Operator Definition and 1000 Threads

Hello I have this Hamiltonian: \mathcal{H} = \alpha S_{+} + \alpha^{*}S_{-} + \beta S_{z} with \alpha, \beta \in \mathbb{C} . The Operators S_{\pm} are ladder-operators on the spin space that has the dimension 2s+1 and S_{z} is the z-operator on spin space. Do you know how to get (if...

\hat{\rho} = \begin{bmatrix} \frac{1}{3} & 0 & 0 \\[0.3em] 0 & \frac{1}{3} & 0 \\[0.3em] 0 & 0 & \frac{1}{3} \end{bmatrix} If I have this statistical operator I get i\hbar\frac{d\hat{\rho}}{dt}=0 So this is integral of motion and...

Hi- I have a basic QM problem I am trying to solve. We are just starting on the formalities of Dirac notation and Hermitian operators and were given a proof to do over Spring Break. I am stuck on how to set up the operators and wave equation in matrix and vector form to complete the proof as...

what is it?

Homework Statement Consider an observable A associated to an operator A with eigenvalues an. Using the formula <A> = ∫ψ*Aψ compute the expectation value of A for the following wave function: \Psi=\frac{1}{\sqrt{3}}\phi_{1}+\frac{1}{\sqrt{6}}\phi_{2}+\frac{1}{\sqrt{2}}\phi_{3} where...

Homework Statement The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra. Homework Equations See above. The Attempt at a Solution I have been looking around, reading the textbook and...

Homework Statement Let V = P_n(\textbf{F}) . Prove the differential operator D is nilpotent and find a Jordan basis. Homework Equations D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} The Attempt at a Solution I already did the proof of D being nilpotent, which was easy. But we haven't covered...

Homework Statement I know that,if (operator)(function)=(value)(samefunction) that function is said to be eigenfunction of the operator. in this case i need to show this function to be eigenfunction of the Lz angular momentum:Homework Equations function: ψ=(x+iy)/r operator: Lz= (h bar)/i (x...

Is there some sort of relationship between the differential dx and the differential operator which means to take the derivative d/dx if x is a dependent variable? My prof said that dx * d/dx = 1 but that doesn't seem to work out in the case I'm looking at, so I must be missing something.

Homework Statement Use the differential operator method to find a fundamental set of solutions {y_1(x), y_2(x)} of the equation d^2 y/dx^2 - 18 dy/dx + 90y = 0. Homework Equations Differential operator method. The Attempt at a Solution I have a huge problem understanding how to...

Hey, I'm working on a proof for a research-related assignment. I posted it under homework, but it's a little abstract and I was hoping someone on this forum might have some advice: Homework Statement Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n. (a) Are...

Can someone explain to me how H(\sum_n w_n |a_n><a_n|) = \sum_n w_n(H|a_n><a_n|-|a_n><a_n|H) I've done this before and I remember being confused about it before then finding out it was something simple.. I should really start filing my notes away for such an eventuality :frown: I can't...

I don't know how to start this problem. Since it's a bi-implication, I need to show each statement implies the other. I started playing around with the definitions of inner product and norm directly, but it's not going anywhere...

Hi everyone: How is the following derived? Just for example: \Deltax\alphae\alpha=\Deltax\alpha(\delta/\deltax\alpha) does it not mean? e\alpha=\delta/\deltax\alpha But How?

Homework Statement Hello, I need some things clarified before I can do my homework. They have to do with the adjoint operator. Say I have an operator P and its adjoint is P(dagger). I noticed that the complex conjugate of the adjoint gives back the operator. Does that mean the adjoint is...

If I have a function f:R\rightarrow L where L is the space of linear operators from an hilbert space to itself, how can i define the derivative of f at a particular point of R? I mean, it is "obvious" that one should try: f'(s_0)=lim_{\Delta s\rightarrow0}\frac{|f(s_0 + \Delta s)...

The sum below has two potential answers: A: 1/-2/3 = -0.1666666667 B: 1/-2/3 = -1.5 Programming languages and most (but not all) calculators claim A is correct. However, B seems to follow operator precedence more accurately as the division is performed before the unary minus. Using...

Homework Statement Calculate the eigenvectors and eigenvalues of the two-dimensional matrix representation of the Hermitean operator \hat{O} given by |v_k'>\left(O|v_k>= {{O_11,O_12},{O_21,O_22}} where all Oij are real. What does Hermiticity imply for the o- diagonal elements O12...

