Eigenstate Definition and 88 Threads

Quantum mechanics, McIntyre, pg 62 For above spin ##1## Stern Gerlach experiment a set of results is "## \begin{array}{c} \mathcal{P}_{1 x}=\left.\left.\right|_{x}\langle 1 \mid 1\rangle\right|^{2}=\frac{1}{4} \\ \mathcal{P}_{0 x}=\left.\left.\right|_{x}\langle 0 \mid...

For the free particle in QM, the energy and momentum eigenstates are not physically realizable since they are not square integrable. So in that sense a particle cannot have a definite energy or momentum. What happens during measurement of say momentum or energy ? So we measure some...

To my understanding any quantum system can be describes as a linear combination of eigenstates or eigevectors of any hermetian operator, and that the eigen values represent the observable properties. But how does the system change with time? I suppose big systems with many particles change with...

The goal I am trying to achieve is to determine the momentum (2D) in a quantum system from the wavefunction values and the eigenergies. How would I go about this in a general manner? Any pointers to resources would be helpfull.

Calculate, with a relevant digit, the probability that the measure of the angular momentum $L ^2$ of a particle whose normalized wave function is \begin{equation} \Psi(r,\theta,\varphi)=sin^2(\theta)e^{-i\varphi}f(r) \end{equation} is strictly greater than ##12(\hbar)^2##...

If a system is in an eigenstate of the hamiltonian operator, the state of the system varies with time only with a "j exp(w t)" phase factor. So, the system is in a "stationary state": no variation with time of observable properties. But the system could in theory (for what I understand) be...

For part a, I have the following $$\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$$ but I am totally lost on how to proceed.

Given any system with discreet energy eigenstates, φn(x)e-iEnt . The φn are functions only of position. But are they also almost always real-valued?Thanks in advance.

I found this: Eigenstate: a quantum-mechanical state corresponding to an eigenvalue of a wave equation. would you please some one explain simply? Thanks

Certainly, ##\left [ A ,B \right ] \neq 0## does not mean that they do not have a same eigenstate. But how to construct a same eigenstate for ##L_x## and ##L_y## if it exists? Since ##L_x Y_l^m = \frac \hbar 2 \left ( \sqrt { l \left ( l+1 \right ) -m \left ( m+1 \right )} Y_l^{m+1} + \sqrt...

Please see this page and give me an advice. https://physics.stackexchange.com/questions/499269/simultanious-eigenstate-of-hubbard-hamiltonian-and-spin-operator-in-two-site-mod Known fact 1. If two operators ##A## and ##B## commute, ##[A,B]=0##, they have simultaneous eigenstates. That means...

The definition of coherent state $$|\phi\rangle =exp(\sum_{i}\phi_i \hat{a}^\dagger_i)|0\rangle $$ How can I show that the state is eigenstate of annihilation operator a? i.e. $$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$

Let's say I have a system whose time evolution looks something like this: This equation tells me that if I measure energy on it, I will get either energy reading ## E_0 ## or energy reading ## E_1 ## , when I do that, the system will "collapse" into one of the energy eigenstates, ## \psi_0 ##...

Homework Statement I'm trying to show the Eigenstate of S2 is 2ħ^2 given the matrix representations for Sx, Sy and Sz for a spin 1 particle Homework Equations Sx = ħ/√2 * \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} Sy = ħ/√2 * \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i...

If any superposition of quantum states is stable, why the preference for one of the eigenstates of the observable at the measurement? What is the attraction of such state?

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...

Homework Statement The Hamiltonian of a spin 1/2 particle is given by: $$H=g\overrightarrow { S }\cdot \overrightarrow { B } $$ where ##\overrightarrow { S }=\hbar \overrightarrow{\sigma }/2## is the spin operator and ##\overrightarrow { B }## is an external magnetic field. 1. Determine...

p\bar{p} pair is a CP eigenstate? As p and \bar{p} are fermions (the pair is assumed to be at S-state), the pair seems to be C's eigenstate with eigenvalue of -1. As they have opposite intrinsic parity, the pair state seems to be P's eigenstate with eigenvalue -1. Then isn't it CP eigenstate...

