Eigenstates Definition and 188 Threads

Hey there, the question I'm working on is written below:- Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by: H = |a'>∂<a''| + |a''>∂<a'| where ∂ is just a real number. Write down the eigenstates...

Homework Statement The hamiltonian of a simple anti-ferromagnetic dimer is given by H=JS(1)\bulletS(2)-μB(Sz(1)+Sz(2)) find the eigenvalues and eigenvectors of H. Homework Equations The Attempt at a Solution The professor gave the hint that the eigenstates are of...

For the free particle the solution to the SE are eigenstates of the momentum. You get something like: ψ = Aexp(ik(x-vt)) + Bexp(-ik(x+vt)) , where k is a constant And my book then says that first term represents a wave traveling to the right and the second a wave traveling to the left. But I...

This might be a silly question, but I'm not sure about the answer... Commuting operators have identical eigenstates. For example: energy and position operators seem to commute: [E,x] = Ex - xE = 0 Does it mean that position and energy operators share identical eigenstates? Because...

When a particle is in superposition of energy eigenstates and has a probability of being found in either state, what does that say about the energy of the particle and conservation of energy. What I mean is, since the energy eigenstates have different energy values, where's the rest of the...

It is said that all non-strange non-baryonic states are eigenstates of G-parity. And all members of an isospin multiplet have the same eigenvalue. Can anyone give me a proof to these two statements, or show me where I can find one? In addition, the composite state consisting of K^{+}K^{-} should...

Homework Statement I need to express the rotation operator as follows R(uj) = cos(u/2) + 2i(\hbar) S_y sin(u/2) given the fact that R(uj)= e^(iuS_y/(\hbar)) using |+-z> as a basis, expanding R in a taylor series express S_y^2 as a matrix Homework Equations I know...

Homework Statement Im trying to understand what happened in this book they have |I>=\frac{|1>}{\sqrt{2}}+\frac{|2>}{\sqrt{2}} and then |II>=\frac{|1>}{\sqrt{2}}-\frac{|2>}{\sqrt{2}} and then they say they superimpose these equations to get...

Homework Statement Observable \widehat{A} has eigenvalues \pm1 with corresponding eigenfunctions u_{+} and u_{-}. Observable \widehat{B} has eigenvalues \pm1 with corresponding eigenfunctions v_{+} and v_{-}. The eigenfunctions are related by: v_{+} = (u_{+} + u_{-})/\sqrt{2} v_{-} =...

Homework Statement (See attachment) Homework Equations x = \sqrt{\frac{\hbar}{2m \omega}} ( a + a^{\dagger} ) x = i \sqrt{\frac{\hbar m \omega}{2}} ( a^{\dagger} - a ) The Attempt at a Solution In part a) I was able to construct a separable Hamiltonian for the harmonic...

Homework Statement From Liboff edition 4: For the state ψ(x,t)=A exp(x-x0)2 / {4a2} * exp(ip0x)/hbar * exp (-iω0t) show that (Δx)2 = a2 then argue the consistency of this conclusion with the change in shape that |ψ2| suffers with a change in the parameter a Homework...

I am very confuse about mass eigenstate. In some books they say'' If neutrinos are massless then lepton mixing is unobservable.Any Cabibbo-like rotation still leaves us with neutrino mass eigenstates''. I do not understand that statement.Why the mixing of some states gives a state being mass...

Homework Statement Consider a particle moving in one dimension and bound to an attractive Dirac δ-function potential located at the origin. Work in units such that m=\hbar=1. The Hamiltonian is given, in real (x) space, by: H=-\frac{1}{2}\frac{d^2}{dx^2}-\delta (x) The (non normalized)...

I believe that in qft the particle states are eigenvectors of the Hamiltonian(which also commutes with the number operator), i.e. H|n> = E|n>. A Fock vector is a product of many particle state vectors. But what are the eigenvectors of psi(x)? thanks

Weak eigenstates d', s', b' are a mixture of mass eigenstates. For example, s'=V_cd*d+V_cs*s+V_cb*b This doesn't seem to be the case for u, c, t quarks. For example, there is no c'=V_su*u+V_sc*c+V_st*t Why is that?

Say you have some operator A with an incomplete set of eigenstates, but the state of the system is such that it happens to be expressible as a sum (possibly infinite, or integral, whatever) of the eigenstates of A, and let's say the eigenvalues are real and whatever is necessary...we may assume...

