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.

Filter by
Sorted by
Tagged with
1
vote
1answer
37 views

Eigenvalue and Eigenfunction for a particle trapped in a 1D infinite asymmetric potential well

As we're barely scratching the surface of Quantum Physics in class, we haven't been taught about asymmetric potential wells. However, I find it fascinating, moreover difficult, to find the eigenvalues ...
0
votes
0answers
48 views

Do eigenstates of the creation operator actually exist?

Regarding the eigenstate of creation operator $\hat{a}^\dagger$, the answer to this question shows that the eigenstate does not exist. However, it is stated in another answer, that the proof has ...
0
votes
1answer
62 views

Diagonalizing eigensystem to find normal modes of coupled oscillator

I've two equations of motion that arose in a coupled oscillators in a magnetic field $\rightarrow$ continuum problem in classical mechanics: \begin{eqnarray*} -\omega^2 X &=& - \omega_0^2 X (2 ...
0
votes
0answers
7 views

How to identify the multimodes of an optical fiber?

I use Lumerical mode software to simulate a photonic crystal fiber of NANF type. The solver gives me a lot of modes (solutions). I want to know how to know if this fiber is a single mode fiber or ...
2
votes
2answers
111 views

Weird quantum linear operator [closed]

For a problem sheet at uni, I need to find eigenvalues and normalised eigenstates of a linear operator. This operator is $\hat{Q}$ and is defined by its action on the normalised eigenstates of the ...
3
votes
0answers
52 views

Diagonalization of a Hamiltonian in a transformed reference frame

Under a time-dependent transformation $V(t)$ of the state vectors $|{\psi}\rangle$ \begin{equation} |\psi'(t)\rangle = V(t) |\psi(t)\rangle \end{equation} The Hamiltonian $H(t)$ has to transform as \...
1
vote
1answer
87 views

Determinant of differential operator $( \partial^2 + m^2)$

For a scalar field in QFT the generating functional is given as: $$ Z[J] = \int \left[ d\phi \right] \exp{\left( i\ S[\phi] + i \int d^4 x\ \phi (x) J(x) \right)} $$ with $ S = \frac{1}{2} \int d^4 x\ ...
0
votes
0answers
31 views

How to get eigenvalue equation with Transfer Matrix Method?

I'm studying a paper on physics entitled "Bound State of the One-Dimensional Dirac Equation for Scalar and Vector Double Square Well Potential" It states in the paper that "It is ...
-3
votes
0answers
45 views

How to get $E$ (Eigenvalue) or $E_n$ from this transcendental complex equation? [migrated]

I want to solve this transcendental complex equation, but I don't know the step by step to get E in form equation and value of E ...
0
votes
0answers
38 views

Fourier transform of positive momentum states only?

Suppose I have the momentum-space eigenstates of a system $\psi_n(p)$. I write the time-evolved momentum states as $\Psi(p,t) = \sum c_n\psi_n(p)e^{-i E_n t}$ where $E_n$ are the corresponding energy ...
1
vote
1answer
73 views

Wavefunction without any basis

I know that we can pick a set of basis (discrete or continuous) to represent the wave function. Can we represent a wave function without using any basis? If no, what is limiting us? If yes, what are ...
1
vote
1answer
97 views

Examples of the physical significance and importance of matrix diagonalization and eigenvalues for first year undergraduates? [closed]

To a student of physics, who is only exposed to the techniques of mathematical physics and read classical mechanics at the undergraduate level, but not quantum mechanics yet, how can we explain the ...
0
votes
1answer
85 views

Normalizing eigenvectors [closed]

Over the course of this quantum class I'm taking I've run into issues with properly normalizing my eigenvectors. Here is my TA's explanation of this particular example is done. I am lost as to where ...
1
vote
1answer
60 views

Confusion Regarding the Derivation of Graphene Dispersion Using Annihilation and Creation Operators

I am going through a text which derives the energy bands in graphene (https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/3/67057/files/2018/09/graphene_tight-binding_model-1ny95f1.pdf) and am stuck on a ...
1
vote
1answer
62 views

Meaning of The Eigenvalues and Eigenvectors of a Quantum Operator

This is more a check to ensure I know the physical meaning of eigenvectors and eigenvalues in quantum mechanics, and to ask the general community if this is wrong: On some observable, represented by ...
1
vote
1answer
56 views

Eigen-States of Creation Operator

I know the annihilation operator has eigen-states $\hat{\alpha} |\alpha \rangle = {\alpha} |\alpha \rangle $ I also know the creation operator $\alpha^{\dagger}$ has no eigenstates. Is the following ...
0
votes
1answer
25 views

Does the first-order energy correction in the degenerate case equals to the eigenvalues of the perturbation matrix?

