Operators Definition and 1000 Threads

Homework Statement Hi, I am solving a system of differential equations and in one of my equations I have this, (D+2)X+(D+2)Y=0 where X and Y are variables, D is my differential operator. My question is, would it be mathematically correct to divide out (D+2) and thus getting X+Y=0, X=-Y ?

Please, can somebody show me why a Hamiltonian like \sum_nh(x_n) can be written as \sum_{i,j}t_{i,j}a^+_ia_j, with t_{i,j}=\int f^*_i(x)h(x)f_j(x)dx? Thank you.

A couple questions: is mass quantized? Energy is quantized, and momentum has eigenvalues for its operator so I took that to mean that momentum is also quantized. If those two are true (might not be! I'm new to this :-p), following E^2 = (pc)^2 + (mc^2)^2 Would that not mean that mass is...

I've found the wave-packet picture quite useful as I work my way through the very basics of quantum mechanics. But I'm having trouble finding a wave-mechanical picture of operators. For example, at least in terms of a free particle, using the wave mechanics treatment (as opposed to the matrix...

Homework Statement Find the following hermitian conjugates and show if they are hermitian operators: i) xp ii) [x , p] iii) xp + px Where x is the position operator and p is the momentum operator. Homework Equations <f|Qg> = <Q^{t}f|g> Q = Q^{t} Hermitian operator p =...

Homework Statement I must show several properties about linear operators using the definition of the adjoint operator. A and B are linear operator and ##\alpha## is a complex number. The first relation I must show is ##(\alpha A + B)^*=\overline \alpha A^*+B^*##. Homework Equations The...

Homework Statement We only briefly mentioned this in class and now its on our problem set... Show that all eigenvalues i of a Unitary operator are pure phases. Suppose M is a Hermitian operator. Show that e^iM is a Unitary operator. Homework Equations The Attempt at a Solution...

I have a question about the formalism of quantum mechanics. For some operator A... \langle x |A|\psi\rangle = A\langle x | \psi \rangle Can this be derived by sticking identity operators in or is it more a definition/postulate. Thanks.

In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?

I have a confusion regarding expressing operators as projectors in Schrodinger and Heisenberg pictures. Please help. Consider a two-state system with |1> and |2> We know that e.g. a raising operator can be expressed as: \hat{\sigma}_+=|2><1| But here's my line of thought: In the...

Unfortunately, I can't find the thread (if someone finds it, please let me know, and I'll merge this post onto that thread), but someone asked why it is that in quantum mechanics, if you have an observable $B$, that the expectation value (average value) $\langle B \rangle$ is given by $$\langle...

Hi all! I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the...

Hi, I'm currently going through Ticciati's book along with the notes from Sidney Coleman's course and I have a question pertaining to Wick diagrams/expansion of S. In their example (section 4.3 of Ticciati and lecture 9 in Coleman's notes) they never seem to contract the adjoint nucleon field...

Reading through my QM text, I came across this short piece on ladder operators that is giving me trouble (see picture). What I am struggling with is how to get to equations 2 and 3 from equation 1. Can someone point me in the right direction? Where does the i infront of the x go?

Homework Statement Are the momentum eigenfunctions also eigenfunctions of e free particle energy. Operator? Are momentum eigenfunctions also eigenfunctions of the harmonic oscillator energy operator? An misplayed system evolves with time according to the shrodinger equation with potential...

Homework Statement Hello. I am supposed to find the commutator between to operators, but I can't seem to make it add up. The operators are given by: \hat{A}=\alpha \left( {{{\hat{a}}}_{+}}+{{{\hat{a}}}_{-}} \right) and \hat{B}=i\beta \left( \hat{a}_{+}^{2}-\hat{a}_{-}^{2} \right), where alpha...

Can someone give an example of a nonlinear operator on a finitely generated vector space(preferably ℝn)? I'd be particularly interested to see an example of such that has the group property as well.

Basically is it possible to take a time derivative of a creation/annhilation operator?

