# 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 $$\nabla^2 \psi = -k^2 \psi$$ ...
1 vote
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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 ...
1 vote
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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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1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...