Questions tagged [eigenvalue]

A linear operator (including a matrix) acting on a non-zero *eigenvector* preserves its direction but, in general, scales its magnitude by a scalar quantity *λ* called the *eigenvalue* or characteristic value associated with that eigenvector. Even though it is normally used for linear operators, it may also extend to nonlinear operations, such as Schroeder functional composition, which evoke linear operations.

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594 views

Eigenvectors and eigenvalues of a metric

Suppose we have an $n$-dim Riemannian space $V_n$ endowed with a metric. More precisely, it is defined a $(0,2)$-type tensor field in $V_n$: $$g_{\mu\nu} = g_{\mu\nu}(x),\quad x\in V_n$$ Questions. ...
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Quantum pure quartic oscillator

It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian $$ \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{...
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Uniqueness of simultaneous eigen states of two linear operators

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigen states ...
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168 views

Probability of a specific energy state

We consider the normalized wave function: $$\psi(x,t) = \sqrt{\frac{2}{3}}\psi_0(x)\exp\left(\frac{-iE_0t}{\hbar}\right) + \sqrt{\frac{1}{3}}\psi_1(x)\exp\left(\frac{-iE_1t}{\hbar}\right) $$ To ...
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Solve eigenvalue problem with known constraint on one of the Eigenvalues

I have the following problem and would appreciate any help. I have a real, symmetric matrix M given by $$M=\begin{pmatrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{12} & m_{...
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Hamiltonian matrix for a delta potential with periodic boundary condition

I'm trying to find the energy eigenvalues of a Dirac delta potential: $$V(x)=-\alpha\delta(x)$$ with periodic boundary condition over some length $L$: $$\psi(x+L)=\psi(x)$$ and only even ...
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Unitary Transformation of Eigenstates

Suppose I have two operators, $A$ and $B$, with eigenstates $A \lvert a \rangle = a \lvert a \rangle$ and $B \lvert b \rangle = b \lvert b \rangle$, where $a$ and $b$ are all unique. Furthermore, ...
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What are the eigenvalues of $L_+$ and $L_-$?

I'm studying angular momentum in quantum mechanics. My question involves the operators $L_+=L_x+iL_y$ and $L_-=L_x-iL_y$; in a problem I have a Hamiltonian, $H$, depending an $L_y$, $L^2$ and $L_z$. ...
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171 views

Finding eigenfunctions using ladder operators

Using $L_1\pm iL_2$ as raising and lowering operators I need to obtain an expression for $Y_{l,l}(\theta,\phi)$ and after this I need to find $Y_{l,m}(\theta,\phi)$ for $(l,m)=(1,1),(1,0),(1,-1),(2,2),...
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If eigenstate for a Hermitian operator are orthonormal why are the energy eigenstates not so?

My Quantum Mechanics notes says: We found the orthonormality relation holds for any Hermitian operator eigenstates: Upon reading this I would assume that this holds for Energy eigenstates too. Then ...
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Does the dimensions of the wavefunction vector in Hilbert space depend on the number of eigenfunctions it is a superposition of?

I've seen people say that wavefunctions represented as vectors in a Hilbert space can (but don't have to) have infinite dimensions. So if a state vector requires X number of basis eigenfunctions ...
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Solving the Schrodinger equation for a free particle in momentum space

Time-independent form of the Schrodinger equation states $$\hat H\psi=E\psi$$ For a Hamiltonian in form of $$\hat H=\frac{\hat p^2}{2m}$$ Which indicates a free particle, In the position space is ...
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Quantum State Representation with Commuting Operators

Let $[A,B]=0$. Then, we can find a set of eigenvectors $\{|a_n,b_n\rangle\}$ common to both $A$ and $B$. According to this, and my own understanding, it makes sense to write an arbitrary quantum state ...
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How should one think about the concept of an eigenfunction in quantum mechanics?

I was working through some problems today and came across this one: Consider a particle in an infinitely deep potential 'well'. That is to say: $V(x) = 0$ for $-a/2<x<a/2$ and $V(x)=\infty$ ...
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248 views

Why do normal modes not exchange energy between each other?

