Eigenstates Definition and 188 Threads

I'm confused about the statement that if operators commute then eigenstates are shared. My main confusion is this one: ##L^2## commutes with ##L_i##. Then these two share eigenstates. But ##L_x## and ##L_y## do not commute, so they don't share eigenstates. Isn't this violating some type of...

If the spectrum of a hermitian operator is continuous, the eigenfunctions are not normalizable. I have been told that these eigenfunctions do not represent possible physical states, but what exactly does that mean? Is there a good interpretation of the physicality of these eigenstates? If a...

Hi to all. Say that you have an eigenvalue problem of a Hermitian matrix ##A## and want (for many reasons) to calculate the eigenvalues and eigenstates for many cases where only the diagonal elements are changed in each case. Say the common eigenvalue problem is ##Ax=λx##. The ##A## matrix is...

Suppose we have an electron in a hydrogen atom that satisfies the time-independent Schrodinger equation: $$-\frac{\hbar ^{2}}{2m}\nabla ^{2}\psi - \frac{e^{2}}{4\pi \epsilon_{0}r}\psi = E\psi$$ How can it be that the Hamiltonian is spherically-symmetric when the energy eigenstate isn't? I was...

Let's suppose that we have an entangled state of two systems ##A## and ##B##: $$ \frac{1}{2}\left(|\psi_1 \phi_1\rangle+|\psi_2 \phi_2\rangle \right) $$ where ##|\psi \rangle## and ##|\phi \rangle## are energy eigenstates of ##A## and ##B## respectively. However the eigenstates##|\phi_1\rangle##...

Just like it says, are all solutions of the 1D time independent Schrodinger equation, by default, energy eigenstates? I'm having a hard time imagining how solutions, with these conditions, that aren't energy eigenstates could exist if they have to satisfy the relation E \psi(x)=\hat{H}\psi(x)

I'm reading about stationary states in QM and the following line, when discussing the time-independent, one-dimensional, non-relativist Schrodinger eqn, normalization or the lack thereof, and the Hamiltonian, this is mentioned: "In the spectrum of a Hamiltonian, localized energy eigenstates are...

Recently I've been studying Angular Momentum in Quantum Mechanics and I have a doubt about the eigenstates of orbital angular momentum in the position representation and the relation to the spherical harmonics. First of all, we consider the angular momentum operators L^2 and L_z. We know that...

When we deal with an infinite-dimensional basis,the normalization condition of this basis becomes <x|x'>=δ(x-x')(here for example the position basis).Same thing for momentum eigenstates <p|p'>=δ(p-p'). Lets look now on the eigenvalue problem of the momentum operator: \hat p | p \rangle =p |...

Look at the following derivation: ## p=\frac{im}{\hbar}[H,r] ## if ##H|\psi\rangle=E|\psi\rangle##, then ## \langle \psi|p|\psi \rangle = \frac{im}{\hbar}\langle \psi|Hr-rH|\psi \rangle = \frac{im}{\hbar}\langle \psi|r|\psi \rangle(E-E)=0 ## What's wrong with my derivation or it is true that...

Homework Statement For an infinite potential well of length [0 ; L], I am asked to write the following function ##\Psi## (at t=0) as a superposition of eigenstates (##\psi_n##): $$\Psi (x, t=0)=Ax(L-x) $$ for ## 0<x<L##, and ##0## everywhere else. The attempt at a solution I have first...

I'm a bit confused as to what is meant by instantaneous eigenstates in the Heisenberg picture. Does it simply mean that if vectors in the corresponding Hilbert space are eigenstates of some operator, then they won't necessarily be so for all times ##t##, the eigenstates of the operator will...

Hi. I am just starting to self-study QFT. Have been looking at the non-relativistic case of particles in a box. Have come across creation operators in the occupation number representation which create a particle in specific momentum state. But I thought in a rigid box there are no momentum...

Hi. Hoping for some help with the following questions - 1 - Are there any momentum eigenstates in a box ? I think the answer is no because if i solve the momentum eigenvalue equation in 1-D i get Aikx but it seems impossible to get this to meet the boundary conditions 2 - As far as I know a...

I saw some control diagrams for emotions on this website http://www.emotionalcompetency.com/sadness.htm and thought it would be cool to model it with a state space formalism. let's take x as a vector x = [anger, sadness, joy, etc...] where anger sadness and joy are quantities probabilities...

