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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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How can I interpret the normal modes of this mechanical system?

How can I interpret the normal modes of this mechanical system? The equations of motion for the system are as follows: $$\left[\begin{array}{ccc} m_{1}\\ & m_{2}\\ & & 0 \end{array}\...
fortega20's user avatar
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The eigenvectors associated to the continuous spectrum in Dirac formalism

I am comfused about the definition of an observable, eigenvectors and the spectrum in the physics litterature. All what I did understand from Dirac's monograph is that the state space is a complex ...
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Decoupling Linearly Coupled Wave Equations with Potentials

I'm currently working numerically with wave equations and I was wondering if one can always decouple two wave equations, with potentials, which are linearly coupled. The system I'm talking about is ...
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Green function in scattering theory

I'm having a bit of trouble with a step in scattering theory. Context: The Schrödinger equation for a two-body scattering problem can be written as: $$ (E - H_0) |\psi\rangle = V |\psi\rangle. $$ Here,...
Lucas's user avatar
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Bogoliubov de Gennes formalism for rotating systems

I have a question relating to the Bogoliubov de Gennes formalism. I am studying Bose Einstein condensates and I want to calculate the excitation energies of a system in one dimension (a ring with ...
ZaraReinm.'s user avatar
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Calculating Eigenkets of Perturbed Matrix for Second-Order Correction

Q: Find the eigenvalues of the 3x3 symmetric matrix $H$ using perturbation theory where all of the elements on the diagonal of $H$ are an order greater than the elements not on the diagonal. We can ...
PineappleThursday's user avatar
2 votes
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What is the physical interpretation of the eigenvalues of the Maxwell stress tensor?

The Wikipedia page on Maxwell stress tensor has a section on the eigenvalues of the Maxwell stress tensor, which is given by $$ \mathrm{Eig}\{\mathbf{T}\} = \left\{ -\left(\frac{\epsilon_0}{2}E^2+\...
Jonathan Huang's user avatar
4 votes
2 answers
100 views

Different definitions of resolvent in matrix model

When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as $$Z=\int[dM]e^{-NTrV(M)},$$ where $V(M)$ is a matrix valued function of $...
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Why does applying Ladder operators change the eigenfunction?

When applying a ladder operator to a spherical harmonic function, it spits out the function with a lower or higher magnetic quantum number. My question is how does this abide by the classical ...
ajox3412's user avatar
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"Eigenvalue" in Statistical Mechanics

In Pathria's "Statistical Mechanics", 3rd ed., on page 41, he is going over a discussion of the canonical ensemble and lays out the following definitions: $$ \mathcal{N}=\sum_rn_r \quad \...
michael b's user avatar
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Difference between the expectation value of an operator and operator applied to wave function?

Expectation value of any operator $\hat{Q}$ is defined as, $$ \left\langle\psi_n\mid\hat{Q}\mid \psi_n\right\rangle $$ and action of the operator $\hat{Q}$ on wavefunction is defined as $$ \hat{Q} \...
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Eigenstates of spin-1 Hamiltonian involving $x,y,z$ components

I am trying to find the energy eigenvalues and eigenstates of the spin-1 system with Hamiltonian operator $$H \enspace = \enspace a J_z^2 + b( J_x^2 - J_y^2 ) \quad , \qquad a, b \in \mathbb{R}$$ or ...
Octavius's user avatar
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2 answers
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Momentum Eigenstates for Particle in a Box [closed]

The following lines as attached as photos taken from Beiser Modern Physics (6th Edition): Now these equations and wavefunctions make no sense to me at all, first of all how are these wavefunctions ...
L lawliet's user avatar
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Momentum Eigenvalues for Particle in a Box

A question from my college exams is as follows: Find out the eigenfunctions and eigenvalues of the momentum of a particle of mass $m$ moving inside an infinite one-dimensional potential well of width ...
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Average Energy and Magnetization of One Site Hubbard Model

