Eigenstates Definition and 188 Threads

I've a doubt about the following definition from PSE thread. The first answer says that the position representation of the position operator ##\hat{x}## is: $$\bra{x}\hat{x} = \bra{x}x$$ I believe there is a typo, it should actually be $$\bra{x}\hat{x} = x \bra{x}$$ Does it make sense ? Thanks.

A very basic doubt about a QM system (particle) with spin 1/2 (qbit). From the Bloch sphere representation we know that a qbit's pure state is represented by a point on the surface of the sphere. Picking a base, for instance the pair of vector/states ##\ket{\uparrow}## and ##\ket{\downarrow}##...

Firstly I have found the eigenstates for both the original well and the new well as the following $$\psi_{n,\frac{L}{2}} = \begin{cases} \sqrt{\frac{2}{L}} \cos{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ odd} \\ \sqrt{\frac{2}{L}} \sin{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ even}...

I'll risk a quick off-topic answer here, since I think it's straightforward QM, not vague "interpretation" stuff. :oldbiggrin: In QM (e.g., Ballentine p81), for a free particle, ##H = \frac12 \, M\, V\cdot V + E_0##. So in the ground state ##|E_0\rangle## we have ##H|E_0\rangle = E_0...

Going back to high school chemistry, i remember being taught that the electrons in an atom can each be identified with four quantum numbers - one for energy, two for angular momentum and one for spin. These numbers are integers except for the spin quantum numbers, which are either 1/2 or -1/2...

Apologies for an additional thread (could not delete the previous one which was not coherent). After reading this paper: https://link.springer.com/article/10.1007/s10701-021-00464-7 "Fast Vacuum Fluctuations and the Emergence of Quantum Mechanics" Gerard ’t Hooft I was struck by a general...

I'm really not sure what the question expects me to do here but here is what I do know. If the state is an eigenstate it should satisfy the eigenvalue equation for example; $$\hat{H} f_m^l = \lambda f_m^l$$ but is the question asking me to use each operator on each state? How do I know if...

I have multiple questions about eigenstates and eigenvalues. The Hilbert space is spanned by independent bases. The textbook said that the eigenvectors of observable spans the Hilbert space. Here comes the question. Do the eigenvectors of multiple observables span the same Hilber space? Here...

Given the unperturbed Hamiltonian ##H^0## and a small perturbating potential V. We have solved the original problem and have gotten a set of eigenvectors and eigenvalues of ##H^0##, and, say, two are degenerate: $$ H^0 \ket a = E^0 \ket a$$ $$ H^0 \ket b = E^0 \ket b$$ Let's make them...

Quantum mechanically, a spin 1/2 particle in a uniform magnetic field has two energy eigenstates ##\ket{up}## and ##\ket{down}## and rotational degrees of freedom (distinct from the energy eigenstates) about the axis of the magnetic field. this can be derived from the Pauli matrix commonly...

The matrix representation of a certain operator in a certain basis is $$\begin{bmatrix} 1 & 0 & 0 \\0 & 0 & -i \\ 0 & i & 0 \end{bmatrix} .$$ The eigenvalue problem leads to this equation $$0=det\begin{bmatrix} 1-\lambda & 0 & 0 \\0 & -\lambda & -i \\ 0 & i & -\lambda \end{bmatrix}...

With this information I concluded that the diagonal elements of ##\hat{A}## are equal to the eigenvalue ##a##, so ##\hat{A} = \begin{bmatrix} a & A_{12} & A_{13} \\ A_{21}& a & A_{23}\\A_{31} & A_{32} & a \end{bmatrix}## but I can't see how to go from this to the commuting relation, since I...

To show that when ##[J^2, H]=0 ## the propagator vanishes unless ##j_1 = j_2## , I did (##\hbar =1##) $$ K(j_1, m_1, j_2 m_2; t) = [jm, e^{-iHt}]= e^{iHt} (e^{iHt} jm e^{-iHt}) - e^{-iHt} jm $$ $$ = e^{iHt}[jm_H - jm] $$ So we have $$ \langle j_1 m_1 | [jm, e^{-iHt} ] | j_2 m_2 \rangle $$ $$ =...

