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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$(E_n - E_m) \omega_{mn} = \sum_l (\omega_{ml}H_{ln}-H_{ml}\omega_{ln})$
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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
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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 ...
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Quantum physics problem [closed]
Challenging Quantum Physics Problem
Consider a particle in a one-dimensional potential well defined by the potential energy function:
$$V(x)=V₀x²+(V₀/4)x⁴$$
where V₀ is a positive constant. This ...
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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 ...
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Why does $Z \otimes Z$ have only two eigenvalues?
The observable $Z = \begin{bmatrix}1 & 0\\\ 0 & -1\end{bmatrix}$ on a single qubit system has two eigenvalues, 1 and -1, which means when measured, the system can give one of two possible ...
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Appearance of Sturm-Liouville Problem in Quantum Mechanics
I am considering the below ODE (ordinary differential equation), which can be studied using techniques from Sturm-Liouville theory.
The context is mathematical, but I was wondering if anyone ...
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Physical interpretation of the bra-ket notation
The bra-ket notation generally consists of 'ket', i.e. a vector, and a 'bra', i.e. some linear map that maps a vector to a number in the complex plane.
Now, using this bra-ket notation we can compute ...
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The eigenvectors obtained in the diagonalization in the paper "Two Soluble Models of an Antiferromagnetic Chain" by Lieb, Schultz and Mattis [closed]
The diagonalization involves few transformations, that transforms the anisotropic XY model in a matrix eigenvalue equation as $(A-B)(A+B)\phi_k= \lambda^2_k \phi_k$ where the matrix $(A-B)(A+B)$ has a ...
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Solving an equation with Laplace operator for a specific solution in spherical and polar coordinates
I am trying to solve an eigenvalue problem related to Laplace operator. I want to find a specific solution that only depends on the radial coordinate $r$ to the equation
$$\nabla^2 \psi = -k^2 \psi$$
...
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Why does the eigenvalues of an angular frequency matrix are the natural frequency? (INTUITION)
lest say we have a system of differential equations of some coupled oscillator such that:
$$\overrightarrow a = [w^2]\overrightarrow x$$
if we find the eigenvalues of $[w^2] = \lambda$ why those ...
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Calculating eigenvalues and eigenstates of an infinite dimensional Hamiltonian
Consider the Hamiltonian, $$H = E_{0} \sum_{m = - \infty}^{\infty}(|m⟩⟨m + 1| + h.c.),$$
where $E_{0}$ is an energy scale, $|m⟩$ are kets which can be used to form a complete basis and h.c. denotes
...
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Green Function associated to a periodic Schrodinger operator
If $V:\mathbb R\to \mathbb R$ is an $L$ periodic function in $\operatorname L^{\infty}$ we can always find two independent solutions for $$\psi''(x)+V(x)\psi(x)=E\psi(x)$$
$\psi^{\pm}(x)=e^{\pm ipx}\...
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Are the eigenvalues of the Dirac operator real?
In 3+1 dimensions, define the Dirac operator as
$$\tag{1} \not D = \gamma^\mu (\partial_\mu -i A_\mu )$$
where $A_\mu$ is a $U(1)$ gauge field. Define the following inner product between spinor fields:...
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Why/if separation of variables gives all eigenvalues of Hamiltonian for Hydrogen atom?
I'm self-studying QM through Griffith's book. I have a question on the time-independent Schrödinger equation with Coulomb's potential energy. The Schrödinger equation was solved to rediscover the Bohr ...
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Eigenvalues of position operator in higher dimensions is vector, not scalar?
If we accept that eigenvalues of operators show correspond with possible values measurements by that operator can take, then it makes sense to define the position operator $X$ on a "deterministic&...
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Angular Momentum Operators and Spherical Harmonics in Higher Dimensions
Suppose we have a $d$-dimensional quantum system with a rotationally symmetric Hamiltonian $\hat{H}$. Extrapolating from the two and three dimensional cases, one might expect that the eigenstates of $\...
