Operators Definition and 1000 Threads

Homework Statement This has been driving me CRAZY: Show that \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle and \langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle Homework Equations Raising/lowering eigenvalue equations: a |n...

Right so I've had an argument with a lecturer regarding the following: Suppose you consider P_4 (polynomials of degree at most 4): A(t)=a_0+a_1t+a_2t^2+a_3t^3+a_4t^4 Now if we consider the subspace of these polynomials such that a_0=0,\ a_1=0,\ a_2=0}, I propose that the dimension of of this...

Hi all, I've been looking over some results from functional analysis, and have a question. It seems that often times in functional analysis, when we want to show something is true, it often suffices to show it holds for the unit ball. That is, if X is a Banach space, then define b_1(X) =...

I am about to start a graduate program in signal processing. A lot of the literature that I've been recently browsing, uses the concept of operators on functions - such as a differential, or Fourier transform operator. I really like this "framework" (for lack of understanding) but have never...

Homework Statement I'm just not sure how to change the operators in summation, can anyone help? Let s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k what is s_{2n}?Homework Equations s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k The Attempt at a Solution s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k or...

Hi, Will give a 30 minute talk in class about (group) and Unitary Operators. Could anybody suggest a suitable soursde suitable for a presentation (keeping the class interested) in first year Grad. level . Thank you

My lecturer has written A | \alpha_n> = a_n |\alpha_n> => A = \sum_n a_n | \alpha_n>< \alpha_n | and B | \alpha_k> = b_k |\alpha_k> => B = \sum_k b_k | \alpha_k>< \alpha_k | Where A is a hermitian operator. I understand he's used the properties of the unitary projector operator here, but is...

Forgive me if I am putting this in the wrong place, but this is my first post here. The question that I have is directed to the more experienced researchers than I am, I guess. In the Hamiltonian formulation of QFTs we write everything in terms of the ladder operators, right? So in practice...

im the very beginner of learning c programe so i would like to ask the follow questions and hope someone can give me some idea:: i know: 3%2 = 1 8%3 = 2 12%4 = 0 but I am not sure about the followings (or my concept is wrong or right): 12%100 = 12 1%2 = 1 can i say when the...

I am confused about operators and eigenstates. What does "an operator has two normalized eigenstates" mean ? Is there a way I can make a physical interpretation ? How are measurements made with these ?

Homework Statement The problem is to show that, \hat{a_{+}}|\alpha>=A_{\alpha}|\alpha+1> using \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha>It's not hard to manipulate \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha> into the form...

If A is an operator, is it correct/allowed to say: Ae^{iA} = e^{iA}A Thanks

im working through a proof and am stuck on the last line. i can't understand why \nabla_a \nabla_b \omega_c - \nabla_b \nabla_a \omega_c + \nabla_b \nabla_c \omega_a - \nabla_c \nabla_b \omega_a + \nabla_c \nabla_a \omega_b - \nabla_a \nabla_c \omega_b=0? any advice?

Can somebody please explain the following? Given the measurements of 2 different physical properties are represented by two different operators, why is it possible to know exactly and simultaneously the values for both of the measured quantities only if the operators commute?

I forgot the exact terminology for these types of operators but here goes. take for example the operators x, and p. the commuter equals i(h bar), and the eigenvectors are Fourier transforms of each other. my question is, how do you go about proving at least one of the properties listed...

Homework Statement Consider the vector space of square-integrable functions \psi(x,y,z) of (real space) position {x,y,z} where \psi vanishes at infinity in all directions. Define the inner product for this space to be <\phi|\psi> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}...

Prove that a normal operator with real eigenvalues is self-adjoint Seems like a simple proof, but I can't seem to get it. My attempt: We know that a normal operator can be diagonalized, and has a complete orthonormal set of eigenvectors. Let A be normal. Then A= UDU* for some...

Hey, My brain seems to have shut down. Let's say I'm working in the space H_a \otimes H_b and I have an operator A and B on each of these spaces respectively. Furthermore, consider a general state in the tensor space |\psi \rangle = | x_1 \rangle |y_1\rangle + |x_2 \rangle |y_2 \rangle...