Hi guys The Laplace Operator The Laplace operator is defined as the dot product (inner product) of two gradient vector operators: When applied to f(x,y), this operator produces a scalar function: My question is how a vector dot product ( del operator vector dot product...

Homework Statement http://desmond.imageshack.us/Himg810/scaled.php?server=810&filename=screenshot20120131at923.png&res=medium The Attempt at a Solution So in particular I want to look at the last part of this problem. That is, "Show that S^n = 0" I know that dim(KerS^k) = k and...

Ket Notation -- Effects of the Projection Operator Homework Statement From Sakurai's Modern Quantum Mechanics (Revised Edition), it is just deriving equation 1.3.12. Homework Equations \begin{eqnarray*}\langle \alpha |\cdot (\sum_{a'}^N |a'\rangle \langle a'|) \cdot|\alpha \rangle...

Homework Statement What physical quantity is represented by the operator i\bar{h}∂/∂t Homework Equations i\bar{h}∂/∂t The Attempt at a Solution It's a one mark question, I just have no idea what it is and I can't find it in my notes D:.

Hi, I have to show that a unitary operator U can be written as U=\frac{\mathbf{1}+iK}{\mathbf{1}-iK} where K is a Hermitian operator. Now how could you possibly have a fraction of operators if those can be represented by matrices? Not sure what to do here.

Homework Statement http://s1.ipicture.ru/uploads/20120130/fcGLnUw5.png The attempt at a solution I have been trying to understand how to obtain the R.H.S. of each property from its L.H.S. but i can't find how, although i know that it's somehow related to differentiating the L.H.S. I am...

why is complex number i involved in defining momentum operator p I mean Px=-ih... What has complex number to do with momentum. I do get however that i in other cases of quantum mechanics has to do with euler's formula that comes from harmonic nature of wave.

Homework Statement Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!...Homework Equations U=exp(iC) C=C* U*U=I U=A+iB exp(M) = sum over n...

This question concerns the outcome when operator valued functions act on an energy eigenstate. Given an eigenstate at t =0, say |Ej > , I have seen or inferred in some of the literature that the following applies : exp(-iHt/h) |Ej > = exp(- iEj t/h) |Ej > Where h = h-bar Ej is energy...

I feel kind of lame, but here's my situation: We start with the operator g_{\mu \nu} \Box - \partial_{\mu}\partial_{\nu} and convert to momentum space to get -g_{\mu \nu} k^{2} - k_{\mu}k_{\nu}. Apparently it's easy to see that this has no inverse? I'm told that if it *did* it would be...

Homework Statement I have a problem with vectors x and y, where x=[-3,0,0,2,5,8] and y=[-5,-2,0,3,4,10] The problem asks me to determine z=y<~x I've searched my book and cannot determine what this operator is telling me to do. Am I supposed to determine the values of y that are not less...

So in the attachment, in fact (6), the formula for the propagator rectangled in red... is the Hamiltonian ACTING on (t-t')? is the Hamiltonian a function of (t-t')? or should it be (this is what I think), to be more clear U(t,t') = exp((-i/h)(t-t')H), so that when acting on a state...

Is there a "magnitude of momentum" quantum operator? Homework Statement Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not? What can you say about the magnitude of the momentum? Homework Equations Ground state...

Homework Statement Find wave functions of the states of a particle in a harmonic oscillator potential that are eigenstates of Lz operator with eigenvalues -1 h , 0, 1 h and have smallest possible eigenenergies. Check whether these states are also the eigenstates of L^2 operator. Eventually...

I'm trying to understand why the del operator is working a certain way. So in my literature there is a term: \nabla \cdot \rho_a \mathbf{v} but then after saying that \rho_a=w_a\rho the term can somehow become \rho (\mathbf{v}\cdot \nabla w_a) I do not understand how nabla and the...

Greetings all, In going through an operator approach to deriving the rules of angular momentum I find myself asking a curious question. Is it possible to fully derive the Angular momentum algebra relying on only a minimal set of classical relations. That is to say, if we take the only...

Please teach me this: Why is Casimir operator T^{a}T^{a} be an invariant of the coresponding Lie algebra? I know that Casimir operator commutes with all the group generators T^{a}. Thank you very much for your kind helping.

