other papers, but no description of what they are.

I may have heard this or understood this incorrectly, so if i am asking the wrong question, feel free to correct me. As I understand it, if you have a degenerate set of simultaneous eigenvectors, you haven't specified a complete set of operators. For example, the hydrogen atom. You typically...

What exactly are the axioms of non-relativistic QM of one spin-0 particle? The mathematical model we're working with is the Hilbert space L^2(\mathbb R^3) (at least in one formulation of the theory). But then what? Do we postulate that observables are represented by self-adjoint operators? Do we...

Homework Statement Homework Equations The Attempt at a Solution I've gone round in circles doing this! I started of by writing it as an integral of (psi* x A_hat2 x psi) w.r.t dx, then using the equation above but I keep coming back at my original equation after flipping it...

Homework Statement This is problem 2.11 from Griffith's QM textbook under the harmonic oscillator section. Show that the lowering operator cannot generate a state of infinite norm, ie, \int | a_{-} \psi |^2 < \infty Homework Equations This isn't so hard, except that I consistently get the...

Homework Statement I'm trying to convert a context free grammar into Chomsky's normal form. These are the productions of my grammar: S -> 0|1|a|b|S+S|S.S|S*|(S) Where 0, 1, a, b are terminals, +, . are binary operators and *,() are unary operators. I know that for a grammar to be in CNF...

How does one obtain the formulae for the angular momentum operators in spherical polar coordinates i.e. \hat{L_x}=i \hbar (\sin{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \cos{\phi} \frac{\partial}{\partial{\phi}} \hat{L_y}=i \hbar (-\cos{\phi}{\phi}...

Homework Statement Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator T whose kernel is K is compact and symmetric. Let \varphi_k(x) be the eigenvectors (with eigenvalues \lambda_k) that diagonalize T . Then: a. K(x,y) \sim \sum_k \lambda_k...

Homework Statement Using the definitions of Lz and L^2, show that these two operators commute. Homework Equations Lz = -ih_bar * d/d(phi) L^2 = -(h_bar)^2 {1/sin(theta) * d/d(theta) * [sin(theta) * d/d(theta)] + 1/sin^2(theta) d^2/d(phi)^2} The Attempt at a Solution I'm actually...

Homework Statement OK, so assuming we have a physical observable with three values, a(1),a(2) and a(3), and there are given matrices for the measurement operators M(a(1))...M(a(3)). How does one actually go about finding a(1),a(2) and a(3) given the matrices?The Attempt at a Solution These...

I have a question about Witten's original 1998 paper on AdS/CFT http://arxiv.org/abs/hep-th/9802150 Since the AdS metric diverges at the boundary, the boundary metric is only defined up to a conformal class Eq. (2.2), ds^2 \to d\widetilde{s}^2 = f^2 ds^2 Similarly, the solution for...

I have encountered some mathematical difficulties when examining a one dimensional system defined by a Lagrange's function L(x,\dot{x}) = M|\dot{x}|^{\alpha} - V(x), where \alpha > 1 is some constant. The value \alpha=2 is the most common, but I am now interested in a more general...

Homework Statement C is an operator that changes a function to its complex conjugate a) Determine whether C is hermitian or not b) Find the eigenvalues of C c) Determine if eigenfunctions form a complete set and have orthogonality. d) Why is the expected value of a squared hermitian...

Homework Statement Hi all. When riding home today from school on my bike, I was thinking about some QM. The general solution to the Schrödinger equation for t=0 is given by: \Psi(x,0)=\sum_n c_n\psi_n(x), where \psi_n(x) are the eigenfunctions of the Hamiltonian. We know that the...

Hey everyone! I have a question regarding the matrix representation of a projection operator. Specifically, does the wavefunction have to be normalized before determining the projection operator? For example: |Ψ1> = 1/3|u1> + i/3|u2> + 1/3|u3> |Ψ2> = 1/3|u1> + i/3|u3> Ψ1 is obviously...

In a recent lecture on String Theory, we encountered the divergent sum 1+2+3+... when calculating the zero mode Virasoro operator in bosonic String Theory. This divergent sum is then set equal to a finite negative constant - the argument for doing so was a comparison with the definition of the...

Hello. I am stuck trying to find an understandable answer to this online: Carry out the following operations on the vector field A reducing the results to their simplest forms: a. (d/dx i + d/dy j + d/dz k) . (Ax i + Ay j + Ax k) b. (d/dx i + d/dy j + d/dz k) x (Ax i + Ay j + Ax k) I...

When quantizing a classical field theory, e.g. Klein-Gordon theory, the Hilbert space of one-particle states is taken to be a set of equivalence classes of (Lebesgue) measurable and square integrable solutions of the classical field equation, but how do you use the classical theory to construct...

Hi Short question: What is the generalization of the BAC-CAB rule for operators? Longer question and context: please read below I was reading Schiff's book on Quantum Mechanics (3rd Edition) and on page 236, he has defined a generalized Runge-Lunz vector for a central force as \vec{M} =...

I tried to use the eigenvalue of the operators but I couldn't get the result. Can anyone help me to understand this relationship? Thank you.

I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient. Dirac rather liberally talks about observables that have a continuous range of...

I read in my notes that S{-}|1> = sqrt(2)h(bar)|0> and similar for all six products of using the raising and lowering operators on |1>, |0>, |-1> I don't understand where the sqrt(2)h(bar) has come from? Cheers Philip

Problem Suppose that f(A) is a function of a Hermitian operator A with the property A|a'\rangle = a'|a'\rangle. Evaluate \langle b''|f(A)|b'\rangle when the transformation matrix from the a' basis to the b' basis is known.The attempt at a solution Here's what I have... I'm not sure if the last...