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What exactly is the relation of the continuous spectra with intervals?
I've read in multiple quantum mechanics books that the name "continuous" of the continuous spectra is said continuous because in many examples it is an interval of values. But I couldn't ...
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Eigenvalue Decomposition of Operators
I have a question about the eigenvalue decomposition of an operator, more specifically about the matrix with the eigenvectors as columns.
If i have an operator that i decompose as follows:
$$
\hat{A} =...
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Would a two-particle pure momentum eigenstate scatter in QED?
Okay, I know a momentum eigenstate is not realizable in practice since it's not normalizable. But say we ignore the normalizability constraint, and just apply the rest of the theory. My question is, ...
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Eigenstates for quantized oscillator [closed]
Hi I am new to solid state physics and am reviewing a prior knowledge section and would like some clarification. The following appeared in the course notes:
From my understanding, Eigenstates are ...
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Numerical computation of a spectrum of the Hamiltonian
I'm working with $L_2(\mathbb{R}^n)$, and I have operators of the form
$$ H = - \frac{\hbar^2}{2 m} \Delta + V(x_1, \dots, x_n) $$
where $V(x)$ can be very involved dependence on $x$.
What's a good ...
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Eigenstates and Eigenvalues Problems in Quantum Mechanics (completeness/normalizable)
A few questions bother me a lot when I read Griffith's book.
Are all associated Legendre functions $\{P_{\ell}^m(\cos \theta)\}$ complete for any function $f(\theta)$?
For the Hamiltonian of a ...
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How limiting a potential affects the energy eigenvalues?
I am asking myself, how constraining a potential changes the energy eigenvalues.
With the WKB-Approximation-Method one can derive that the dependence of the eigenenergies regarding a potential $V(x) \...
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Show that a simultaneous eigenfunction of $\hat{L}_x$ and $\hat{L}_y$ leads to a contradiction [duplicate]
It is easy to show via the Canonical Commutation Relation $ [\hat{x}, \hat{p}_x] = i\hbar $ that the existence of a simultaneous eigenfunction of $\hat{x}$ and $\hat{p}_x$ leads to a contradiction, ...
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Is there any method for folding a Hamiltonian matrix to lower dimension?
I want to solve a tight-binding Hamiltonian which is $6\times6$. I'm only interested in two of the six bands which lie near zero energy at $\vec{k}=(0,\frac{4\pi}{3\sqrt{3}a})$. Is there any way to ...
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Approximate Hamiltonian from first-order perturbation theory
In the book Quantum Information, Computation, and Communication by Jaksch and Jones, the authors claim (on page 117) that in first-order perturbation theory, we can simply neglect the off-diagonal ...
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Can eigenvalues of the density matrix in the Lindblad equation be negative?
Can the density matrix in the Lindblad equation for an open mixed quantum system have (real part) negative eigenvalues?
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Why do an observable's eigenstates always form a basis?
Given an observable $Â$, any state can always be written as a linear combination of its eigenvectors, in other words its eigenvectors form a basis of the Hilbert space of all possible states.
I know ...
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Is it possible to build eigenstates of linear combinations of $\hat{P}$ and $\hat{X}$?
For the quantum harmonic oscillator, the position operator $\hat{X}$ has eigenstates saisfying $\hat{X}|x\rangle = x | x \rangle$. The momentum operator meanwhile acts like $\langle x | \hat{P} | \Psi ...
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Eigenvalues of Hamiltonian of two interacting spins
I want to compute the eigenvalues of the following Hamiltonian for a system of two interacting 1/2 spin particles :
$$
\begin{aligned}
\hat{H} & =A \overrightarrow{\hat{S}}_{(1)} \cdot \...
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Why do systems of $n$ coupled oscillators have $n$ normal modes?
Consider a linear system of $n$ differential equations with constant coefficients corresponding to a physical scenario where I have $n$ coupled oscillators (like $n$ masses attached by springs in ...
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How does the phase factor change after measurement?