I have been trying to implement the exercises from Section 2 part B (for which $t = 0$, and only considering the effects of U) given in this set of lecture notes - Numerical Studies of Disordered ...
CuriousMind's user avatar
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Time evolution using non-Hermitian (not a PT symmetric) Hamiltonian

I am currently dealing with non-Hermitian hamiltonian and dynamics using it. In general the diagonalizable non-Hermitian matrix might have complex eigenvalues and the eigenvectors may not be ...
user101134's user avatar
1 vote
1 answer
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Understanding equation for eigenvalues of a Hamiltonian

I'm reading the paper Hamiltonian Truncation Study of Supersymmetric Quantum Mechanics. I'm not understanding a claim they make about the eigenvalues of a certain Hamiltonian. In particular, how eqn 3....
Gleeson's user avatar
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Pauli matrix exponentials [closed]

Just a short query to confirm my understanding. Given the Pauli-X operator $\hat{X}$ and it's eigenstates $|+\rangle:=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle:=\frac{1}{\sqrt{2}}(|0\...
John Doe's user avatar
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On the validity of energy eigenvalues obtained when solving the Schrödinger equation for a particle in a 1D box

I'm having trouble understanding the legitimacy of solving the Schrödinger equation for a particle confined in an infinite square well. Aren't we supposed to solve it for the whole space and not just ...
Arjun's user avatar
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8 votes
3 answers
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Property of the Hamiltonian's discrete spectrum

I have found a statement online saying that there must be an eigenvalue of the Hamiltonian inside the range $(E-\Delta H,E+\Delta H)$. Where the mean value and variance are defined for a random (...
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Eigenvalues and Normal Modes in SHM

I'm reading Symmetries part from the textbook provided by MIT OCW Physics3 8.03SC course, but have a question about the condition to find normal modes of SHM. In the book they mentioned $S$ - symmetry ...
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What kind of physical process would correspond to an operator that doesn’t result in an eigenvalue equation: $ \hat{A}ψ=a ψ$?

I'm studying quantum mechanics and I'm trying to understand the concept of operators. They can be represented in general by the equation: $$ \hat{A}ψ=ψ'. $$ Here the wavefunction is changed to $ψ'$ ...
bananenheld's user avatar
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2 votes
1 answer
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Why is the spectrum of the momentum operator continuous for states containing 2 or more particles?

Consider a free theory and let $P^\mu = (H, \mathbf{P})$ be the 4-momentum operator. Since $P_\mu P^\mu = m^2$ is a Lorentz scalar, we get the relation $H^2 - |\mathbf{P}|^2 = m^2$. Here $H$ must be ...
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2 votes
2 answers
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Change of basis in bra-ket notation [duplicate]

In the post Change of Basis in quantum mechanics using Bra-Ket notation , the accepted answer explores the relationship between an arbitrary operator $\hat{x}$ and another named $\hat{u}$, such that $\...
JBatswani's user avatar
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When an eigenvalue is degenerate, are there always other operators which distinguishes the degenerate states? [duplicate]

Familiarity with QM tells us that when an eigenvalue of an operator $\hat{A}$ is degenerate i.e. more than one eigenfunction of $\hat{A}$ has the same eigenvalue, there is usually another operator (or ...
Solidification's user avatar
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1 answer
66 views

TISE solutions should be combinations of eigenstates. Why this is not the case? [closed]

I would really appreciate some help with a question I have about the TISE (Sch. tipe independent equation). This is a linear equation and linear combination of the solution should be solution too. The ...
Domenico Giardino's user avatar
0 votes
1 answer
121 views

An operator with integer eigenvalues?

As is well-known, number operator $N=a^{\dagger}a$ with the commutation relation $[a,a^{\dagger}]=1$ has non-negative integer eigenvalues. I am looking for a similar expression for an operator ($A(a^{\...
Arian's user avatar
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1 answer
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What is the spectrum of a broken square drum?