The del operator is often informally written as (d/dx, d/dy, d/dz) or \hat{x}\frac{d}{dx}+\hat{y}\frac{d}{dy}+\hat{z}\frac{d}{dz}, a pseudo-vector consisting of differentiation operators. Could there be a pseudo-matrix operator like it? What would one be differentiating with respect to- that is...

I'm trying to show: a(p)|q1,q2,...,qN> = \sumNi=1(2pi)32Ep\delta(3)(p-qi)x|qi,...,qi-1,qi+1,...,qN> I'm pretty sure you have to turn the ket into a series of creation operators acting on the vacuum |0>, but then not sure what relations need to be invoked for it to be clear. Any help...

Consider a particle in one dimention. there is a dirac delta potential such as V=-a delat(x). The wave functions in two sides(left and right) are Aexp(kx) and Aexp(-kx) respectively. So if the momentum operator acts on the wave functions, would give two complex numbers while the momentum...

Hi all I am trying to go through the Griffiths Intro to QM book and I'm afraid I'm already stumped! He determines the momentum operator by beginning with the following equation: <x>=\int_{-\infty}^\infty {x|\psi(x)|^2} He takes the time derivative and manipulates the integral: (I'm...

Homework Statement The Laplace operator Δ is defined by: Δ= Show in polar coordinates r and Θ, that the Laplace operator takes the following form: http://upload.wikimedia.org/wikipedia/en/math/0/7/a/07a878276cffd0c680f3f827204aba24.png Homework Equations x=rcos(Θ), y=rsin(Θ), r ≥ 0, Θ ∈...

I have seen how people implement the prefix and postfix ++ overloading, which are as follow: Number& operator++ () // prefix ++ { // Do work on this. return *this; } Number operator++ (int) // postfix ++ {...

(1) Let f(x)=x^3+y^3+z^3-3xyz, Find grad(f). grad(f)=(3x^2-3yz, 3y^2-3xz, 3z^2-3xy). (2) Identify the points at which grad(f) is a) orthogonal to the z-axis b) parallel to the x-axis c) zero.I have managed to solve for (1), but don't have a clue how to solve for the second part. I have not...

Hi, Was wondering if anyone could give me a hand. I need to prove that the Cayley Transform operator given by U=(A-i)(A+i)^-1 is UNITARY, ie that UU*=U*U=I where U* is the adjoint of U (I am given also that A=A* in the set of bounded operators over a Hilbert space H). My solution so...

Hello. The method here is to add an item to a heap. As the title states, I am getting the error "The operator > is undefined for the argument type(s) E, E" in the parenthesis after the while. I assume this is not the correct way to compare E values. Does anybody know what would be the correct...

Homework Statement The operator F is defined by Fψ(x)=ψ(x+a) + ψ(x-a), where a is a nonzero constant. Determine whether or not F is a Hermitian operator. Homework Equations ∫(x+a)d/dx + (x-a)d/dxψ The Attempt at a Solution f = (1=ax) + (1-ax)ψ What are the steps I need...

Hallo, I was wondering what is the physical significance of scalar field \Phi (x) as an quantum operator. \Phi (x) have canonical commutation relation such as [ \Phi (x) , \pi (x) ] so it must be an opertor, thus what are his eigenstates? Thanks, Omri

Homework Statement The square of the angular momentum operator is (in SPC); http://img6.imageshack.us/img6/67/54712598.png Show that Y(\theta,\phi) = Csin^{2}\thetae^{2i\phi} Is an eigenfunction. C is a constant. Homework Equations The Attempt at a Solution I am not able to get it in the...

I want to understand positive operator valued measures in QM, in particular why they are considered "observables". Anyone know a good place to start reading about this? I know some functional analysis and some measure theory, but I haven't made it all the way to the spectral theorem.

the definition of a green's function is: LG(x,s)=δ(x-s) the definition of a right inverse of a function f is: h(y)=x,f(x)=y→f°h=y how does it add up?

Homework Statement Concicer the dilation operator D = \vec{r} * \vec{p} Compute [D,\vec{r} ] and [D, \vec{p}] Homework Equations p = - i * hbar The Attempt at a Solution I think the question is really if [D, \vec{r}] commutes I got this: D = \vec{r} * \vec{p}...