I've been looking around for an answer to this and have had trouble learning about why this is. Does it have to do with the fact that the modes are eigenvalues?
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Propagator of Hermitian operator

Is the propagator of a hermitian operator always unitary? I am asking this because the propagator of in the Schrödinger equation is unitary and my book says this is to be expected since the ...
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4answers
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An simple eigenvalue problem in elementary Quantum Mechanics [closed]

if $\hat{A}$ is act on $\psi$ with an eigenvalue of $a$ $$\hat{A}\psi=a\psi$$ then how we can calculate $$\left(\tfrac{1}{\sqrt{2}}\right)^\hat{A}\psi=??$$
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Why does the first radial excitation of a particle in a 2D annulus $a<r<b$, for $b\gg a$ lie between the second and third azimuthal excitations?

Consider the quantum mechanics of a massive particle confined by infinite potential walls to a 2D annulus $a<r<b$, for which the Hamiltonian's eigenfunctions obey the stationary Schrödinger ...
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On different Hilbert spaces and energy spectra in the “Particle in a box” problem

Disclaimer In this question I suspect some of the used words are not precise so there is a possibility for misunderstanding here. If you know how to say more correctly or precisely - EDIT! Sorry in ...
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760 views

Can the wavefunction of a free particle always be written as some linear combination of more basic wavefunctions?

In various explanations I've read, from what I've gathered all particles have a wavefunction $\Psi(\mathbf{r},t)$ where $\mathbf{r}$ is the cartesian coordinates in however many dimensions you're ...
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Finding eigenvectors of an observable [closed]

I am trying to find the eigenvectors of the operator: $$ S_{u}=\frac{\hbar}{2}\left( \begin{matrix} \cos{\theta} & e^{-i\phi}\sin{\theta} \\ e^{i\phi}\sin{\theta} & -\...
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Quantum entropy taking the basis into account

The usual von Neumann entropy only takes the information of the eigenvalues into account: $S(\sum_j p_j |j\rangle \langle j |) = -\sum_j p_j \log p_j$. It is invariant under unitary transformations ...
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QM eigenstate expansion

Why sometime we use the integral to expand the eigenstates and sometime we use the sum to expand? now i am read the modern quantum mechanics J.J.Sakurai text and confusing
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Single particle operator in second quantization

I want to understand why we write in the formalism of second quantization for a single particle operator \begin{equation} \hat H=\sum_i \varepsilon_i \hat a_i^{\dagger} \hat a_i \end{equation} where ...
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1answer
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Show eigenvalue does not depend on magnetic quantum number $m$

We have a scalar operator $A$, being invariant under rotations which commutes with the angular momentum, i.e. $$[A,J_i]=0 \text{ where } i=x,y,z$$ $$[A,J^2]=0 $$ So eigenfunctions of $A$ can be ...
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1answer
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What is the form of the $n$-th order term of the perturbation series of an eigenvalue?

Suppose I have a matrix given by a sum $A=D+\epsilon B$, where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as power series in $\epsilon$. The leading order is just the ...
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2answers
345 views

Momentum eigenfunction

After solving the eigenvalue equation for the momentum operator, I get $u(x)=Ce^{ipx/\hbar}$, just like in Gasiorowicz's chapter 3. And then it says there: "...and the eigenvalue $p$ real, so that ...
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Feynman diagrams for eigenvalue perturbation theory

I posted this question in MathOverflow but was not lucky with the answers, so wil try here. Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, ...
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3answers
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How does an operator on the form $e^{\beta \hat{H}}$ act on a eigenstate?

The notion of operators in exponentials is a bit confusing to me. I know that that in some cases one can use the Taylor series of $e^x$, but how do you work with them when that's not the case?
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Eigenvalues, Hermitian operators and observables in quantum mechanics

Consider a hermitian operator. So a) in a space of infinite dimension its eigenvectors are a base. b) in a finite-dimensional space the matrix that represents the hermitian operator is always ...
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Are there multiple equivalent ways of writing an eigenstate?

So my problem here is that I'm confused about how to solve for the eigenstates corresponding to certain eigenvalues. For my problem I have the Hamiltonian $$ H=E_0 \begin{pmatrix} 3 & 5i \\ -5i ...
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Proof of Bloch theorem

On many websites, it is written that translation operator $\hat{T}$ commutes with Hamiltonian $\hat{H}$, and thus they share the same eigenstates: $\hat{H} |\psi> = E |\psi>$, $\hat{T} |\psi&...
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1answer
327 views

Calculating eigenvalues (Schrödinger stationary states)

I'm trying to determine the stationary states and the corresponding energies. My system has an angular momentum that is given by quantum number $l=1$, and the eigenvectors to $L_z$ are given as $|+1\...
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Quantum harmonic oscillator: How can we know that the lowest eigenvalue for the operator $A^{\dagger}A$ is zero and not a positive number? [duplicate]

With operator methods we can set the Hamiltonian of the harmonic oscillator in the following form: $$\hat{H}=\hbar \omega(A^{\dagger}A+1/2).$$ My question is that how can we know that the lowest ...
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627 views

How to use numerical methods to determine the eigenvalues of TISE?