According to Griffiths, the degenerate perturbation theory says that the first-order corrections to the energies are the eigenvalues of the perturbation matrix. Griffiths solves for the eigenvalues in ...
0
votes
1answer
37 views

Irreducible unitary representation of $\text{SU}(2)$ and multiplicity of $J_3$ eigenvalues

I searched a lot about this, but I can't find anything near to an answer. I'm trying to find an irreducible unitary representation of $\text{SU}(2)$ Lie group, so writing the generic element as $e^{i\...
1
vote
1answer
40 views

What is modal analysis?

I know what modal analysis is, and I know how to conduct one. I can get the eigenvalues and vectors (modes). Not a problem. However, I am lost at trying to understand the philosophy of what it is. By ...
0
votes
0answers
21 views

Good basis for coupled modes

Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system looks something like this: $$H=-\frac{J}{2} \sum_{m}b_{...
0
votes
1answer
41 views

Eigenvalues and eigenstates of a pair of spin-1/2 systems

I came across a problem in the context of degenerate perturbation theory. $ \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\braket}[2]{\langle{#1}|{#2}\rangle} \...
1
vote
1answer
69 views

Getting the time evolution relationship in QM

Considering the standard evolution for a generic quantum state $\psi(t)$, setting $\hbar=1$ we have: $$| \psi(t) \rangle = U |\psi(0) \rangle \hspace{5em} \text{where}\hspace{1em} U=\exp[-iH(t-t_0)] $...
0
votes
1answer
43 views

On Unitary Equivalent Observables

In J.J. Sakurai's Modern Quantum Mechanics, he introduces the concept of 'Unitary Equivalent Observables'. If $|a^{'}\rangle$ and $|b^{'}\rangle$ are the orthonormal bases eigenkets of two non-...
2
votes
2answers
47 views

Is $E=0$ included in the energy spectrum of the free particle in 1d?

In finding the eigenfunctions, $\psi_E$'s, of the free-particle Hamiltonian in 1d, $$ H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}, $$ with eigenvalues $E$'s, subject to the conditions that they are ...
0
votes
1answer
55 views

Physical meaning of eigenvalues in the heat equation problem

Let's consider the heat equation on a $\Omega \subset \mathbb{R}^2$ manifold with a boundary $\Gamma$, with initial and boundary conditions \begin{align} \dot{u}(\mathbf{r}, t) &= \Delta u(\mathbf{...
1
vote
2answers
145 views

If two operators are related to each other by a unitary operator, are their eigenvalues the same?

If we consider 2 linearly independent basis as follow: $$\{ |\psi_1 \rangle , |\psi_2 \rangle ... |\psi_n \rangle\}$$ $$\{ |\phi_1 \rangle , |\phi_2 \rangle ... |\phi_n \rangle\}$$ And they are ...
-2
votes
4answers
148 views

Imaginary oscillation frequencies of quadratic bosonic Hamiltonians [closed]

Consider the Hamiltonian \begin{equation} H = a^\dagger a + c^\dagger c + \text{i} \xi \left( a^\dagger - a \right) \left( c + c^\dagger \right), \end{equation} where $a$ and $c$ are bosonic ...
1
vote
1answer
25 views

Superposition of two same energy levels

Spatial part of the ground state energy eigenfunction for a quantum particle confined to one dimension with a harmonic oscillator potential energy is given by: $u_0(x)=(\frac{m\omega}{\pi \hbar})^{1/4}...
0
votes
0answers
14 views

Simple method for calculating and plotting the band structure of a known periodic potential?

A thousand pardons if this is trivial, but I've been stuck here for hours. I'm trying to compute the spectrum and eigenfunctions (i.e. band structure) of the eigenvalue equation $u'' + k^2\epsilon\...
0
votes
1answer
64 views

Eigenvalues in Floquet theory

After calculating Floquet Hamiltonian and then it's eigenvalues I stumbled upon a problem with ordering of eigenvalues. I am using eigen library for c++ and for every Floquet Hamiltonian for given ...
1
vote
1answer
61 views

Finding the probability of measuring a particular eigenvalue of an operator for a system after time evolution

Consider a quantum system with Hamiltonian H and consider the measurement of an observable $a_n$ associated with a different operator A. Initially the system is an eigenstate $|\phi_n \rangle$ with ...
1
vote
1answer
35 views

Spectrum of the angular momentum and angular momentum squared

I am trying to understand how to build the spectrum of the angular momentum; of course since different components of the angular momentum do not commute with each other we must chose only one ...
1
vote
2answers
49 views

Freedom in choosing elements/entries of an eigenvector

I want to understand why there is freedom in choosing entries of an eigenvectors on some instances. I will take up a particular Hamiltonian to explain this. $$H=H_0 \left[ {\begin{array}{ccc} 1 &...
0
votes
1answer
56 views

Doubt in a solved example from Quantum Mechanics: Concepts and Applications by Nouredine Zettili [closed]

Question 3.7 b) from Quantum Mechanics: Concepts and Applications by Nouredine Zettili, on page no. 188 (solved examples) - I understand all the solutions mentioned therein but can't figure out why ...
1
vote
1answer
38 views

Does degenerate matter have anything to do with the degeneracy of eigenvalues and eigenstates?