On page 298 of Shankar's 'Principles of Quantum Mechanics' the author makes the statement : ""In an arbitrary ##\Omega## basis, ##\psi(\omega)## need not be even or odd, even if ##| \psi \rangle ## is a parity eigenstate. "" Can anyone show me how this is the case when in the X basis...

When we say energy levels of the hydrogen atom. Are that energies of the atom or of an electron in the atom? Also corresponding states? http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html Why energies are negative? E_n \propto \frac{-1}{n^2}

What is known about the eigenstates of the ##\phi^4## theory in QFT? Is there an informal understanding of how these states are organized in the nonperturbative regime? For example, are there known to be any bound states in any dimensions? How does the energy of a multiparticle eigenstate (if...

I couldn't quite answer, so looked at the solution. I just want to ensure I am undertsanding the answer correctly. The answer is given here on page 3. Q2a: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps4_sol.pdf Am I right in concluding that...

Momentum eigenstates can be written in form of e^(2*pi*x) how?? and also i have question how momentum is conserved as consequences of periodicity of wave function.

I do not know what I'm doing wrong but I'm working on the problem of finding the normalization constants for the energy eigenstate equation for a 1D plane wave that is traveling from the left into a potential barrier where E < V at the barrier. This is from Allan Adams' Lecture 12 of his 2013...

I started with the first of the relevant equations, replacing the p with the operator -iħ∇ and expanding the squared term to yield: H = (-ħ^2 / 2m)∇^2 + (iqħ/m)A·∇ + (q^2 / 2m)A^2 + qV But since A = (1/2)B x r (iqħ/m)A·∇ = (iqħ / 2m)(r x ∇)·B = -(q / 2m)L·B = -(qB_0 / 2m)L_z and A^2 =...

A measurement X collapses the wave function randomly into an eigenstate of X. Then if a different measurement Y is made the wave function will randomly collapse into an eigenstate of Y. So for example if you measure position, the wave function will collapse into a narrow peak. Now if you measure...

I simply use the equation above, and the eigenvalus whish yield: ##\hbar^2 [ s_1(s_1+1) + s_2(s_2+1) + m_1m_2 + \sqrt{s_1(s_1+1) - m_1(m_1+1)}\sqrt{s_2(s_2+1) - m_2(m_2-1)} + \sqrt{s_2(s_2+1) - m_2(m_2+1)}\sqrt{s_1(s_1+1) - m_1(m_1-1)}## Very straight forward. My issue is that I don't know...

Homework Statement Consider a particle which is confined in a one-dimensional box of size L, so that the position space wave function ψ(x) has to vanish at x = 0 and x = L. The energy operator is H = p2/2m + V (x), where the potential is V (x) = 0 for 0 < x < L, and V (x) = ∞ otherwise. Find...

PF has gotten so far over my head, I keep expecting to see this: :frown:

Homework Statement For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form $$L(p)=exp(i\phi J^{(21)})exp(i\theta...

Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates. I've previously shown that ##\hat{H} = \sum_j...

Homework Statement I am given the Rashba Hamiltonian which describes a 2D electron gas interacting with a perpendicular electric field, of the form $$H = \frac{p^2}{2m^2} + \frac{\alpha}{\hbar}\left(p_x \sigma_y - p_y \sigma_x\right)$$ I am asked to find the energy eigenvalues and...

Does saying "states of definite mass" is the same as saying "mass eigenstates"?

What are the different eigenstates of molecules that are most often used in chemistry?

Neutrinos were flavor eigenstates at the time of their decoupling from baryonic matter. Since they were not pure mass eigenstates, how do you take this fact into account if you try to study how they evolved as the universe expanded? Could we determine if the heaviest neutrino could be non...

Hi, I have come across two definitions of eigenstates (and eigenvalues), both of which I understand but I don't understand how the two are related: 1) An eigenstate is one where you get the original function back, usually with some multiple, which is called the eigenvalue. 2) An eigenstate...

I have actually read so much about density matrix and eigenstates today. I just want to know what particular situations when mixed states are eigenstates. Can this occur? Mixed states and eigenstates have one thing in common.. they have a value.. but I know mixed states aren't eigenstates...

As a hydrogen atom approaches a Neutron star, is the probability distribution of eigenstates of the electron in that atom influenced by the gravitational field of the star?

