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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41 views

Measuring a state in a basis other than eigenbasis

Suppose I have a state expressed in its eigenbasis as follows. $\rho = \sum_i\lambda_i\vert i\rangle\langle i\vert$. It is now measured in some other basis $\{\vert x\rangle\}$ that is distinct from ...
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Is symmetrization $xp-px$ required for commutation $[H,x]=0$?

Given a Quantum Hamiltonian: $$\hat{H}=ax^2+bp^2$$ It does not commute with either $x$ or $p$. Suppose we have a Hamiltonian :$$H = k \hat{p}\hat{x}$$ why do we need it to be: $$H = k (\hat{p}\hat{x} -...
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Approximate eigenvectors of complex value eigenvalue [migrated]

I was studying some basic theory to approximate differential equations near their fixed points and some question regarding eigenvectors and eigenvalues occurred to me, here it is: If you have $2\...
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Electromagnetic Variational Principle and Concentration of Optical energy

While reading a book on photonic crystals, I came across the Electromagnetic Variational Principle. If the electric and magnetic fields are represented by the real part of $$\mathbf{H}(\mathbf{r}, t)...
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Collapse of wavefunction to its eigenfunction upon measurement

In quantum mechanics, it is postulated that to every observable, we have an associated operator. It is further postulated that when we do a measurement on a system, the measured value is one of the ...
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(How) can you tell that a given operator's eigenspectrum will feature degeneracy?

I am speaking about operators representing physical observables and am not interested in purely mathematical objects (if that's relevant to answering the question). I know that a degenerate ...
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Translation operator eigenvalues can be real and arbitrary?

Consider the translation in space operator in $1D$: $$D(a)=e^{-ia\hat{p}/\hbar}$$ It is unitary - $D(-a)=D^{\dagger}(a)=D^{-1}(a)$ - which implies that $D(a)$ has eigenvalues on the unit circle like ...
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Numerical way of finding energy spectrum of $N$-body Schrodinger equation

For a single particle trapped in a potential, one can discretize the Time Independent Schrodinger Equation and hence find the eigenvalues of the corresponding Hamiltonian by diagonalising numerically. ...
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How to choose boundary conditions for numerical solution of Schrodinger's equation whose solutions are expected to die out “at infinity”?

I am using the "Shooting method" for solving the TISE with a "reasonably arbitrary" potential in 1D,with boundary conditions such that the eigenfunctions $\psi_n\to0$ as $x\to\infty$(And another ...
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Floquet bandstructure calculation

In this paper "Photonic Floquet Topological Insulators" the authors calculate the bandstructure of a time-periodic Hamiltonian. They create a time-dependent tight-binding Hamiltonian via the Peierl's ...
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Finding the eigenvalues of a matrix with particular symmetry [migrated]

I have a matrix for which I want to get some analytical equations of the eigenvalues. The matrix is given as \begin{align} \mathbf A &= \begin{pmatrix} \epsilon_a & 0 & 0\\ 0 & \...
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Number Operator on the Product State of Identical Bosons

Suppose that we have a single photon (or any elementary boson) with the state $$\Phi_{1} = |n\rangle.$$ Suppose also that there is a two-particle system whose state is given by $$\Phi_{2} = |n\rangle_{...
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For $[A,B]=0$, if an eigenfunction of $A$ not an eigenfunction of $B$, does that imply degeneracy of one operator?

When two operators $A$ and $B$ commute, there can be functions which are eigenfunctions of $A$ but not that of $B$. For example, in case of the one-dimensional harmonic oscillator, any linear ...
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1answer
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Wavefunction of a particle on a ring ($E > V$) using WKB method

For a particle on a ring (with radius $R$ and changing angle $\theta$) with only kinetic energy ($V=0$) we get the expressions for the wavefunction (normalized) and eigenvalues $$\Psi_n (\theta) = \...
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What is a physical example of an observable with degenerate eigenvalues? [closed]

If eigenvalues of an observable have the physical meaning of a possible result after a measurement, what's the interpretation of degenerate eigenvalues, and what is an example of such an observable?
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If two operators commute, does it mean that every eigenfuction of one is also an eigenfunction of the other?

I have trouble interpreting the result of a problem. If we have a function $$\psi ( \theta , \phi) = e^{-3i\phi}cos \theta $$ and two operators $$A=\frac{\partial}{\partial \phi} $$ $$B=\frac{\partial}...
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Eigenvalues of a two particle system in a coupled vs. uncoupled basis

Consider a system of two distinguishable spin-1/2 particles with Hamiltonian \begin{align} H &= \frac{\alpha}{4} \vec{\sigma}_1 \cdot\vec{\sigma}_2.\\ \end{align} where $\vec{\sigma}_1 = (\sigma_x\...
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Is it wrong to assume that $|{-\vec r}⟩ = - |{\vec r}⟩$?