I am working on a homework problem from quantum mechanics. In order to solve the problem I need to derive the raising and lowering operators. In order to to this I did the following: S+operator = <1,i | S+operator | 1,j > where i = 1, 0, -1 with i = 1 corresponding to row one etc...

Homework Statement I need to find the momentum space function for the ground state of hydrogen (l=m=0, Z=n=1) Homework Equations \phi(\vec{p}) = \frac{1}{(2\pi\hbar)^{3/2}}\int e^{-i(\vec{p}\cdot\vec{r})/\hbar}\psi(\vec{r})d^3\vec{r}...

as an example, say I am operating on r^2 with 2/r*d/dr : do I differentiate r^2 first, then times by 2/r or times by 2/r and then differentiate. Confused as they give different answers

If A and B are hermitian operators, then AB is hermitian only if the commutator=0. basically i need to prove that, but i don't really know where to start ofther than the general <f|AB|g> = <g|AB|f>* obv physics math is not my strong point. thanks :)

I have the following two equations #1 d(A(t))/dt=A(t)B where A is some matrix that depends on parameter t, and B is another matrix, d is the differential this can be simplified to by multiplying both sides by the left inverse of A(t), A^-1(dA(t))=B*t which allows me to solve...

Homework Statement (from "Advanced Quantum Mechanics", by Franz Schwabl) Show, by verifying the relation \[n(\bold{x})|\phi\rangle = \delta(\bold{x}-\bold{x'})|\phi\rangle\], that the state \[|\phi\rangle = \psi^\dagger(\bold{x'})|0\rangle\] (\[|0\rangle =\]vacuum state) describes a...

Partial differential equations represented as "operators" Homework Statement Partial differential equations (PDEs) can be represented in the form Lu=f(x,y) where L is an operator. Example: Input: u(x,y) Operator: L=∂xy + cos(x) + (∂y)2 => Output: Lu = uxy+cos(x) u + (uy)2 Homework...

If the Hamiltonian is given by H(x,p)=p^2+p then is it Hermitian? I'm guessing it's not, because quantum-mechanically this leads to: H=-h^2 \frac{d^2}{dx^2}-ih\frac{d}{dx} and this operator is not Hermitian (indeed, for the Sturm-Liouville operator O=p(x)\frac{d^2}{dx^2}+k(x)\frac{d}{dx}+q(x)...

Seen a couple of pieces on gauge theories of finance equating arbitrage opportunities to curvature, for this to work the curvature must therefore not be constant, instead it would be some sort of a function of capital flows

Does anyone know if there is a relationship between the requirement in Quantum Computing that logic gates be reversible and the requirement in Quantum Mechanics that observables have to be self-adjoint?

I don't unterstand de function F(\hat{J}) where J is the operator \hat{J}=(\hat{J_1},\hat{J_2},\hat{J_3}) and the components of J do not commute. In case when F a function of only one component we have the definition F(\hat{J_1})|m>=F(m)|m> where \hat{J_1}|m>=m|m>, but how do you define the...

Homework Statement Prove that the sum of any two positive operators on V is positive.Homework Equations The Attempt at a Solution This problem seems pretty simple. But I could be wrong. Should I name two positive operators T and X such that T=SS* and X=AA*? I have a bad history of seeing a...

Statement: One of the key elements in being in what people call group is that elements must be associative. So this means if we take any three elements from what we propose to be a group, they should be associative, a*(b*C) = (a*b)*c Question: Suppose we do have a group with elements a, b, c...

Homework Statement Suppose V is a complex inner-product space and T ∈ L(V) is a normal operator such that T9 = T8. Prove that T is self-adjoint and T2 = T.Homework Equations The Attempt at a Solution Consider T9=T8. Now "factor out" T7 on both sides to get T7T2 =TT7. Now we represent T as a...

Homework Statement Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.Homework Equations The Attempt at a Solution Let c be an eigenvalue. Now since T=T*, we have <TT*v, v>=<v, TT*v> if and only if TT*v=cv on both sides...

Homework Statement Show that if V is a real inner-product space, then the set of self-adjoint operators on V is a subspace of L(V). Homework Equations The Attempt at a Solution Let M be the matrix representing T. Since we are dealing with real numbers, and T is self-adjoint, T=T* so M=MT...