Homework Statement A One dimensional box contains a particle whose ground state energy is ε. It is observed that a small disturbance causes the particle to emit a photon of energy hν=8ε, after which it is stable. Just before emission a possible state of the particle in terms of energy...

I was studying quantum states in quantum field theory and I came across the formula for defining eigenstates: |n> = [(a†)n / sqrt(n!)] * |0> However, my book did not actually define ground state |0> (meaning the book did not give some function or numbers or anything like that to define what...

Homework Statement Consider the following state constructed out of products of eigenstates of two individual angular momenta with ##j_1 = \frac{3}{2}## and ##j_2 = 1##: $$ \begin{equation*} \sqrt{\frac{3}{5}}|{\tiny\frac{3}{2}, -\frac{1}{2}}\rangle |{\tiny 1,-1}\rangle +...

Hi, I have learned about how to find the 4 spin states of 2 spin 1/2 particles, and how to find them by using the lowering operator twice on |1/2, 1/2> to find the triplet, then simply finding the orthogonal singlet state, |0, 0>. I started to attempt finding the states of 3 spin 1/2...

I have been looking at the solution to a question and I don't understand how the eigenstates are calculated. The question concerns a 3-state spin-1-system with angular momentum l=1. The 3 eigenstates of L3 are given as ## \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} ## , ## \begin{pmatrix} 0 \\ 1...

Does Energy Eigenstates refer to each orbital from the ground state or within one orbital in terms of the kinetic and potential energy of the electron in the orbital or from energy bands of molecular system? What is the term for each case called?

How large could position eigenstates indeterminacy be so as to be indistinguishable from classical state? For example. If a particle is smeared by 10 Planck length.. could we tell or could we consider it as classical state? What is the most accurate device that has probe the smallest region...

Homework Statement A potential has V(x)=2V0 for x<0 , V(x)=0 for 0<x<a and V(x)=V0 for x.>a with V0>0. The next 3 questions apply to this potential. 1 - there are 2 energy eigenstates for each energy E>V0 but smaller than 2V0 True or false ? 2 - there is one normalizable state for each E>0 but...

Homework Statement I have a hamiltonian: \begin{pmatrix} a &0 \\ 0&d \end{pmatrix} + \begin{pmatrix} 0 &ce^{i w t} \\ ce^{-iwt}&0 \end{pmatrix}=\begin{bmatrix} a & c e^{i w t} \\ c e^{-i w t}&d \\ \end{bmatrix} Where the first hamiltonian can be labeled with states |1> and |2>...

If a wavefunction can be in different states each with a different eigenvalue when operated on by an Hermitian operator then am I correct in thinking that the expectation value of that operator when acting on a superposition of states could give a value that does not equal any single one of the...

Homework Statement The state of an electron is, |Psi> =a|l =2, m=0> ⊗ |up> + Psi =a|l =2, m=1> ⊗ |down>, a and b are constants with |a|2 + |b|2 = 1 choose a and b such that |Psi> is an eigenstate of the following operators: L2, S2, J2 and Jz. The attempt at a solution I am really not sure...

Homework Statement Determine the eigenstates of ##\hat{\mathbb{S}}_x## for a spin##-1## particle in terms of the eigenstates ##|1,1\rangle, \ |1,0\rangle,## and ##|1,-1\rangle## of ##\hat{\mathbb{S}}_z.##Homework EquationsThe Attempt at a Solution Not sure exactly how to set this problem...

How many energy eigenstates can "fit" inside a one-dimension box of length L?

Homework Statement Consider a two-dimensional space spanned by two orthonormal state vectors \mid \alpha \rangle and \mid \beta \rangle . An operator is expressed in terms of these vectors as A = \mid \alpha \rangle \langle \alpha \mid + \lambda \mid \beta \rangle \langle \alpha \mid +...

if I derive a hermitian relation use: [1] \left \langle \Psi _{m} | H |\Psi _{n}\right \rangle =E_{n}\left \langle \Psi _{m} |\Psi _{n}\right \rangle and [2] \left \langle \Psi _{n} | H |\Psi _{m}\right \rangle =E_{m}\left \langle \Psi _{n} |\Psi _{m}\right \rangle if i take the complex...