I found a problem that says the following: The 'even' operator is defined as: $$\Pi|{\vec r}\rangle = |{-\vec r}\rangle$$ Show that $\Pi$ is Hermitian and find $\Pi^2$ and its eigenvalues. All ...
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Moment of inertia tensor and symmetry of the object

What information does the moment of inertia tensor give on the structure of an item. I was told that its eigenvectors give the principal axes of the object. Do you know more about this?
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Ground state of hydrogen molecule using Heitler–London method (H-L)

I am reading The Theory of Magnetism I, by Mattis. In Chapter 2, a hydrogen molecule is studied in the following way: We have a Hamiltonian of a hydrogen molecule: $$H = H^0_1 + H^0_2 + H^\lambda \...
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Angular Momentum and Spherical Top

The Schrödinger equation for 2 nuclei at fixed distance $R$ can be transformed to relative and centre of mass coordinates and gives rise to the Eigenvalue problem $$\frac{{J}^2 \Psi_{jm}}{2\mu R^2} = ...
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How come the eigenvalues of the Hamiltonian represent energy levels when the Hamilton doesn't represent the energy of the system?

Like in the Hamiltonian for a particle in an electromagnetic field. This is not a conservative field so the Hamiltonian doesn't represent the energy of the system. And yet the time independent ...
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Does no-level-crossing theorem (aka avoided crossing) always hold in perturbation theory?

In perturbation, J.J. Sakurai Modern Quantum Mechanics Second Edition page 310 stated a no-level-crossing theorem stated that "a pair of energy levels connected by perturbation do not cross as ...
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Guessing eigenvalue solution

I am reading The Theory of Magnetism I, by Mattis. In Chapter 2, he proposes the following eigenproblem: $$ \left ( \begin{matrix} V & U \\ U^\dagger& V \end{matrix} \right ) \left ( \begin{...
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How do you find the Triangle Inequality from an Inertia Matrix?

If you have an inertia matrix of the form $$\begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}=I$$ If the matrix ...
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Determining What Eigenfunction Occurs With Wavefunction Collapse

Suppose an operator $O$ has eigenfunction normalized $f$ corresponding to eigenvalue $n.$ Of course, any function $cf$, with $c$ on the unit circle, is also a normalized eigenfunction. Thus, if a ...
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Quantum observables in nonstandard Hilbert space

Consider a Hermitian $(n \times n)$-matrix $A$, and a Hilbert space $\mathbb{C}^n$, foreseen with a nonstandard inner product. (An inner product $s(\cdot,\cdot)$ is standard if for any two vectors $x =...
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The energy eigenstates $\psi_n(x)$ are eigenstates of parity operator?

For one dimensional system system described by symmetric potential energies with the property $V(x)=V(-x)$, the energy eigenstates $\psi_n(x)$ are eigenstates of parity operator? Is the above ...
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Looking for a freeware software/app that can solve eigenvalue problems symbolically

I'm taking a quantum mechanics course and my homework involves extremely tedious algebra to solve symbolic eigenvalue problems. I'm looking for a software that I can give matrices with symbolic ...
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Confusion about calculating first order correction to energy eigenstate / state vector

I am trying to determine the first order correction to the ground state for a particle in an infinite square well with a given perturbation, $$V'(x) = \frac{2\pi^2 h^2}{mL^3} (\epsilon x- \frac{\...
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4×4 Cofactor Transpose Matrix calculation gone wrong, in Shankar's Principles of Quantum Mechanics

In Appendix A$.1$, Shankar, R; Principles of Quantum Mechanics, the cofactor transpose of a $3\times3$ matrix $M$ is given as (to be referred to as the first procedure by me) $$\overline{M}=\begin{...
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Spin-Spin Interaction: Ground State Degeneracy

I'm given the hamiltonian $$\hat{H}=\sum_{i=1}^{L-1}\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}_{i+1}$$ (reminiscent of a para- or ferromagnetism situation?) where $\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}...
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Are eigenfunctions always normalizable?

If $y$ is an eigenfunction which corresponds to the eigenvalue $a$ of the operator $A$: $$A\langle y|=a\langle y|$$ Can we assume that $$\int y^*y=1$$ ?
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Is the mass an eigenvalue of Dirac equation?