What is an eigenstate in relation to the Schodinger equation? We've been working with this stuff but I don't exactly understand what that is. I know of linear algebra eigenstates or eigenfunctions but I don't know if they are directly related.

Homework Statement The red box only Homework EquationsThe Attempt at a Solution I suppose we have to show L_3 (Π_1) | E,m> = λ (Π_1) | E,m> and H (Π_1) | E,m> = μ (Π_1) | E,m> And I guess there is something to do with the formula given? But they are in x_1 direction so what did they have...

Imagine a spatial frame of reference attached to a point-like particle. It has x=0 since it is at the origin and p=0 since it is at rest. Having definite position and momentum is normally considered a violation of the uncertainty principle. How would you resolve this paradox? 1. Position frames...

what does it mean that a particle is a pure eigenstate? could someone explain this to me simply ?

Homework Statement Prove that if a particle starts in a momentum eigenstate it will remain forever in a eigenstate given the potential c*y where c is a constant and y is a spatial variable. Homework Equations (h/i)d/dx is the momentum operator and a momentum eigenstate when put in the...

Let's say you have two operators A and B such that when they act on an eigenstate they yield a measurement of an observable quantity (so they're Hermitian). A and B do not commute, so they can't be measured simultaneously. My question is this: You have a matrix representation of A and B and...

Why is the probability of finding a particle in an eigenstate of position zero and not one? When we say we have located a particle at a particular position - why is it always in a superposition of position eigenstates about that position. But still the probability should not be zero. I need...

How do I know if some given state is and eigenstate of some given operator?

Hello :-) I have a small question for you :-) 1. Homework Statement The Operator e^{A} is definded bei the Taylor expanion e^{A} = \sum\nolimits_{n=0}^\infty \frac{A^n}{n!} . Prove that if |a \rangle is an eigenstate of A, that is if A|a\rangle = a|a\rangle, then |a\rangle is an...

Hi, I'm trying to learn some QFT at the moment, and I'm trying to understand how interactions/nonlinearities are handled with perturbation theory. I started by constructing a classical mechanical analogue, where I have a set of three coupled oscillators with a small nonlinearity added. The...

It would be really appreciated if somebody could clarify something for me: I know that stationary states are states of definite energy. But are all states of definite energy also stationary state? This question occurred to me when I considered the free particle(plane wave, not a Gaussian...

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...

Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...

Homework Statement Is state ψ(x) an energy eigenstate of the infinite square well? ψ(x) = aφ1(x) + bφ2(x) + cφ3(x) a,b, and c are constants Homework Equations Not sure... See attempt at solution. The Attempt at a Solution I have no idea how to solve, and my book does not address this type...

say we have some wavefunction |psi> and we want to find the probability of this wavefunction being in the state |q>. I get that the probability is given by P = |<q|psi>|^2 since we're projecting the wavefunction onto the basis state |q> then squaring it to give the probability density. However...

suppose that the momentum operator \hat p is acting on a momentum eigenstate | p \rangle such that we have the eigenvalue equation \hat p | p \rangle = p| p \rangle Now let's project \langle x | on the equation above and use the completeness relation \int | x\rangle \langle x | dx =\hat I we...

So, I was examining the ground state of a Bose-Hubbard dimer in the negligible interaction limit, which essentially amounts to constructing and diagonalizing a two-site hopping matrix that has the form H_{i,i+1}^{(n)} = H_{i+1,i}^{(n)} = - \sqrt{i}\sqrt{n-i+1}, with all other elements zero...

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...

Let's take a quantum state ##\Psi_p##, which is an eigenstate of momentum, i.e. ##\hat{P}^{\mu} \Psi_p = p^{\mu} \Psi_p##. Now, Weinberg states that if ##L(p')^{\mu}\,_{\nu}\, p^{\nu} = p'##, then ##\Psi_{p'} = N(p') U(L(p')) \Psi_{p}##, where ##N(p')## is a normalisation constant. How to...