We now that if [A,B]=0, they have the same eigenstates. But consider a harmonic oscillator with the spring constant k1. If we change k1 to k2, then [H1,H2]=0 and the above expression implies that the eigenstates should not change while they really change! Could you please tell me if i am wrong?

Hi, I am hoping someone could clear up a few things about neutrinos oscillations for me. For the sake of this dicussion let's set up the neutrino mixing equations in such a way that the flavor eigenstates are a superposition of mass eigenstates. So now for example we have...

QM says that states which are simultaneous eigenstates of two commuting observables are allowed. If you don't have such states to start with you can construct them with the Gramm-Schmidt orthogonalization procedure. Consider the excited states of a nucleus. (They can be considered eigenstates...

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...

The first thing I remember hearing about in QM was the time-independent 1-D schrodinger equation, Hψ = (\frac{-\hbar^2}{2m}\frac{d^2}{dx^2} + V)ψ(x) = Eψ(x) . This is an eigenvalue equation, the Hamiltonian operator H operating on the energy eigenstate ψ to produce the product of the energy...

Homework Statement Let |0,0\rangle be the simultaneous eigenstate of \mathbf{J}^2 and J_z with eigenvalues 0 and 0. Find J_x|0,0\rangle \quad\quad J_y |0,0\rangle \quad\quad [J_x,J_y]|0,0\rangle 2. The attempt at a solution It seemed reasonable to write J_x and J_y in terms of ladder...

I'm not exactly looking for help finding the eigenvalues of the spin operator, I'm mainly wondering if there is a better technique to do it. Homework Statement Find the eigenvalues and corresponding eigenstates of a spin 1/2 particle in an arbitrary direction (θ,\phi) using the Pauli...

A spin-3/2 particle (labeled 1) and a spin-1/2 particle (labeled 2) are interacting via the Hamiltonian \lambda \vec S_1 \cdot \vec S_2 + g(S_{1z} + S_{2z})^2 where S1 is the spin operator of particle 1 (the spin-3/2) and S2 is the spin operator of particle 2 (the spin-1/2). λ and g...

Can an electron in a hydrogen atom every been in an energy eigenstate?

Hi everybody! Two question for you: 1) Take a free particle, moving in the x direction. Its (time indipendent) wave function, in terms of the momentum is \psi(x)=\frac{e^{i\frac{p}{\hbar}x}}{\sqrt{2\pi\hbar}}. Now, i know the momentum of the particle: p. So i should not know anything about its...

Could somebody please explain to me the difference between flavour and mass eigenstates. The question is "Neutrinos can be produced from charged pion decay, What force is involved, and at the time of creation is the neutrino in a flavour or mass eigenstate, and why?" And why would it be...

"Continuous eigenstates" vs "discrete eigenstates" There's this thing that's bothering me: if I have an Hamiltonian with a discrete and continuous spectrum, every book I read on quantum mechanics says that eigenvectors of discrete eigenvalues are orthogonal in the "Kronecker sense" (their...

Why do we associate the energy eigenstates with the "wavefunction" and "position"? This is something that has bothered me for a long time but that I never got around to asking. I suspect I'll feel like an idiot for not knowing this but anyway... One of the first things I was introduced to...

Homework Statement I'm having some trouble understanding exactly how the eigenstates of this matrix are being presented. A= ( 0 -i i 0 ) <- matrix I can find the eigenvalues to be +/- 1 which gives the eigenvectors to be x=-iy or x=iy. The eigenvectors are then being...

To find the probability of a particle being at position x we use <\Psi|\Psi> where the complex conjugate ensures that the answer is real. This means that we're looking at the square of the wave function to determine the probability of finding the particle. Now to determine the probability...

Homework Statement I have some questions about the eigenstates of an operator. A state is an eigenstate of an operator if the application of the operator on the state results in a constant complex multiple of the state. The constant complex multiple will be the eigenvalue. For instance...

Homework Statement A particle in the infinite square well with V(x)=0 for 0<x<a and V(x)=infinity otherwise has the initial (t=0) wave function: psi(x,0)=Ax for 0<x<a/2 psi(x,0)= A(a-x) for a/2<x<a 1) Sketch psi and psi^2 (DONE) 2) Determine A [DONE - 2*sqrt(3)*a^(-3/2)] 3) Find psi(x,t)...