Writing the Dirac equation as: $$(i \hbar\gamma^{\mu}\frac{\partial}{\partial x^{\mu}})\psi = m \psi$$ it seems that $m$ is an eigenvalue of the operator of the left side, and we need to find the ...
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Proof that coherent states are eigenstates of annihilation operator [closed]

My goal is to prove that, for $|\lambda\rangle=N\exp(\lambda\hat{a}^\dagger)|0\rangle$ is an eigenvector of the operator $\hat a$. I took 2 approaches, but both make sense to me and I get different ...
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1answer
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Why is $|\alpha\rangle$ not eigenstate of $a^{\dagger}$ for $\alpha^*$

I know that even if we have: $$a |\alpha \rangle = \alpha |\alpha\rangle$$ We don't have: $$a^{\dagger} |\alpha \rangle = \alpha^* |\alpha\rangle$$ Actually as explained in the second answer here ...
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Is this an eigenfunction of a ladder operator? [duplicate]

$a^{\dagger}$ and $a$ are ladder operators: $a^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle \,\,\text{and}\,\, a|n\rangle= \sqrt{n}|n-1\rangle $ Is the state $|n\rangle$ an eigenfunction of $a\...
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Quantum Eigenvalues, measurements, chances not matching up

I've got an assignment with a given quantum state, denoted $\phi=\frac{1}{\sqrt{2}} \begin{bmatrix}0 \\ 1 \\ -1\end{bmatrix}$ and an operator for the observable B, given by $B=b\begin{bmatrix}0 & ...
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The diagonal representation of Pauli $y$ matrix?

The textbook says the eigenvalue of the Pauli y matrix is 1 and -1, the corresponding eigenvectors are, $$\sqrt{\frac{1}{2}} \begin{bmatrix} 1\\ i \end{bmatrix} , \sqrt{\frac{1}{2}} \begin{bmatrix} 1\...
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What is an interaction (Quantum Mechanics) and is the wavefunction collapse an objective phenomenon?

First of all, I'm an undergrad. student of engineering physics, so I must explicit my lack of formal knowledge on the subject and total confusion with its implications. I also understand it may be ...
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Hydrogen wave function and angular momentum operators [closed]

I'm working my way through an old exam assignment for my Introduction to Quantum mechanics and a few (parts of) questions have me a little confused. The assignment is about the hydrogen atom and its ...
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Relation between irreducible representations and eigenstates

When in quantum mechanics the Hamiltonian possesses some symmetry, then knowing the irreducible representations of the group to which the symmetry belongs gives information about the eigenstates. As ...
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Eigenstates and eigenvalues of the operator $xp+ px$ [duplicate]

It is a hermitian operator, right? So, what are its eigenstates and eigenvalues? It is known that it is the infinitesimal generator of the dilation group in 1d. But apparently, a nontrivial dilation ...
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93 views

Momentum operator and eigenvalue

Can we take the eigenvalue we obtained after momentum operator (of quantum mechanics) operates on the state vector of a system as the momentum of the system?
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Completeness relation, commuting operators

I have a question about some formulars our professor wrote on the black board. Let $\hat{Q}_{1},...,\hat{Q}_{N}$ be operators, which are a CSCO. We know now that there exists a set of eigenvectors $\{...
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Eigenvalues of the momentum operator in position basis

We know that the definition of the momentum operator $\hat{P_x}$ in an state space $\mathcal{E}$ is: $$\hat{P_x}|\psi\rangle=P_x|\psi\rangle$$ where $P_x \in \mathbb{R}$. However we also know that ...
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1answer
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Does the energy eigenvalue have a position dependence?

I would like to transform the time independent Schrodinger equation from position to momentum space, but I am stuck on one point: Does the energy eigenvalue act as a unchanged diagonal matrix or does ...
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doubt about Bogoliubov for diagonalize matrix

I have the following equations: $$\begin{pmatrix} \dfrac{d}{dt}C \\ \dfrac{d}{dt}C^{*} \end{pmatrix}= - \dfrac{1}{i} \begin{pmatrix} A& B \\ -B^{*} & -A^{*} \end{pmatrix} \begin{...
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Eigenfunctionals and their application in physics

Is there any sensible meaning of the term eigenfunctionals? The object I want to describe is a solution to the following equation $$ {\mathscr D}_x F[g] = f(x) F[g] $$ where $ {\mathscr D}_x$ is an ...
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1answer
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Eigenvalue for complex variable

I was trying to reproduce the results of an exercise where they calculate the normal modes of oscillation. $$\begin{pmatrix} \dfrac{d}{dt}C \\ \dfrac{d}{dt}C^{*} \end{pmatrix}= - \dfrac{1}{i} \...

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