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# 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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### Finding the ground state of a Hamiltonian Matrix

I have numerically constructed a Hamiltonian matrix. I am currently finding the ground state by full diagonalisation of the matrix (with the GSL library) and finding the most negative eigenvalue and ...
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### