Operators Definition and 1000 Threads

If we have two linear operators A, and B, where A+ is the adjoint of A, how do we prove the property, (AB)+=(B+)(A+)

OK, I stumbled upon a problem, but I feel somehow stupid about writing the exact problem down, so I'll ask a more "general" question. I have to see if three linear operators A, B and C from the vector space of all linear operators from R^2 to R^3 are linearly independent. The mappings are all...

So, say I have a composite hilbert space H = H_A \otimes H_B, can I write any operator in H as U_A \otimes U_B? Thanks

1.What does it mean for an operator to be hermitian? Note: the dagger is represented by a ' 2. How do I show that for any operator ie/ O' that O + O' , i(O-O') and OO' are hermitian? Thanks in advanced

I am taking a QM course and we are using griffiths intro to QM text, 2nd edition. I like the text but I find it lacking when it comes to explaining ladder operators. I need to see how to use them in a very detailed step-by-step problem. Does anyone know of any good textbooks or websites that...

Homework Statement Find the matrices which represent the following ladder operators a[SIZE="3"]+,a_, and a[SIZE="3"]+a- All of these operators are supposed to operate on Hilbert space, and be represented by m*n matrices. Homework Equations a[SIZE="3"]+=1/square root(2hmw)*(-ip+mwx)...

In my question I have to find what the commutation of a electrons kinetic and potentials energys are, in 3 Dimensions. I have started by finding the kinetic operator T and the potential energy from coloumbs law. I have then applied commutation brackets and I'm at the stage where I'm solving the...

Homework Statement A is a non-Hermitian operator. Show that i(A-A^t) is a Hermitian operator.Homework Equations \int \psi_1^*\L\psi_2 d\tau=\int (\L\psi_1)^*\psi_2 d\tau \int \psi_1^*A^t\psi_2 d\tau=\int (A\psi_1)^*\psi_2 d\tauThe Attempt at a Solution \int \psi_1^*i(A-A^t)\psi_2 d\tau =\int...

Homework Statement Within the framework of quantum mechanics, show that the following are Hermitian operators: a) p=-i\hbar\bigtriangledown b) L=-i\hbar r\times\bigtriangledown Hint: In Cartesian form L is a linear combination of noncommuting Hermitian operators. Homework Equations...

Homework Statement I've just initiated a self-study on quantum mechanics and am in need of a little help. The position and momentum operators do not commute. According to my book which attemps to demonstrate this property, (1) \hat{p} \hat{x} \psi = \hat{p} x \psi = -i \hbar...

Cohen-Tanoudji defines a "standard basis" of the state space as an orthonormal basis {|k,j,m>} composed of eigenvectors common to J² and J_z such that the action of J_± on the basis vectors is given by J_{\pm}|k,j,m>=\hbar\sqrt{j(j+1)-m(m\pm 1)}|k,j,m\pm 1> But isn't is automatic that such...

hello everyone, while studying QM you learn the physical meaning of commutating operators, namely they have simultaneous eigenstates. For observables it means, that they can be simultaneusly exactly mesured. What is the physical meaning of anticommuting and not anticommuting operators...

Just a quickie: If two operators commute, what can be said about their eigenfunctions? The only thing I can glem from the chapter in my textbook about this is that the eigenfunctions are equal? Is this right, or have I misread it?

I know that with an Hermitian operator the expectation value can be found by calculating the (relative) probabilities of each eigenvalue: square modulus of the projection of the state-vector along the corresponding eigenvector. The normalization of these values give the absolute probabilities...

The general uncertainty relation between two observables A and B. (\Delta A)^2(\Detla B)^2 \geq -{1\over 4}<[A, B]>^2 I have to prove the above relation using the definition of expection values etc. The reference I use (Liboff) have this relation given as an exercise. But Gasiorowicz's book...

Homework Statement Show that for the eigenstate |l,m> of L^2 and Lz, the expectation values of Lx^2 and Ly^2 are <Lx^2>=<Ly^2>=1/2*[l(l+1)-m^2]hbar^2 and for uncertainties, show that deltaLx=deltaLy={1/2*[l(l+1)-m^2]hbar^2}^(0.5) Homework Equations eigenvalues of L^2 are l(l+1)hbar^@...

Let be the Kelin-gordon equation (m=0) with a potential so: (-\frac{\partial ^{2}}{\partial t^{2}}+V(x) )\Phi=0 my question is if you consider the wave function above as an operator..is the K-G operator of the form: <0|T(\Phi(x)\Phi(x')|0> T=time ordered I think that in both...

I know this is a simple part of Quantum Mechanics, but I seem to be having trouble with it, I'm not sure if my math is just wrong or if I'm applying it wrong. The questions that I have are: Prove the following for arbitrary operators A,B and C: (hint-no tricks, just write them out in...

Having a lot of trouble with this one. I'm given that the Hamiltonian of a certain particle can be expressed by H = A(a+a) + B(aa+), where A and B are constants and a+ and a are the raising and lowering operators, respectively. I'm supposed to find the energies of the stationary states for the...

hello...this might look very stupid but I am totally confused... Let have operators A, B, C. Let [A,B]=[A,C]=0 and [B,C] not 0... When two operators commute, they have the same base in (Hilbert) space. So base in A representation is the same as in the B representation and also the basis...

Hey. I am pretty confused on how to use differentation operators (dy/dx,d/dx), what does it mean in equations and how do I know when it means i should find the derivative of something. Word problems are confusing me on how to use these and when to find derivatives. I always thought...

If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!

If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!

Let \mathcal{L} = \frac{d}{dx} p(x) \frac{d}{dx} + q(x) be a self-adjoint operator on functions f : [a,b] \rightarrow \mathbb{C}. Under what circumstances is the operator Hermitian with <u|v> = \int_a^b u^*(x) v(x) dx ? Can someone give me a hint on this one? I know that hermitian operators...

I worked these problems out: [x, H] = xH - Hx = 0 [p, H] = pH - Hp = non-zero H is the harmonic oscillator Hamilitonian, x and p are the position and momentum operators, respectively. My question is, why doesn't p commute with H, but x does?

Hi again, Question: \hat{A} is an Hermitian Operator. If \hat{A}^{2}=2, find the eigenvalues of \hat{A} So We have: \hat{A}\left|\Psi\right\rangle=a\left|\Psi\right\rangle But I actually don't know how to even begin. \hat{A} is a general Hermitian operator, and I don't know where...

I was wondering, what is the difference between an operator and a relation? For example, instead of saying 2+3 I can say Add(2,3). Or the \frac{df(x)}{dx} operator can be written as D(f(x)). I fail to see any difference between an operator and a relation. What do you guys think?

asking here because i originally asked in the wrong place :) this question is two parts, both dealing with telling if combinations of hermitian operators are hermitian. the first combination is PX + XP, where P stands for the momentum operator, (h bar /i)(d/x), and X is the "x operator"...

Can someone explain to me the concept of Casimir operators for someone who's not too familiar in abstract mathematics. E.g. What is the quadratic Casimir operator and why is it part of a maximally commuting set of operators?

I'm having the worst time with a homework problem, in which I am asked to establish the operators which correspond to the independent observables of a three level system( states |1> , |0>,|-1> ) . I know that the operators should be 3x3 matrices, so I tried to express an arbitrary 3x3 Hermitian...

Let U=e^{iH} where H is an operator. 1. If H= \left(\begin{array}{cc}a & b\\c & d\end{array}\right) in its matrix representation. Then what is U in its matrix representation. Im confused, is it U= \left(\begin{array}{cc}e^{iH(1,1)} & e^{iH(1,2)} \\e^{iH(2,1)} & e^{iH(2,2)}...

Hi, I've seen the following in quantum info textbooks and papers and I was just wondering if anyone knows if it has any phsical interpretation or significance? The space of operators that act on a HIlbert space is isomorphic to the tensor product of the original Hilbert space with its dual...

Can someone give me an example of a nonlinear operator? My textbooks always proves that some operator is a linear operator, but I don't think I really know what a nonlinear operator looks like. One of my books defines an operator like \hat{B} \psi = \psi^2. I see that this is a nonlinear...

Ask for Help: Is this operator reasonable in physics? Is this kind of the operator is reasonable in physical sense? $\sum\limits_{\bf k_x,k_y} \alpha k_x c^\dag_{k_x,k_y}c_{k_x,k_y}$ where $\alpha$ is contant, k_x , k_y is wavevector along the x and y direction.

Hi, I have done most of the problem in this word document (attached). I have some trouble though. In my QM class, we assumed that the z component of angular momentum Lz satisfies, Lz Ylm = m hbar Ylm and the ladder operator L+ and L- were defined as L+_ = Lx +- iLy. We were able to find the...

The angular momentum is the generator of spatial rotations. Are the commutation relations for angular momentum the result of the fact that rotations (all rotations, also classical) do not commute or are they the result of the quantization rules for quantum mechanical angular momentum? Are...

i need to prove the next statement: let S and T be linear operators on a vector space V, then det(SoT)=det(S)det(T). my way is this: let v belong to V, and {e_i} be a basis of V v=e1u1+...+e_nu_n then T(v)=e1T(u1)+...+enT(un)...

I'm new to quantum mechanics, i.e. the type of QM you don't learn through books by Brian Greene :biggrin: . I know there aren't any derivations for an operator associated with an observable and they are usually defined in a certain form. So why do they have those particular forms. Was it trial...

My memory is fading. Can somebody please remind me how I would go about determining in each of the following cases whether the operator A is Hermitian or not? Case 1. A\psi(x) = \psi(x+a) Case 2. A\psi(x) = \psi^*(x) where the star indicates complex conjugation.

I need someone to check some homework problems that I've done so far regarding inverse differential operators. 1) 9y"-4y=sinx yp=-1/13 sinx 2) y"-4y'-12y=x-6 yp=-1/12(x-6) 3) y'''+10y""+25y'=e^x yp=36 4) y""+8y'=4 yp=1/2 x 5) y"-9y=54 yp=-6 6) y"-y'-12y=e^(4x) yp=1/7...

Hi, I try to understand the proof for the uncertainty principle for two Hermitian operators A and B in a Hilbert space. My questions are rather general so you don't need to know the specific proof. The first thing I couldn't get into my head was the definition of uncertainty (\Delta...

Hi there, Was wondering if anyone could point me in the right direction for this one? Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal? Thanks

Hi, I'm looking for a general power series for a function of F of n operators. As normal, the operators do not necessarily commute. My first guess was: F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i p^j + b_{ij}p^i x^j However, I don't think this is correct as it is possible to...

purpose of each of the "operators", divergence, gradient and curl? Hi. Can anybody give me a reasonably simple explanation of what the purpose of each of the "operators", divergence, gradient and curl? (I've been looking but I never found something simple to understand) I know how to evaluate...

hi, In a discussion of the historical motivations for a move from calculus to operators, my QM book says... "Many mathematicians were uncomfortable with the 'metaphysical implications' of a mathematics formulated in terms of infinitesimal quantities (like dx). This disquiet was the stimulus...

Let \mathcal{H} be a Hilbert space over \mathbb{C} and let T \in \mathcal{B(H)}. I want to prove that \|Tx\| = \|x\| \, \Leftrightarrow \, T^{\ast}T = I for all x \in \mathbb{H} and where I is the identity operator in the Hilbert space. Since this is an if and only if statement I began...

Okay, I've read and re-read the section on tensor operators and the Wigner-Eckart theorem in Sakurais book, but I'm still confused. Could anyone explain to me how to think about vector and tensor operators and the significance of the Wigner-Eckart theorem? :confused: Thanks.

We're doing differential operators in my Differential Equations class right now, and our professor assigned the following problem to us: (D-x)(D+x) Which inevitably gives us the following terms as part of the final answer: Dx-xD The answer in the book tells me that Dx-xD = 1, and some...

I'm not sure where to start with these proofs. Any suggestions getting started would be appreciated. 1. Show that is A,B are linear operators on a complex vector space V, then their product (or composite) C := AB is also a linear operator on V. 2. Prove the following commutator...

I was wondering: is every eigenstate of L^2 also an eigenstate of Lz? I know that commuting operators have the same eigenfunctions but if [A,B] = 0 and a is a degenerate eigenfunction of A the the corresponding eigenfunctions of A are not always eigenfunctions of B.