Operators Definition and 1000 Threads

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...

The fraction of members of a set of numbers \{k_i\}_{i=1}^N that are equal to a specific number k can be written as \frac 1 N\sum_{i=1}^N\delta_{kk_i} Now consider an ensemble of N identical systems. Because of the above, the operator f_k^{(N)} defined by...

Let V be the vector space of all real integrable functions on [0,1] with inner product <f,g>=\int_0^1 f(t)g(t)dt Three linear operators defined on this space are A=d/dt and B=t and C=1 so that Af=df/dt and Bf=tf and Cf=f I need to find the eigenvalues of these operators: For A...

Homework Statement None of the operators Lz, Ly and Lx commute. Show that [Lx,Ly] = i(h-cross)Lz Homework Equations Where Lx is defined as Lx=-ih ( y delta/delta x - z delta/delta y) Where Ly is defined as Ly=-ih ( z delta/delta x - x delta/delta z) Where Lz is defined as Lz=-ih (...

I'm not exactly sure if this belongs in introductory or advanced physics help. Homework Statement In my book, the author was explaining the proof of the Uncertainty relation between po position and momentum. It simply stated that [x,p]= ih(h is reduced) But when I tried to verify it I got...

I'm reading a paper on NMR, and the authors keep referring to the operators I_x, I_y, I_z . What are these operators? I keep finding them mentioned in other papers, but no description of what they are.

I may have heard this or understood this incorrectly, so if i am asking the wrong question, feel free to correct me. As I understand it, if you have a degenerate set of simultaneous eigenvectors, you haven't specified a complete set of operators. For example, the hydrogen atom. You typically...

What exactly are the axioms of non-relativistic QM of one spin-0 particle? The mathematical model we're working with is the Hilbert space L^2(\mathbb R^3) (at least in one formulation of the theory). But then what? Do we postulate that observables are represented by self-adjoint operators? Do we...

Homework Statement Homework Equations The Attempt at a Solution I've gone round in circles doing this! I started of by writing it as an integral of (psi* x A_hat2 x psi) w.r.t dx, then using the equation above but I keep coming back at my original equation after flipping it...

Homework Statement This is problem 2.11 from Griffith's QM textbook under the harmonic oscillator section. Show that the lowering operator cannot generate a state of infinite norm, ie, \int | a_{-} \psi |^2 < \infty Homework Equations This isn't so hard, except that I consistently get the...

How does one obtain the formulae for the angular momentum operators in spherical polar coordinates i.e. \hat{L_x}=i \hbar (\sin{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \cos{\phi} \frac{\partial}{\partial{\phi}} \hat{L_y}=i \hbar (-\cos{\phi}{\phi}...

Homework Statement Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator T whose kernel is K is compact and symmetric. Let \varphi_k(x) be the eigenvectors (with eigenvalues \lambda_k) that diagonalize T . Then: a. K(x,y) \sim \sum_k \lambda_k...

Homework Statement Using the definitions of Lz and L^2, show that these two operators commute. Homework Equations Lz = -ih_bar * d/d(phi) L^2 = -(h_bar)^2 {1/sin(theta) * d/d(theta) * [sin(theta) * d/d(theta)] + 1/sin^2(theta) d^2/d(phi)^2} The Attempt at a Solution I'm actually...

Homework Statement OK, so assuming we have a physical observable with three values, a(1),a(2) and a(3), and there are given matrices for the measurement operators M(a(1))...M(a(3)). How does one actually go about finding a(1),a(2) and a(3) given the matrices?The Attempt at a Solution These...

I have encountered some mathematical difficulties when examining a one dimensional system defined by a Lagrange's function L(x,\dot{x}) = M|\dot{x}|^{\alpha} - V(x), where \alpha > 1 is some constant. The value \alpha=2 is the most common, but I am now interested in a more general...

Homework Statement C is an operator that changes a function to its complex conjugate a) Determine whether C is hermitian or not b) Find the eigenvalues of C c) Determine if eigenfunctions form a complete set and have orthogonality. d) Why is the expected value of a squared hermitian...

Homework Statement Hi all. When riding home today from school on my bike, I was thinking about some QM. The general solution to the Schrödinger equation for t=0 is given by: \Psi(x,0)=\sum_n c_n\psi_n(x), where \psi_n(x) are the eigenfunctions of the Hamiltonian. We know that the...

Hey everyone! I have a question regarding the matrix representation of a projection operator. Specifically, does the wavefunction have to be normalized before determining the projection operator? For example: |Ψ1> = 1/3|u1> + i/3|u2> + 1/3|u3> |Ψ2> = 1/3|u1> + i/3|u3> Ψ1 is obviously...

In a recent lecture on String Theory, we encountered the divergent sum 1+2+3+... when calculating the zero mode Virasoro operator in bosonic String Theory. This divergent sum is then set equal to a finite negative constant - the argument for doing so was a comparison with the definition of the...

Hello. I am stuck trying to find an understandable answer to this online: Carry out the following operations on the vector field A reducing the results to their simplest forms: a. (d/dx i + d/dy j + d/dz k) . (Ax i + Ay j + Ax k) b. (d/dx i + d/dy j + d/dz k) x (Ax i + Ay j + Ax k) I...

When quantizing a classical field theory, e.g. Klein-Gordon theory, the Hilbert space of one-particle states is taken to be a set of equivalence classes of (Lebesgue) measurable and square integrable solutions of the classical field equation, but how do you use the classical theory to construct...

Hi Short question: What is the generalization of the BAC-CAB rule for operators? Longer question and context: please read below I was reading Schiff's book on Quantum Mechanics (3rd Edition) and on page 236, he has defined a generalized Runge-Lunz vector for a central force as \vec{M} =...

I tried to use the eigenvalue of the operators but I couldn't get the result. Can anyone help me to understand this relationship? Thank you.

I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient. Dirac rather liberally talks about observables that have a continuous range of...

I read in my notes that S{-}|1> = sqrt(2)h(bar)|0> and similar for all six products of using the raising and lowering operators on |1>, |0>, |-1> I don't understand where the sqrt(2)h(bar) has come from? Cheers Philip

Problem Suppose that f(A) is a function of a Hermitian operator A with the property A|a'\rangle = a'|a'\rangle. Evaluate \langle b''|f(A)|b'\rangle when the transformation matrix from the a' basis to the b' basis is known.The attempt at a solution Here's what I have... I'm not sure if the last...

I'm pretty sure this is correct, but could someone verify for rigor? Problem Two observables A_1 and A_2, which do not involve time explicitly, are known not to commute, yet we also know that A_1 and A_2 both commute with the Hamiltonian. Prove that the energy eigenstates are, in general...

I have two questions: 1. Let V be a finite dimensional inner product space. Show that if U: V--->V is self-adjoint and T: V---->V is positive definite, then UT and TU are diagonalizable operators with only real eigenvalues. 2. Suppose T and U are normal operators on a finite dimensional...

Hi there! :) I'm trying to understand a theorem, but it's full with analysis (or something) terms unfamiliar to me. Is there an intuitive interpretation for the sentence: 'An operator being limited is equivalent to continuity in the topolgy of the norm'? Also, how can I partially...