I am reading The Principles of Quantum Mechanics 4th Ed by Paul Dirac, specifically where he introduces his own Bra-Ket notation. You can view this book as a google book. http://books.google.com.au/books?id=XehUpGiM6FIC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false...

Linear operator A is defined as A(C_1f(x)+C_2g(x))=C_1Af(x)+C_2Ag(x) Question. Is A=5 a linear operator? I know that this is just number but it satisfy relation 5(C_1f(x)+C_2g(x))=C_15f(x)+C_25g(x) but it is also scalar. Is function ##A=x## linear operator? It also satisfy...

"Proof" that all operators are linear I've "proven" that all operators acting on a Hilbert space are linear. Obviously this isn't true, so there must be a fault in my reasoning somewhere. I having trouble finding it though, and would appreciate input by someone who can. Let |\psi\rangle =...

I know that L and S commute, but does this mean they do not anticommute? [L,S]=0. \left\{L,S\right\}=0?

In the paper http://link.aps.org/doi/10.1103/PhysRevA.85.062329, the authors separate the position and momentum operators into classical motion and quantum fluctuations: \hat{X}_i \equiv \bar{X}_i + \hat{q}_i; \quad \hat{P}_i \equiv \bar{P}_i + \hat{\pi}_i Can someone point me to a reference...

Hi, I'm currently reading the book "Quantum Field Theory for Mathematicians" by Ticciati and in section 2.3 he mentions that the Lorentz action on the free scalar field creation operators \alpha(k)^\dagger is given by U(\Lambda)\alpha(k)^\dagger U(\Lambda)^\dagger = \alpha(\Lambda...

Can anyone tell me why it is necessary to express a field as annhilation and creation operators? I just don't see why we need a field to explain the creation of particles in relativity, after all two colliding particles with enough energy produce some more.

Homework Statement Just starting third level Uni. stuff & am faced with linear operators from Quantum Mechanics & need a little help. OK, an operator, Ô, is said to be linear if it satisfies the equation Ô(α f1 + β f2) = α(Ô f1) + β(Ô f2) Fine but I have an equation I can't wrap my...

It is well known that unbounded operators play a crucial role in the mathematical formulation of quantum mechanics. In some sense, unbounded operators are inevitable. Indeed, we can prove that if A and B are self-adjoint operators such that [A,B]=ih, then A and B can never both be bounded. My...

If you have an operator a represented in some basis l1>, l2> you find its matrix elements by doing Aij = <ilAlj> But more oftenly you are interested in the expectation value of A. So you take: <ψlAlψ>. My teacher tends to call these numbers matrix elements too. But which matrix element...

There must be some lapse in my understanding of this. I understand that you can have an eigenstate of a system with an angular momentum magnitude value and a value for one component of the angular momentum (z). Using the lowering and raising operators we can create states (or deduce that states...

Homework Statement Homework Equations The Attempt at a Solution I have totally no idea how to solve this question. But I find it somehow similar to the Larmor precession problem. Therefore I try to solve my problem by referring to that. Are there any mistakes if I do it like...

One basic operator is addition. In order to add the any number x number of times, multiplication was invented. In order to multiply any number x number of times, exponentiation was invented. What if we want to raise a number to a power x number of times? How come we didn't invent that? Also...

Imagine we have two operators A and B on a complex hilbert space H such that: [A,B] \psi = (AB-BA) \psi=c \psi \ \ \ \ \psi \epsilon H \mbox{ and } c \epsilon C Then can we say that [A,B] is the same as cI when I is the identity operator?Why? Thanks

Homework Statement Given that u(x,y) and y(x,z) are both continuous, differentiable functions show that (\frac{\partial u}{\partial z})x=(\frac{\partial u}{\partial y})x(\frac{\partial y}{\partial z})x Homework Equations Only equations given above The Attempt at a Solution I...

I've searched for this but found nothing,so I ask it here. What are canonically conjugate operators? Is [A,B]=cI a definition for A and B being canonically conjugate? Thanks

Homework Statement I am struggling to understand the meaning behind: an operator of an operator: Let A and B be 2 volterra integral operators. If A and B are 2 integral operators, what does the following mean: (\textbf{A}) (\textbf{B}^{-1}(f))(x) Homework Equations We have the...

