Operators Definition and 1000 Threads

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...

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...

Currently I am working through a script concerning QFT. To introduce the concept of canonical filed quantisation one starts with the (complex valued) Klein-Gordon field. I think the conept of quantising fields is clear to me but I am not sure if one can claim that the equations of motion for the...

Homework Statement I have a differentiation operator on P_3, and: S = {p \in P_3 | p(0) = 0}. I have to show that 1) D : P_3 -> P_2 is not one-to-one. 2) D: S -> P_3 is one-to-one. 3) D: S -> P_3 is not onto. The Attempt at a Solution For #1, I want to show that our...

I solved my last problem, however, I have another question with regards to logical operators in MATLAB. Suppose I have four column arrays "one," "two," "three," and "four." Each array contains 500 scalar values. How can I say: If anyone of these scalar values are equal to zero then...

Does anyone know how to differentiate an exponential, which has an operator in its power? I found it quite a trouble in Peskin's QFT (page 84, formulas (4.17), (4.18)). Here we have these two formulas of Peskin: U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }e^{-iH\left(...

Hi everyone How do I show that the expression \sum_{b'}|\langle c'|b'\rangle|^{2}|\langle b'|a'\rangle|^{2} = \sum_{b'}\langle c'|b'\rangle\langle b'|a'\rangle \langle a'|b'\rangle \langle b'|c'\rangle equals the expression |\langle c'|a'\rangle|^{2} = |\sum_{b'}\langle c'|b'\rangle...

It seems to me that in the quantization of a classical field, one first takes the Fourier transform of the field to put it in frequency space: F \left(X, \omega \right) = \int f(X,t)e^\left(-i \omega t\right) then multiply by the annihilation and creation operators for a given wavelength: F...

Homework Statement Let A and B be nxn positive self-adjoint matrices such that for all x \in Cn, x*Ax = x*Bx. Prove that A = B. Equivalently, prove that if A, B are positive operators on H such that <Ax,x> = <Bx,x> \forall x \in H, then A = B. Hint: See Lemma 2.12. Homework Equations...

Homework Statement folks, I have a small problem understanding a function as to what its doing: I have run this program in C++. I will comment the lines of code as per my understanding. Your insight would be useful unsigned int myfunc(unsigned int n) { // for n here I took 1200...

surfing the web and arxiv i found the strange formula lnA= \frac{d^{n}}{ds^{n}} \frac{s^{n-1}}{n! A^{s}} my question is .. where does this formula come from ?? here 'n' is supposed to be a finite parameter we must define to avoid the divergences, is it valid for non-renormalizable or...

In the Anderson model, it cost an energy Un_{\Uparrow}n_{\Downarrow} for a quantum dot level to be occupied by two electrons. Here n_{\Uparrow} is the second quantized number operator, counting the number of particles with spin \Uparrow. I need the term Un_{\Uparrow}n_{\Downarrow} in first...

Homework Statement In R^{3} ||x||= a_{1}*|x_{1}|+ a_{2}*|x_{2}|+ a_{3}*|x_{3}|. where a_{i}>0 What is ||A||(indused norm = sup||Ax|| as ||x||=1). (Suppose we know the coeffisients of the matrix/operator A)?? Homework Equations The Attempt at a Solution

Homework Statement Hi guys! Many time reader, first time poster... I've struggled big time with the following. Any advice at all would be great. I'm so muddled, it's just not funny any more... (plus I'm not really familiar with who to write the mathematic script so please be patient) I...

[SOLVED] raising and lowering operators Homework Statement http://img125.imageshack.us/img125/2923/85098487ch9.jpg The Attempt at a Solution I expand a+ and a-, introduce the wavefunction and then substitute the values given at the very end to give...

i'm just not sure on this little detail, and its keeping me from finishing this problem. take the arbitrary operator \tilde{p}^{n}\tilde{y}^{m} where p is the momentum operator , and x is the x position operator the expectation value is then <\tilde{p}^{n}\tilde{y}^{m} > is this the same...

Hi there, Can anyone give me an hint/idea of how to prove Hilbert-Schmidt operators are compact? More specifically, if X is a seperable Hilbert space and T:X->X is a linear operator such that there exists an orthonormal basis (e_{n}) such that \sum_{n} ||T(e_{n})||^{2}<\infty then show that T...