I've successfully used numerical methods to derive the TISE solution for a Simple Harmonic Oscillator potential problem. I'm able to plot graphs of the Wave equation. Is there a way which I can make ...
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1answer
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Are eigenvalues in quantum mechanics related to eigenfunctions (in the PDE sense) or to linear algebra and eigenvectors?

I'm in 10th grade and a beginner in the amazing world of quantum physics, I want to become a mathematician but I like quantum mechanics as well. The eigenvalues in Schrödinger wave equation used to ...
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Are there any non-physical eigenstates of interacting QFT?

While reading about the Källén-Lehmann representation I came across the definition of eigenstates in general QFT. As $\vec{p}$ (total momentum) and $H$ commute they can be simultaneously diagonalized, ...
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A mismatch in quantum mechanics

One of the basic rules of Quantum mechanics is that after a measurement of an observable, the wavefunction is an eigenstate and any subsequent measurement will give the same result. This is not so in ...
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How does one obtain observables from a wave function?

I'm beginning to study the quantum chemistry (my background is computer science and computational mathematics) and I'm not sure if I understand well the basic concepts, like wave function and ...
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What is the problem of having an inertia tensor not satisfying the triangle inequality?

It is well known that rigid body inertia tensors are 3 by 3 positive semidefinite matrices, which is the same as saying that their eigenvalues are all non-negative. A little less known is the fact ...
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Examples of problems that are easier by knowing eigenvalues/vectors [closed]

First at all I have to say that I've done this questions a couple of hours ago in the Math section of this page, but I think that there could possible be more answers here then there because I'm ...
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Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
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2answers
793 views

Momentum eigenfunctions

Is $e^{-\frac{|x|}{a}}$ an eigenfunction of momentum? If we apply the momentum operator $\hat{P}=-i\hbar\frac{\partial }{\partial x}$ we get: $$ -i\hbar\frac{\partial }{\partial x}e^{\frac{|x|}{a}}...
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Has the asymptotic theory of eigenvalues of infinite matrixes already been applied to vibrations analysis?

My question is reffering to the masses/springs model of a material, like the one presented in this article http://www.laserpablo.com/baseball/Kagan/UnderstandingCOR-v2.pdf. If one treates a long ...
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Waveguide eigenmode weak-form (Comsol)

Maxwell equations are: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},\,\,\,\,\,\,\,\,\nabla \times \mathbf{H} = +\frac{\partial \mathbf{D}}{\partial t}$$ Subject to the ...
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Why are the energy levels of a simple harmonic oscillator equally spaced?

The energy level of a simple harmonic oscillator is $E_n=(n+\frac{1}{2})\hbar\omega$. Is there any physical explanation why these levels are equally spaced ($= \hbar\omega$)? Maybe this link can be ...
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Can you write the Einstein equation as an eigenvalue equation for analytical solutions (to local problems) on non-flat spacetimes

In reading about various local solutions to Einstein's field equation it is easy to forget that they almost all assume a flat background spacetime (at least asymptotically). Considering this made me ...
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1answer
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Boundary conditions for the numerical particle in a box example

I want to solve the one-dimensional Schrödinger equation for the particle in a box example, and want to force the wavefunctions to zero on the boundaries. I am using the matrix, \begin{equation} \hat{...
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3answers
278 views

Trouble with position operator in quantum mechanics

I'm having some trouble with understanding the derivation of the action of the $X$ operator. It seems to be a result of the notation used and not a property of itself. The usual argument is to ...
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Eigenvalue/Eigenstate of Hamilton with 2 Spin particles [closed]

I got a task, I don't quite know how to solve. I've got the following Hamiltonian: $$ \hat H = \frac{B}{\hbar^2}\hat{\mathbf S}_1\cdot \hat{\mathbf S}_2+\frac{C}{\hbar}\left(\hat S_{1z}+\hat S_{...