I came across degenerate matter (not the first time) after learning about degeneracy in eigenstates and eigenvalues. Are the two connected? Or is this just another use for the term?
0
votes
1answer
39 views

Expanding the quantum mechanical propagator in terms of the (non-degenerate) eigenvalues of the Hamiltonian

Could anyone please help me with this derivation? I am struggling to see how the Propagator Can be expanded out into the form This is a non-degenerate two-level system. Any help would be greatly ...
0
votes
2answers
79 views

Orthogonal eigenfunctions [closed]

I have to show that two eigenfunctions of an electron in a 1 dimensional infinite square well with different parity and different quantum numbers are orthogonal. I am attempting this by integrating ...
0
votes
0answers
32 views

Coordinate transformation via diagonalisation of the metric

Consider the following metric: \begin{equation} g_{\mu \nu}=\frac{1}{169}\left(\begin{array}{ccc} 160 & -12 & -36 \\ -12 & 153 & -48 \\ -36 & -48 & 25 \end{array}\right) \end{...
1
vote
4answers
59 views

Problem understanding eigenfunctions and operators

I've started studying quantum physics and I'm getting confused on some key concepts. What I understand so far: we have a mathematical object called a wave function from which we can extract quantities ...
0
votes
0answers
24 views

Measurement after entanglement: eigenvalue confusion

The following is written in my notes: Let $\hat{O}$ be a hermitian operator associated with an observable $O$, with eigenvectors $\mid\phi_n\rangle$ and eigenvalues $\lambda_n$: $$ \hat{O}=\sum_n \...
2
votes
3answers
99 views

Determine the energy eigenstates and eigenvectors [closed]

Question: Let $\{\vert\psi_1\rangle, \vert\psi_2\rangle \}$ be an orthonormal basis. Define Hamiltonian $\hat{H} = \alpha \left( \vert\psi_1\rangle \langle\psi_2\vert + \vert\psi_2\rangle\langle\psi_1\...
0
votes
5answers
99 views

How to pick out eigenvectors after solving for eigenvalues?

I'm currently doing a bit of quantum mechanics, and I can't figure out how to pick out eigenvectors. Let me explain through an example. An operator $A= \begin{bmatrix} 1 &0 &0 \\ 0&0 &...
26
votes
7answers
2k views

Why do we use Eigenvalues to represent Observed Values in Quantum Mechanics?

One of the postulates of quantum mechanics is that for every observable $A$, there corresponds a linear Hermitian operator $\hat A$, and when we measure the observable $A$, we get an eigenvalue of $\...
0
votes
0answers
63 views

Commutation rules of field operators and Dirac delta

My question concerns the commutation rules between bosonic fields operators in the case in which the bosons can assume only discrete positions. I have organised this post in two sections, in the ...
1
vote
0answers
5 views

Transform the constrained qubit-qutrit absolute separability Hilbert-Schmidt probability question into unconstrained form

A quantum-mechanical "density" matrix is Hermitian (self-adjoint), positive definite, having trace 1. In terms of its four ordered eigenvalues ($\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \...
0
votes
2answers
64 views

Solving for the momentum from eigenfunctions

When you have a solution to a time-dependent Schrodinger Equation, $$\Psi(x,t)=\exp\left({-\frac{i\hbar^2k_0^2t}{2m}}\right)\sin(k_0x), \tag{1}$$ and want to know the distribution of momentum at time ...
2
votes
2answers
130 views

Meaning of Eigenvalues for position operator

To each observable in quantum mechanics there is an operator corresponding to it. I don't understand what's the meaning of the eigenvalues of the $\hat{x}$ operator. Since $\hat{x}$ is hermitian, ...
1
vote
1answer
59 views

Doppler as an eigenvalue of the Lorentz transformation

It is a known fact that $$ \gamma (1\pm\beta) = \sqrt\frac{1\pm\beta}{1\mp\beta} $$ is an eigenvalue of the Lorentz transformation (which is a linear transformation). This is also (as stated in the ...
0
votes
1answer
91 views

Shared eigenbasis of commuting Operators

Suppose I have two Hamiltonian pieces $H_1$ and $H_2$ such that $[H_1,H_2]=0$. Then we know that the two pieces have shared eigenbasis. Assume both $H_1$ and $H_2$ have eigenvalues 2 and -2. Let $|\...
0
votes
1answer
187 views

Eigenvalues and Eigenstates of Number operator

I have been working through a problem. It has asked me to determine the eigenstates and corresponding eigenvalues of the number operator in a quantum harmonic oscillator; $$\hat{n}=\hat{a}_+\hat{a}_-$$...

1
2 3 4 5
12