Apparently, if I have a Hamiltonian that contains an operator, and that operator commutes with the Hamiltonian, not only can we "simultaneously diagonalize" the Hamiltonian and the operator, but I can go through the Hamiltonian and replace the operator with its eigenvalue everywhere I see it...

let be X,Y and Z operators of some algebra so [X_i , Y_j]= \epsilon _ijkX_k where i,j and k range over X, Y and Z then i define the change of coordinates X= rcos(u) Y= rsin(u) Z=Z again r, u and Z are new operators, the problem is , how can i find for example what is...

Is there a fundamental difference between operators and functions? For example we could have F(x,y)=x+y or we could write SUM(x,y) where SUM is a defined operation in some program. Could operators be considered a particular type of function?

My book on quantum physics says that if two Hermitian operators commute then it emerges that they have common eigenfunctions. Is that true? If A,B hermitian commuting operators and Ψ a random wavefunction then: [A,B]Ψ=0 => ABΨ=BAΨ If we assume that Ψ is B`s eigenfunction: b*AΨ=BAΨ...

About the invariance of similar linear operators and their minimal polynomial Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is...

Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions. τ denotes a linear operator contained in L(V) ι...

Homework Statement Can anyone please explain why any term which is a product of 4 raising and lowering operators with a lowering operator on the extreme right (eg. A-A+A+A-) has zero expectation value in the ground state of a harmonic oscillator? Homework Equations The Attempt...

1. Explain why any term (such as AA†A†A†)with unequal numbers of raising and lowering operators has zero expectation value in the ground state of a harmonic oscillator. Explain why any term (such as AA†A†A) with a lowering operator on the extreme right has zero expectation value in the...

Homework Statement I am currently studying for my quantum physics exam and I am trying to derive the Transformation function: ⟨x'│p' ⟩=Nexp{(ip' x')/ℏ} Homework Equations ⟨x'│p' ⟩=Nexp{(ip' x')/ℏ} The Attempt at a Solution Now I get how to get to p'⟨x'│p' ⟩=-iℏ d/dx' ⟨x'│p' ⟩...

Some time ago, someone in this forum asked how you measure momentum. One of the answers said that if it's a charged particle, you can let it pass through a bubble chamber and a magnetic field, and measure the curvature of the bubble trail. But this isn't really a direct measurement of the...

Hi Suppose \Lambda is a Lorentz transformation with the associated Hilbert space unitary operator denoted by U(\Lambda). We have U(\Lambda)|p\rangle = |\Lambda p\rangle and |p\rangle = \sqrt{2E_{p}}a_{p}^{\dagger}|0\rangle Equivalently, U(\Lambda)|p\rangle =...

Hi everyone I'm trying to express each term of the Hamiltonian H = \int d^{3}x \frac{1}{2}\left[\Pi^2 + (\nabla \Phi)^2 + m^2\Phi^2\right][/tex] in terms of the ladder operators a(p) and [itex]a^{\dagger}(p). This is what I get for the first term \int d^{3}x...

Homework Statement Show [a+,a-] = -1, Where a+ = 1/((2)^0.5)(X-iP) and a- = 1/((2)^0.5)(X+iP) and X = ((mw/hbar)^0.5)x P = (-i(hbar/mw)^0.5)(d/dx)2. The attempt at a solution It would take forever to write it all up, but in summary: I said: [a+,a-] = (a+a- - a-a+) then...

Hello, I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following, c = \frac{(a + ib)}{\sqrt{2}}...

For example, if I were to prove that all symmetric matrices are diagonalizable, may I say "view symmetric matrix A as the matrix of a linear operator T wrt an orthonormal basis. So, T is self-adjoint, which is diagonalizable by the Spectral thm. Hence, A is also so." Is it a little awkward to...

perhaps it is just a nonsense but can we express or could we express g_{ab }(x) = \int exp(iux)a(+)f(+) + \int exp(-iux)a(-)f(-) the idea is, we express the metric g_ab in terms of the creation an anhinilation operators we write also \pi _ab (conjugate momenta) as a sum of...