Homework Statement A system at time t = 0 is in the state |ψ> = a|E1> + b|E2>, where |E1> and |E2> are (normalised) energy eigenstates with two different energies E1 and E2, and a, b are real numbers. Write down the state |ψ, t> for the system at time t. What are the probabilities at time t to...

Not really sure how to go about this. Our lecture said "it can be shown" but didn't go into any detail as apparently the proof is quite long. I'd really appreciate it if someone could show me how this is done. Thanks. (Not sure if this is relevant but I have not yet studied Hilbert spaces).

How did they get (u,v) in the second line and then alpha+? http://i111.photobucket.com/albums/n149/camarolt4z28/2010-11-20113911.jpg?t=1290274889

Homework Statement The eigenstates of the infinite square well are not energy eigenstates and are not momentum eigenstates. Homework Equations The Attempt at a Solution I don't understand how this can be? If the eigenstates of the infinite square well are energy eigenstates...

Homework Statement 1. Is state \psi_{0,2,1}-\psi_{5,0,1} an eigenstate of L_{x} 2. Is state \psi_{1,3,1}-\psi_{4,2,0} an eigenstate of L^{2} Homework Equations Stationary state of Hamiltonian defined by: [itex]\psi_{n,l,m}[/tex] where the subscripts denote quantum numbers. The...

I know that the set of eigenstates of an operator forms a complete set, a basis. Which means that any state can be written as a linear sum of the eigenstates. But is the word 'any' correct? I haven't seen a proof of the above completeness theorem, so i don't know what the mathematical...

If a (charged) particle is in superposition of two different positions A and B is it interacting with itself by electric forces (and gravitational force)? Is it true that the particle is in both places A and B simultaneously?

let L be angular momentum operator. [L^2 , Lz] = 0 [L^2 , Lx] = 0 (I haven't prove this, but appearantly it's correct according to lecturer) does it imply that [Lx , Lz] = 0? this is just one interesting thoughts that cross my mind because I recalled that if 2 matrix [A,B] =0, A and B...

This is taken from a text problem, but I am putting it in this section because I think my question goes beyond the problem itself: If a particle has a wave function psi = A*R(r)*cos2 (theta), for example, then if I want to find the probability that its angular momentum is l I would find the...

My question is that is the fact, that all particles are either boson or fermions, only an empirical fact, or can it be argued theoretically too. The reason why I'm asking this is that I have not encountered anyone stating honestly, that it is an empirical fact only. But on the other hand, I...

I'm a self learner with a decent math background trying to follow this lecture. I have been able to follow him perfectly up to this point but at 16:00 Professor Balakrishnan states that "<n|a|n> = 0" because |n-1> is perpendicular to |n>. My question is why and how do we know that an...

any references? i think it is a basic problem

Homework Statement I have to show that the Fock states are eigenstates for the operators {{\hat{a}}^{\dagger }} and/or {\hat{a}} And I'm not totally sure how to show this. Homework Equations ? The Attempt at a Solution I know that if I use the operators on a random Fock...

First order correction to particle-in-box eigenstates for Dirac perturbation Homework Statement Calculate the first three nonzero terms in the expansion of the correction to the ground state \psi^{1}_{1} for a Dirac delta perturbation of strength alpha at a/2 (box from 0 to a). Homework...

So I've been having a specific major hang-up when it comes to understanding basic quantum mechanics, which is the position operator. For the SHO, the time independent Schroedinger's equation looks like E\psi = \frac{\hat{p}^2}{2m}\psi + \frac{1}{2}mw^2\hat{x}^2\psi Except that...

I'm teaching myself QM and have a question about a problem in my text. (If this makes it homework, sorry to post in the wrong forum). Basically the question is about three observables A,B,C. They obey the following rules: [A,B]=0, [A,C]=0, but [B,C] not equal to 0. So this is like the...

Why can we say that: <x'|e^{i\hat{x}}|x>=e^{ix'}\delta(x'-x) where where \hat{x} is an operator? I mean if \hat{x}|x>=x|x> we may write <x'|\hat{x}|x>=x<x'|x>=x\delta(x'-x) but in the expression at the top, we have an exponential operator (something I've never come across...