Given a square drum with sides length equal to $L$, the squared raised frequencies are $(\pi m/L)^2 + (\pi n/L)^2 $ with $m,n \in \mathbb{N}^*$. Here we have four boundary conditions (no vibration on ...
Naima's user avatar
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1 vote
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Eigenvalues of Hermitian Operators [duplicate]

In quantum mechanics, it's well-known that observables are associated as the eigenvalue of a Hermitian operator. My question is, is the converse also true? i.e. the eigenvalue of a Hermitian operator (...
Jovan Alfian Djaja's user avatar
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2 answers
134 views

Proof/Explanation for why the Hamiltonian operator's eigenvalues are the permitted energy values? [closed]

I'm looking for a proof as to why the Hamiltonian operator's eigenvalues in quantum mechanics are the permitted energies of a quantum particle. I am looking for an intuitive explanation as well as a ...
XXb8's user avatar
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1 vote
2 answers
102 views

Eigenenergies shift of a two-level atom in a semi-classical electric field [closed]

Let's consider a two-level atom, with resonance frequency $\omega_a$, interacting with a semi-classical field, with frequency $\omega$ and Rabi-frequency $\Omega$. We could for example write our ...
Nicolas Schmid's user avatar
1 vote
2 answers
169 views

Relation between eigenvalue equation for an operator and for its square

Consider a time indipendent Schrodinger problem: $$\hat{H}\psi_E(p) = E \psi_E(p)$$ with suitable boundary conditions. We know that $\psi_E$ are the eigenfunctions of $\hat{H}$. If we now consider the ...
LolloBoldo's user avatar
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-4 votes
3 answers
119 views

Something of Schroedinger Equation bugs me [closed]

$\hat{H}\psi(x) = E\psi(x)$ $\psi(x) = c_{1}u_{1}(x) + c_{2}u_{2}(x) + ... = \sum_{n}c_{n}u_{n}(x)$ Here $\psi(x)$ is a superposition state of the eigenfunctions, and $u_{n}(x)$ is eigenfunction, $c_{...
QFT's user avatar
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0 answers
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Spectrum of Klein-Gordon operator in AdS Black Hole

I'm working on obtaining the spectrum of the Klein-Gordon operator in $AdS_2$ for black hole coordinates. To accomplish that, I first consider the problem in hyperbolic space $H_2$ and then Wick ...
MarcelRomp's user avatar
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0 answers
17 views

How can I understand the meaning of a Hermitian eigenvalue problem? [duplicate]

Suppose we have the $\textit{mode functions}$ given as $\textbf{u}_m(\textbf{x})$ which are defined by the following eigenvalue equation: \begin{equation} \nabla^2\textbf{u}_m(\textbf{x}) = -k_m^2\...
Rasmus Andersen's user avatar
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1 answer
80 views

Is it possible to analytically find the eigenvalues and eigenvectors of the following matrix?

$M$ is a $R \times R$ dimensional square matrix. The elements of $M$ are \begin{align} [M]_{nm} &= R \hspace{3 cm} when \hspace{.2 cm} n=m, \\ &= \frac{1-e^\frac{2\pi i R (n-m)}{f}}{1-e^...
Abu Saleh Musa's user avatar
2 votes
2 answers
336 views

Eigenfunctions of Momentum Operator [closed]

Suppose we have the 1-d wave function $\psi(x)=A\sin\left(\frac{p_0x}{\hbar}\right)$ and we want to know wheter this is an eigenfunction for $\hat{p}=-i\hbar\dfrac{d}{dx}$ The argument usually goes ...
Johann Wagner's user avatar
0 votes
1 answer
122 views

Commutable operators and eigenfunctions

My lecture notes says Since $[\hat{L^2} ,\hat{L_{z}}] = 0$ Then $Y(\theta,\phi)$ solves the below equations simultaneously: $$\hat{L^2}Y(\theta,\phi) = A Y(\theta,\phi)$$ $$\hat{L_{z}}Y(\theta,\phi) = ...
jensen paull's user avatar
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0 answers
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Degeneracy in a 2D planar cavity (box potential)