So, obviously one can diagonalize any self-adjoint transformation on a finite dimensional vector space. This is pretty simple to prove. What I'm curious about is integral operators. How does this proof need to be adapted to handle integral operators? What goes wrong? What do we need to account...

Homework Statement I have my quantum mechanics final creeping up on me and I just have a question about something that doesn't appear to be covered in the text. Let's say you have a wave function of the following form for a linear harmonic oscillator: \Psi = c_1 | E_1 \rangle + c_2 | E_2...

We know that a|n> = √n | √(n-1)> and a' |n> = √(n+1) | n + 1 > so, If we use this to find <n|a'a|n> it would be equal to n <n|a'a|n> = n Am I correct? I'm not really sure about my calculations. I operate with a first so. <n|a'a|n> <n|a' √n | √(n-1)> = n ...

Description 1. Prove that operators i(d/dx) and d^2/dx^2 are Hermitian. 2. Operators A and B are defined by: A\psi(x)=\psi(x)+x B\psi(x)=d\psi/dx+2\psi/dx(x) Check if they are linear. The attempt at a solution I noted the proof of the momentum operator '-ih/dx'...

In my QM textbook, there's an equation written as: \vec{J} = \vec{L}\otimes\vec{1} + \vec{S}\otimes\vec{1} referring to angular momentum operators (where \vec{1} is the identity operator). I don't really understand what the outer product (which I'm assuming is what the symbol \otimes means...

Homework Statement Assume C=[A,B]≠0 and [C,A]=[C,B]=0 Show eAeB=eA+Be\frac{1}{2}[A,B] Homework Equations All are given above. The Attempt at a Solution I recently did a similar problem (show eABe-A = B + [A,B] + \frac{1}{2}[A,[A,b]]+...) by defining a function exABe-xA and...

should compatible operators have the same eigenvalues??

Homework Statement operators: K=LM and [L,M]=1 α is an eigenvector of K with eigenvalue λ. Show that x=Lα and y=Mα are also eigenvectors of K and also find their eigenvalues. Homework Equations K=LM [L,M]=1 Kα=λα The Attempt at a Solution I tried, but its not even worth...

1. Homework Statement I have two operators A^2B^2+B^2A^2 1/2(A^B^+B^A^) 2 By what factor do the two operators differ? 3. The Attempt at a Solution I believe I either have to find the inner products of them and relate them somehow or use commutation? Not sure which one! I don't...

Homework Statement I have two operators \hat{A}^{2}\hat{B}^{2}+\hat{B}^{2}\hat{A}^{2} 1/2(\hat{A}\hat{B}+\hat{B}\hat{A}) ^{2} By what factor do the two operators differ? The Attempt at a Solution I believe I either have to find the inner products of them and relate them somehow...

Homework Statement Prove that i d/dx and d^2/dx^2 are Hermitian operators Homework Equations I have been using page three of this document http://www.phys.spbu.ru/content/File/Library/studentlectures/schlippe/qm07-03.pdf and the formula there. The Attempt at a Solution I have...

Why all operators in QM have a Hermitian Matrices ?

Homework Statement What is the correct interpretation of < \frac{\partial {A}}{\partial t} >, where A is an operator?Homework Equations for a wave function \phi and operator A, <A> = \int_{V}\phi^{*}(A\phi)dVThe Attempt at a Solution I thought it could mean < \frac{\partial {A}}{\partial t}...

Hello, I've read that Dirac introduced the idea of the creation and annihilation operators in the solution to the quantum harmonic oscillator problem, but can anyone tell me where he did this? In a paper, or maybe in a book? I've had a little search online, but I've yet to discover...

Can someone please explain to me how do we get the following: [P(x), L(y)]= i h(cut) P(z) P(x) is the momentum operator with respect to x and L(y) is the angular momentum operator with respect to y. I have also attached the solution. I am stuck at the underlined part. I do not know how...