I'm pretty sure this is correct, but could someone verify for rigor? Problem Two observables A_1 and A_2, which do not involve time explicitly, are known not to commute, yet we also know that A_1 and A_2 both commute with the Hamiltonian. Prove that the energy eigenstates are, in general...

I have two questions: 1. Let V be a finite dimensional inner product space. Show that if U: V--->V is self-adjoint and T: V---->V is positive definite, then UT and TU are diagonalizable operators with only real eigenvalues. 2. Suppose T and U are normal operators on a finite dimensional...

Hi there! :) I'm trying to understand a theorem, but it's full with analysis (or something) terms unfamiliar to me. Is there an intuitive interpretation for the sentence: 'An operator being limited is equivalent to continuity in the topolgy of the norm'? Also, how can I partially...

How do you use the S(+) and S(-) operators on integer kets, |1>, |-1>, |0>? I'm told the outcome of the ones which aren't zero will be something like h(bar)/sqrt(2) * |ket> Confused!? I thought operators are 2 x 2 matrices... Any help much appreciated, Philip

Homework Statement http://img252.imageshack.us/img252/4844/56494936eo0.png 2. relevant equations BL = bounded linear space (or all operators which are bounded). The Attempt at a Solution I got for the first part: ||A||_{BL} =||tf(t)||_{\infty} \leq ||f||_{\infty} so ||A||_{BL} \leq 1...

Hi all, I found a commutation relation of lowering operator(J-) and spherical operator in Shankar's QM (2ed, page 418, Eq 15.3.11): [J_-,T_k^q] = - \hbar \sqrt{(k+q)(k-q+1)} T_k^{q-1} I wonder how the minus sign in the beginning of the right hand side come out? I have googled some...

How do you work out the commutator of two operators, A and B, which have been written in bra - ket notation? alpha = a beta = b A = 2|a><a| + |a><b| + 3|b><a| B = |a><a| + 3|a><b| + 5|b><a| - 2|b><b| The answer is a 4x4 matrix according to my lecturer... Any help much appreciated...

Hi there! Doesn't seem like a hard problem.. Homework Statement Show that the 3 isospin operators, defined by T_{+}\left\vert p\right\rangle =0, T_{-}\left\vert n\right\rangle =0, T_{+}\left\vert n\right\rangle =\left\vert p\right\rangle, T_{-}\left\vert p\right\rangle =\left\vert...

Someone told me that any operator can be decomposed in a Hermitian and Antihermitian part. Is this true? How? By addition?

Homework Statement Homework Equations The Attempt at a Solution As a group we're stuck on this as a result of the lecturer saying that he wouldn't help us because we should work as a group and find other ways other than asking him about it. Which is fair enough - but none of us...

Homework Statement If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well? Homework Equations The Attempt at a Solution

Homework Statement I have three questions concerning the electric field: 1- When calculating an electric flux for a spherical charge distribution my proffessor always writes "4 pi r2 E(r) = flux", where E(r) is the electric field. I don't understand this. I've tried to calculate the...

Homework Statement Hi all. The title says it all: Say we have an operator - e.g. the Hamiltonian. Does this have units? I.e. does the Hamiltonian have units of Joules or nothing? Personally I think it is nothing, since it is an operator, but I need confirmation.

operators are those functions which are having domain any set or any function the range is also a function. In simple words operators is a machine which is having domain and range as a set of functions. random operators are those spectiol type of fuctions which are define on a measured space.

Hi all, I'm trying to extract a complete set of states, by applying the spectral theorem to the following differential operator: L = -\frac{d^2}{dx^2} + \mathrm{rect}(x) where rect(x) is the (discontinuous) rectangular function: http://en.wikipedia.org/wiki/Rectangular_function I...

The doubt: It's not a problem, but a doubt. We know that in general quantum physics at undergraduate level, we write pΨ = (ħ/i) dΨ/dx. My doubt is that if we derived this equation from Schrodinger's equation only, so we must operate p on a wave-function only, which satisfies Schrodinger's...

Hello, I have the missfortune of having to calculate a commutator with some powers of the creation and the annihilation operators, something like: \left[ a^m , (a^{\dagger})^n \right] I have managed to derive \left[ a^m , (a^{\dagger})^n \right]= m a^{m-1} \left[ a , a^{\dagger}...

Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V.Homework Equations U is invariant under a linear operator T if u in U implies T(u) is in U.The Attempt at a Solution Assume {0} does not equal U does not...

Hello, what's the difference between Hermitian and self-adjoint operators? Our professor in Group Theory made a comment once that the two are very similar, but with a subtle distinction (which, of course, he failed to mention :smile: ) Thanks!

So...I've got an operator. Omega = (i*h-bar)/sqrt(2)[ |2><1| + |3><2| - |1><2| - |2><3| ] Part a asks if this is Hermitian, and my answer, unless I'm missing something, is no. Because the second part in square brackets is |1><2| + |2><3| - |2><1| - |2><3| which is not the same as Omega...

1. 1) Consider a spin 1/2 system... a) write expressions for the operators Sx Sy Sz in the basis composed of eigenkets of Sz b) Write eigenvalues of Sx Sy Sz c) Write eigenvectors of Sx and Sy in this basis 2) Write a matric corresponding to the operator S_ in the basis composed of the...

Could someone prove the following (if it is precisely correct): Since Fock space is the closure of the finite linear span of finite excitations of the vacuum state $\Omega$, then the operator $\hat{O}$ is densely defined if and only if $\hat{O} \Omega$ has finite norm. Or more...

Hi all! I came upon an expression like that: ' \frac{\delta f(x)}{\delta x} ' several times but can't figure out what it's used for. In Wikipedia it's posted that this derivative type is used when we consider infinitesimally small argument 'x'. So, does this mean: \frac{\delta...