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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Analytic approximation of bound state energies for finite square well
Considering a fairly deep finite potential well given by:
$$U(x)= \begin{cases}
U_0 \ \ , \ \ |x|>\frac{a}{2}\\
0 \ \ , \ \ |x|\le \frac{a}{2}
\end{cases}$$
We know that the energies of the bound ...
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Eigenvalues of Cooper Pair Box with Matlab
I wanted to evaluate the eigenvalues of the Cooper Pair Box Hamiltonian in number basis
$$
\hat H=\sum_n 4E_c(n-n_g)^2 |n\rangle\langle n|-E_J/2(|n\rangle\langle n+1|+|n+1\rangle\langle n|)
$$
as a ...
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Can the non-degenerate perturbation theory formula for higher-order energy corrections be used in case of degenerate perturbation theory?
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Now from $\underline{\textbf{non-...
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Degree of degeneracy of energy levels and irreducible representations of Hamiltonian symmetry group
In case the Hamiltonian of a system has some non-trivial symmetry regarding the physical space, let's assume symmetry that can be described by a finite symmetry group (e.g. a point group symmetry) one ...
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The problem with commutation relation
Quantum mechanics states that operators of general coordinate $Q$ and conjugate momenta $P$ obey commutation relations of the type $QP-PQ= i a I$, where $a$ is a constant factor and $I$ is the ...
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How to measure the distribution of eigenenergies of a cold atomic cloud in an optical trap?
I have a cloud of cold atoms in an optical trap.
For example a BEC or thermal gas of 87Rb (you can adjust the number from 100 to 10^6) in a harmonic trap created by some far detuned laser.
You want to ...
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Graphical representations of the vector model of quantum angular momentum
This question is in reference to the book "Introduction to Modern Physics" by Richtmyer and Kennard, particularly in their discussion of the graphical representation of quantized angular ...
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Non-uniquness in buckling problems
I have a doubt. In linear elasticity (geometrically linear and also using linear constitutive law between stress and strain tensors), we have unique solutions. There is a uniqueness theorem taught in ...
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Transformation of eigenvalue equation
Let $f(x) = e^{i \omega x^0} u(x^1, x^2, x^3)$ be a scalar function on some spacetime manifold in coordinates $(x^{\mu})$. It satisfies the eigenvalue equation
$$\partial_0 f = i \omega f$$
Since $\...
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Use of the eigenfunction expansion of Green's function in physics and physical significance
In physics, we find extensive use of Green's function in almost all branches of physics. To find the Green's function of an ODE, we usually solve an equation of the form
$$L_xG(x,y)=\delta(x-y)\tag{1}$...
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Goldstein Chapter 6 Question
I have a question about a potential error in the $3^{\mathrm{rd}}$ edition of Goldstein's Classical Mechanics. In their exposition in Chapter 6 of small oscillations, the authors obtain the usual ...
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Why in QM the solution to Laguerre equations are ONLY Laguerre polynomials?
I am studying eigenfunction methods to solve Fokker-Planck equations and I got stuck with a calculation that is related to some typical problems in QM. In particular, the radial part of an hydrogen ...
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The eigenvalues of quantum harmonic oscillator [closed]
Can someone explain to me what is the green curve in the graphical rapresentation of energy levels for a quantum harmonic oscillator? I've always encounter this type of photo and nobody explains what ...
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Energy scale for random matrix theory
Consider a random matrix $M$ of size $d$ ($d$ being a very large integer) sampled from the GUE, that is, we take $M=G+G^*$ where the coefficients of $G$ are iid complex normal variables with zero mean ...
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How does the eigenvector interpretation of Lorentz transformations generalise to higher dimensions?
When I came across this YouTube video, I immediately realised how Lorentz transform corresponds to a linear transformation where the vectors representing the speed of light ($\vec{c}$ and $-\vec{c}$) ...
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Eigenvalue under Lorentz metric
I'm currently reading Hawking and Ellis and came across a statement regarding eigenvalues of a matrix under the Lorentz metric, represented by $diag\{1,1,1,-1\}$. A matrix $T$ is defined as follows:
\...
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Trouble understing a passage from Principles of quantum mechanics by Paul Dirac?
I'm currently reading Principles of Quantum Mechanics by Paul Dirac, specifically the 4th edition of 1958. There is a passage I'm having trouble to understand, I'll put the text here. For reference it'...
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Associated Hermite polynomials
In quantum mechanics, one of the most common problems to solve is the quantum harmonic oscillator.
\begin{equation}
\mathcal{H}=\frac{1}{2}p^2+\frac{1}{2}x^2
\end{equation}
The most elegant way to ...
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Symbol denoting parity eigenvalue
What is the symbol reserved for designating the parity of a parity eigenstate?
For example an eigenstate $\phi$ of the squared angular momentum operator $\hat{\mathbf{L}}^2$ is characterized by a ...
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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}\...
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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,...
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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 ...
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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 ...
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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
$$ \mathrm{Eig}\{\mathbf{T}\} = \left\{ -\left(\frac{\epsilon_0}{2}E^2+\...
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Different definitions of resolvent in matrix model
When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as
$$Z=\int[dM]e^{-NTrV(M)},$$
where $V(M)$ is a matrix valued function of $...
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Why does applying Ladder operators change the eigenfunction?
When applying a ladder operator to a spherical harmonic function, it spits out the function with a lower or higher magnetic quantum number.
My question is how does this abide by the classical ...
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"Eigenvalue" in Statistical Mechanics
In Pathria's "Statistical Mechanics", 3rd ed., on page 41, he is going over a discussion of the canonical ensemble and lays out the following definitions:
$$
\mathcal{N}=\sum_rn_r \quad \...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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\...
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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 ...
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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 $ψ'$ ...
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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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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 $\...
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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 ...
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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 ...
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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^{\...
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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 ...
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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 (...
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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 ...
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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 ...
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