Homework Statement Calculate ΔSx and ΔSy for an eigenstate S^z for a spin-1/2 particle. Check to see if the uncertainty relation ΔSxΔSy ≥ ħ|<Sz>|/2 is satisfied. Homework EquationsThe Attempt at a Solution I'm confused on where to start. As I am with most of this quantum stuff. From what...

Homework Statement Find the eigenvector of the annhilation operator a. Homework Equations a|n\rangle = \sqrt{n}|{n-1}\rangle The Attempt at a Solution Try to show this for an arbitrary wavefunction: |V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle a|V\rangle = a\sum_{n=1}^\infty c_{n}|n\rangle...

Homework Statement I am trying to solve the model analitically just for 2 sites to have a comparison between computational results. The problem is my professor keeps saying that the result should be a singlet ground state and a triplet of excited states, but when I compute it explicitally I...

Hello! If we consider a single-particle system, I understand that the measurement of an observable on this system will collapse the wave function of the system onto an eigenstate of the (observable) operator. Therefore, we know the state of the system immediately after the measurement. But as...

Homework Statement Calculate ##\triangle S_x## and ##\triangle S_y## for an eigenstate of ##\hat{S}_z## for a spin##-\frac12## particle. Check to see if the uncertainty relation ##\triangle S_x\triangle S_y\ge \hbar|\langle S_z\rangle|/2## is satisfied. Homework Equations ##S_x =\frac12(S_+...

Homework Statement For the three-dimensional harmonic oscillator H_{xyz} = \frac{p_x^2}{2m}+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}+\frac{1}{2}m \omega^2 x^2 + \frac{1}{2}m\omega^2 z^2 + \frac{1}{2}m\omega^2 z^2 Consider: | \alpha_1 > = \frac{1}{\sqrt{2}} (|n_x = 0, n_y = 0, n_z = 0> + |n_x = 0...

Homework Statement As the homework problem is written exactly: Consider the quantum mechanical system with only two stationary states |1> and |2> and energies E0 and 3E0, respectively. At t=0, the system is in the ground state and a constant perturbation <1|V|2>=<2|V|1>=E0 is switched on...

Hi, suppose that the operators $$\hat{A}$$ and $$\hat{B}$$ are Hermitean operators which do not commute corresponding to observables. Suppose further that $$\left|A\right>$$ is an Eigenstate of $$A$$ with eigenvalue a. Therefore, isn't the expectation value of the commutator in the eigenstate...

Homework Statement Given the a|n> = α|n-1>, show that α = √n : Homework Equations The Attempt at a Solution <n|a^{+}\hat {a}|n> = \alpha <n|a^{+}|n-1> = | \hat a|n>|^2 \alpha = \frac{<n|a^{+}\hat {a}|n>}{<n|a^{+}|n-1>} Taking the complex conjugate of both sides: \alpha* =...

Hi All, I was going through a paper on quantum simulations. Below is an extract from the paper; I would be obliged if anyone can help me to understand it: We will use eigenstate representation for transverse direction(HT) and real space for longitudinal direction(HL) Hamiltonians. HL=...

I thought it was the coherent state, but since that is an eigenstate of the annihilation operator, and the annihilation operator is not hermitian, then it has no corresponding observable, and I'm assuming that one can observe frequency. Thanks.

Homework Statement Hey dudes So here's the question: Consider the first excited Hydrogen atom eigenstate eigenstate \psi_{2,1,1}=R_{2,1}(r)Y_{11}(\theta, \phi) with Y_{11}≈e^{i\phi}sin(\theta). You may assume that Y_{11} is correctly normalized. (a)Show that \psi_{2,1,1} is orthogonal...

In a simple case of hydrogen, we can have simultaneous eigenstate of energy, angular momentum L_z, \hat{\vec{L}^2} . I'm thinking of constructing a state that is an eigenstate of energy but not the angular momentum: \left | \Psi \right > = c_1\left |n,l_1,m_1 \right > + c_2\left |n,l_2,m_2...