Homework Statement T a linear operator on inner product space V and W a T-invariant subspace of V. Then if T is self-adjoint then Tw is self-adjoint. Homework Equations Thm: T is self-adjoint iff \exists an orthonormal basis for V consisting of e-vectors of T. The Attempt at a...

Homework Statement Prove that if T in L(V) is normal, then Ker(Tk) = Ker(T) and Im(Tk) = Im(T) for every positive integer k. Homework Equations The Attempt at a Solution Since T is normal, I know that TT* = T*T, and also that ||Tv|| = ||T*v|| and <Tv, Tv> = <T*v, T*v>. Ker(T) is the...

Homework Statement we shall describe a simple model for a linear molecule, say, CO2. the states |L>, |C>,|R> are the eigenstates of D operator (corresponds to dipole moment) D|L>=-d|L> , D|C>=0 , D|R>= +d|R>. When the electron is localized exactly on the carbon atom, its energy is E1...

Homework Statement I'm not interested in the proof of this statement, just its geometric meaning (if it has one): Suppose T \in L(V) is self-adjoint, \lambda \in F, and \epsilon > 0. If there exists v \in V such that ||v|| = 1 and || Tv - \lambda v || < \epsilon, then T has an...

Homework Statement Prove or give a counterexample: the product of any two self-adjoint operators on a finite-dimensional inner-product space is self-adjoint.Homework Equations The only two equations I've used so far are: \left\langle T v, w\right\rangle = \left\langle v, T^{*}w\right\rangle and...

Homework Statement Make P2(R) into an inner-product space by defining <p, q> = \int_0^1p(x)q(x)dx. Define T in L(P2(R)) by T(a_0 + a_1*x + a_2*x2) = a_1*x. (a) Show that T is not self-adjoint. (b) The matrix of T with respect to the basis (1, x, x2) is \left( \begin{array}{ccc} 0 & 0 & 0\\ 0...

For a system of N non-interacting bosons we start with the tensor product of single particle states \otimes_{n=1}^N | \alpha_i \rangle and then, due to the indistinguisability of the particles, symmetrize to obtain the occupation number state | n_1,n_2,\ldots,n_k\rangle = \frac{1}{\sqrt{N...

Suppose we know the matrix elements of an operator with respect a given cartesian reference frame L. If we know the sequence of rotations going from L to some other reference frame L', what is the expression for the operator in the new reference frame. Let R be the required rotation and...

[SOLVED] Quantum Field Theory: Field Operators and Lorentz invariance Hi there, I am currently working my way through a book an QFT (Aitchison/Hey) and am a bit stuck on an important step in the derivation of the Feynman Propagator. My problem is obviously that I am not a hard core expert...

Supposing we have a vector space V and a subspace V_1\subset V. Suppose further that we have two different direct sum decompositions of the total space V=V_1\oplus V_2 and V_1\oplus V_2'. Given the linear projection operators P_1, P_2, P_1', P_2' onto these decompositions, we have that...

Is there a simple expression for the ladder operators, in terms of x and -i\hbar\partial_x, for the infinite potential well? After some attempts, I couldn't figure out any nice operators that would map functions like this \sin\frac{\pi n x}{L} \mapsto \sin\frac{\pi(n\pm 1)x}{L}.

When defining a field operator, textbooks usually say that one can define an operator which destroys (or creates) a particle at position r. What does this really mean? Are they actually referring to destroying (or creating) a state who has specific quantum numbers associated with the geometry...

Now, here is the problem. (Capital letters indicate operators, lower letters are states, * indicates Hermitian conjugate) Say we know that state | p > = cos(a) |0> + sin(a) |1> (0<a<PI, a is in R) Two operators : M1= |0><0| , M2=|1><1|, apperatantly they satisfy the...

Ive tried this quantum mechanic problem but I am not getting the right anwser: a-operator = [x-operator + i (complex #)] (p-operator) / (square root of 2) and a-operator ^ t = x-operator - i (p-operator) / square root of 2 where x operator is the position operator and p operator is...

Let T be a linear operator on a finite dimensional vector space V, over the field F. Suppose TU = I, where U is another linear operator on V, and I is the Identity operator. It can ofcourse be shown that T is invertible and the invese of T is nothing but U itself. What I want to know is an...

Can anyone explain to me how to expand this expression for curl which I find in the GR book I'm reading (by Hobson, Efstathiou and Lasenby, page 71)? In a section entitled Vector Operators in Component Form they state the curl as a "rank-2 antisymmetric tensor with components": (curl)ab =...