Suppose that we have a nondegenerate quantum system with an orthonormal basis of eigenstates $|\psi_n\rangle$ with eigenvalues $E_n$. Consider a wavefunction $\psi(x,t)=\sum_n c_n|\psi_n(x,t)\rangle$. ...
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Velocity operator, its expression and eigenvalues
Cohen Tannoudji pp 215
Third postulate :The only possible result of the measurement of a physical quantity $\mathscr A$ is one of the eigenvalues of the corresponding observable $\mathbf A$.
pp 225
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Simultaneous eigenkets of two Hermitian and anti-commutative linear operators [closed]
I'm working through the problems at the end of the first chapter of the third edition of Sakurai and Napolitano's Modern Quantum Mechanics, and I've hit a snag with problem 1.18:
"Two Hermitian ...
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Does the value of Maximum Lyapunov exponent depend on the eigenvalues of the system?
I am currently reading this paper where on page 8, the authors say that:
Negative eigenvalues correspond to unstable systems.
This correlates with Figure 8 on page 12.
Does it mean that there is a ...
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Mass matrix for inverse seesaw mechanism
I am having trouble in diagonalizing analytically the mass matrix associated to the inverse seesaw mechanism with lagrangian:
$L= -y_1\bar{L}\tilde{H}N_{R_1} - -y_2\bar{L}\tilde{H}N_{R_2} + N_{R_1}^...
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Question about $\delta$-function in Fermi's Golden Rule
I am aware that Fermi's Golden Rule states the rate $k_{if}$ of a process which moves a quantum system from an initial state $i$ to a final state $f$ can be expressed as $$\frac{2\pi}{\hbar}|V_{if}|^2\...
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"Physically distinct" Hermitian operators with the same eigenspaces
I understand that Hermitian operators can be decomposed in terms of their eigenbasis:
\begin{equation}
H = \sum_i\lambda_i|i\rangle\langle i|
\end{equation}
where the $\lambda_i$ are all real. I've ...
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When spectrum of eigenvalues of Schrodinger equation is continuous or discrete? [duplicate]
There is a point which I do not understand about the Schrödinger equation. I will try to explain the issue.
Consider the Schrödinger equation:
$\hat H = \frac{ \hat p^2}{2m} + U(\hat x)$
and we are ...
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What does the finding the eigenvalue of a wavefunction physically mean?
In https://arxiv.org/abs/physics/0602145 (page 8), there is a passage which says that the discrete nature of the energy levels of an electron in a hydrogen atom comes from the fact that the solutions/...
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Meaning of eigenvalue of the position operator $\hat{x}$?
Apologies for asking a question which may be too basic. I understand at the conceptual level that a measurement collapses a wavefunction into a single spike, which will then evolve again immediately ...
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Spectrum of $f(T)$, where $T$ is a self-adjoint operator
Consider on a Hilbert space $\mathcal{H}$ a self-adjoint operator $T$ with spectrum given by $\sigma(T)=\{\lambda_n\}_{n \in \mathbb{N}} \subseteq \mathbb{R}$ (let's suppose for simplicity that the ...
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Energy gap of mean field model for transverse ising chain
Polynomials of spin operators with real coefficients appear not infrequently in Hamiltonians and in mean field theory, and there are often tricks to find their eigenvalues.
For example, the polynomial ...
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Quantum Particles on a circle and Circulant matrices [closed]
I have $n$ particles on a circle with the Hamiltonian
\begin{equation}
H = \sum_{n=1}^N \frac{p_n^2}{2m} + \frac{1}{2}m\omega^2 \sum_{n=1}^N (x_{n+1}-x_n)^2
\end{equation}
I need to find the energy ...
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Boundedness of a Hamiltonian and when does a Hamiltonian have a spectrum?
In the context of Quantum Field Theory we put restrictions on the potentials we can use. One argument is boundedness. If the potential is unbounded, for example $V(\phi) = \phi^3$, then `the field can ...
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