Assume a finite 2D planar cavity. One can write the energy of a photon in this cavity as $$ \begin{equation} E(k_x, k_y)=\hbar c \sqrt{k_x^2+k_y^2+k_z^2}, \end{equation} $$ where $k_z$ is fixed (hence ...
Andris Erglis's user avatar
1 vote
1 answer
126 views

Finding the zero-order wave-functions of the perturbed state

If I have an unperturbed Hamiltonian $\hat{H}_0$ which is four-fold degenerate corresponding to orthonormal eigenfunctions $\phi_1$, $\phi_2$, $\phi_3$, $\phi_4$ and I have some perturbing Hamiltonian ...
Thomas's user avatar
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1 vote
1 answer
44 views

Non-Hermitian PT-symmetric Interacting Hamiltonian with Real Spectra

The following hamiltonian is $\mathcal{PT}$-symmetric $$\mathcal{H} = -J \sum_{j = 1}^{2N} [ 1 + (-1)^j \delta ] [ c^{\dagger}_{j} c_{j+1} + h.c. ] + \imath \gamma \sum_{j = 1}^{2N} (-1)^j c^{\dagger}...
Snpr_Physics's user avatar
9 votes
3 answers
1k views

Why is the energy of the harmonics in a vibrating string not infinitesimal?

When you pluck a guitar string, initially the vibration is chaotic and complex, but the components of the vibration that aren't eigenmodes die out over time due destructive interference. This ...
silver's user avatar
  • 217
0 votes
1 answer
62 views

$(E_n - E_m) \omega_{mn} = \sum_l (\omega_{ml}H_{ln}-H_{ml}\omega_{ln})$

It's an expression from Landau Lifshitz Statistical physics, where $\omega$ is density matrix. I have troubles with going from eigenvalues $E_n$ and $E_m$ to Hamiltonian matrix elements, besides ...
Sombercy's user avatar
4 votes
2 answers
125 views

Eigenvalues of an time-ordered exponential operator

Let's consider a simple 1-qubit time-dependent Hamiltonian: $$H(t) = \delta(t) \sigma_x + \sigma_z \ ,$$ where $\delta(t)$ is some time-continuous (real-valued) function. Evolving $H(t)$ continuously ...
Mohan's user avatar
  • 83
1 vote
1 answer
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One-body density matrix has occupation numbers greater than 1

TL;DR - I computed the one-body density matrix (OBDM) stochastically via a method in a paper listed below, and it generates non-physical occupation numbers that 1) either has some negative values or 2)...
AlphaBetaGamma96's user avatar
0 votes
1 answer
56 views

Eigenvalues of Superpositions [closed]

First, consider the following superposition of two orthonormal spin states, $|1\rangle$ and $|0\rangle$: $$|\Psi\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |0\rangle)$$ $|\Psi\rangle$ would be an ...
Lory's user avatar
  • 1,073
1 vote
1 answer
106 views

Multipole expansion and spectral decomposition

We can always write a Hermitian operator in the form of its spectral decomposition: $\hat{A}=\sum_i \lambda_i | \chi_i(\boldsymbol{r})\rangle\langle\chi_i(\boldsymbol{r}')|$ where $\lambda_i$ are the ...
Guiste's user avatar
  • 464
0 votes
0 answers
43 views

Eigenvalues of a Hamiltonian for a system consisting of a linear chain of atoms

I was solving some one-dimensional chain of Rydberg Atoms(in Rydberg Blockade Configuration), which is basically a chain of atoms having alternate Rydberg atoms, there I encountered a certain type of ...
Bishal Sarkar's user avatar
1 vote
0 answers
45 views

Why energy eigenstates are extrema of the energy functional? [duplicate]

We have the energy functional of a system: $$E[\psi] = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}$$ and over numerous textbooks it is said that the eigenstates of the ...
Dorukhan Salepci's user avatar
0 votes
1 answer
165 views

Change of Basis and Eigenvalues

This is probably a silly question, but I just confused myself, and I'd love some clarification. I know that the definition of eigenvalues does not depend on a basis, and therefore they are manifestly ...
weirdmath's user avatar

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