If the wavenumber eigenstates are |k> and the position eigenstates are |x>, then my notes say we can write |k>=∫-∞∞ek(x)|x>dx i.e express a wavenumber eigenstate in terms of a superposition of position eigenstates. Now they state that ek(x)=eikx/√(2π). I don't understand how we can say that the...

Hello, Given an electromagnetic wave that is, from a classical point-of-view, not circular polarized. Does that correspond in QM to photons with the ZERO spin eigenstate? Thanks in advance.

Homework Statement Consider the case of an atom with two unpaired electrons, both of which are in s-orbitals. Write the full basis of angular momentum eigenstates representing the coupled and uncoupled representations Homework Equations l=r×p lx=ypz-zpy ly=zpx-xpz lz=xpy-ypx l+=lx+ily...

If an operator (O) acts on a function ψ and transforms the function in a scalar manner as described below, it is said to be in an eigenstate: Oψ=kψ in this case, O is the operator and k some scalar value. My question is essentially if k=0, can this still be a valid eigenstate? for example, O...

Homework Statement Suppose that a state |Ψ> is an eigenstate of operator B, with eigenvalue bi. Homework Equations i. What is the expectation value of B? ii. What is the uncertainty of B? iii. Is |Ψi an eigenstate of B2 or not? iv. What is the uncertainty of B2? part B : Suppose, instead...

When a particle's position is measured, does the wavefunction collapse to the eigenstate of the measurement (a delta function), or do you account for the accuracy and precision of the measurement device by making the state be a mixture of eigenstates corresponding to the device's imperfect...

Homework Statement Consider the following set of eigenstates of a spin-J particle: |j,j > , ... , |j,m > , ... | j , -j > where \hbar^2 j(j+1) , \hbar m are the eigenvalues of J^2 and Jz, respectively. Is it always possible to rotate these states into each other? i.e. given |j,m> and...

Hi guys, I'm currently writing a document about decoherence and I'm trying to make a point about phase factors and how they are altered by an interaction with a surrounding environment. However, i don't want to only state this, i want to show it through basic QM. Is it possible to say the...

Homework Statement This isn't exactly a homework question, but I figured this would be the best subforum for this sort of thing. For the sake of a concrete example, let's just say my question is: Express the position operator's eigenstates in terms of the number operator's eigenstates...

If two operators commute my book says that "we can choose common eigenstates of the two." And I have seen it phrased like this in multiple other books. Does this mean that in general the eigenstates differ, but we can choose a set that is the same or what does it exactly mean in comparison to...

Hi everyone, my question seems pretty simple, but I couldn't find any answers. OK, so the neutrino flavour eigenstates are different to the neutrino mass eigenstates. And this is why neutrino oscillation is possible. But for the charged leptons (the electron, muon and tauon) the mass...

The way I understand it is when particles are entangled, when you measure one the entangled pair is instantly in a measured state. This question really goes to Copenhagen vs. MWI. If Eigenstates of the wave function are entangled, that seems to support MWI. If these Eigenstate are not...

Hello fellow physicsforumists. I am currently looking at the standard model and one of the key ingridients is to rotate the gauge eigenstates to the mass eigenstates by a transformation acting on their family index. The problem is that I can't really see what we are doing. The mass...

Hi, Both Ballentine in "Quantum Mechanics - a modern development" pages 160-162 and Sakurai in "Modern Quantum Mechanics" pages 193-196 use essentialy the same argument to show the existence of a set of eigenvectors of J² and J_z with integer spaced values of the J_z eigenvalues for fixed J²...

I haven't found a comparison like this in any book that I have been reading so let me explain. I decided that I will derive an equation for an energy levels of a particle in an infinite potential well in two ways. 1st I tried to derive it using the interval ##0<x<d## where ##d## is a width of...

Homework Statement Assume a Hilbert space with the basis vectors \left| 1 \right\rangle, \left| 2 \right\rangle and \left| 3 \right\rangle, and a Hamiltonian, which is described by the chosen basis as: H=\hbar J\left( \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\...

Homework Statement Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could...

Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right...

This might sound like a totally dumb question but anyway: QCD lagrangian in the limit of mu=md has flavor SU(2) symmetry with respect these two quarks. And we say that these quarks' wave functions are eigenstates of I and Iz. The question is why? Thanks.