Hey everyone, I've been doing some quantum mechanics but I think I have yet to fully grasp the meaning of eigenstate. What I mean is, I understand that an eigenstate ##x## is such that, if we have an operator ##\hat{A}##, it satisfies ##\hat{A} x=\lambda x## and so ##\hat{A}## represents a...

Hello everyone, I am wondering if the eigenstates of Hermitian operators, which represent possible wavefunctions representing the system, are always stationary wavefunctions, i.e. the deriving probability distribution function is always time invariant. I would think so since these eigenstates...

This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...

In chapter 10 of his book "Quantum Field Theory of Point Particles and Strings", Hatfield treats what he calls the Schrodinger representation of QFT. He starts with a free scalar field and introduces field operators ## \hat \varphi(\vec x) ## and its eigenstates ## \hat \varphi(\vec...

For a time-dependent Hamiltonian, the Schrodinger equation is given by $$i\hbar\frac{\partial}{\partial t}|\alpha;t\rangle=H(t)|\alpha;t\rangle,$$ where the physical time-dependent state ##|\alpha;t\rangle## is given by $$|\alpha;t\rangle =...

Suppose I want to measure the momentum of a quantum system. What I do is I take the momentum operator and expand my wavefunction in term of the eigenfunctions of that operator, then I operate on the wavefunction with the operator and the reusult of the measurment is that the wavefunction...

Are Everett branches (or relative states) the eigenstates or the Hilbert subspaces (or others?)? Once in a branch (or world), what law of QM would be broken if you can cut off the branch you are sitting on and revert back to the global state vector (isn't the quantum eraser, etc. about...

Homework Statement The problem states consider A_hat=exp(b*(d/dx)). Then says ψ(x) is an eigenstate of A_hat with eigenvalue λ, then what kind of x dependence does the function ψ(x) have as x increases by b,2b,...? Homework EquationsThe Attempt at a Solution Started out by doing...

Homework Statement Two observables ##A_{1}## and ##A_{2}## which do not involve time explicitly, are known not to commute, ## [A_{1},A_{2}]\neq0, ## yet we also know that ##A_{1}## and ##A_{2}## both commute with the Hamiltonian: ## [A_{1},H]=0\text{, }[A_{2},H]=0. ## Prove that the energy...

Hello everyone! For my quantum mechanics class I have to study the problem of two quantum oscillator coupled to each other and in particular to find the eigenstates and eigenergies for a subspace of the Fock space. I know that, in general, to solve this kind of problem I have to diagonalize the...

I am relatively well versed when it comes to systems of spin, or doing the maths for them at least, but am unsure whether all of the {L2, Lz, (other required quantum numbers)} basis eigenstates for a general system of n particles of spins si, where si is the spin of the ith particle, can...

Recently I have reviewed by reference books to get a better understanding of the fundamentals of QFT and there is one thing I still do not understand. In the QFT derivation of the path integral formula, it seems that every book and online resource makes the assumption that for the field operator...

Hi, guys. In Povh's book, page 198, he says: "The strong force conserves the strangeness S and so the neutral kaons are in an eigenstate of the strong interaction." I do not see why this must be the case. My atempt to understand it: $$ŜĤ_s |K_0 \rangle = Ĥ_sŜ |K_0 \rangle$$ So $$Ŝ(Ĥ_s |K_0...

Problem: The particle in a 1D box [0, a] Eqs.: The general solution of the time-independent Schrödinger eq. may be written as ψ(x) = Acos(kx) + Bsin(kx), E = ħ2k2/2m. Imposing the boundary conditions ψ(0) = ψ(a) = 0 , we get immediately A = 0, ka = nπ (for any positive integer n). Using x' = x...

In a typical collider experiment, two particles, generally in approximate momentum eigenstates at ##t=-\infty##, are collided with each other and we measure the probability of finding particular outgoing momentum eigenstates at ##t=\infty##. Firstly, what does it mean for the particles to be in...

Homework Statement It is known that ##M_1,M_2, M_3## commute with each other but I don't see how the second line is achieved even though it says that it's using that ##M_1## and ##M_2## commute?

Homework Statement An extra electron is added to one atom of a tri-atomic molecule. The electron has equal probability to jump to either of the other two atoms. (a) Find the eigen-energies for the system. Assume that the new electron energy ##\bar{E_{0}}## is close to the non-hopping case...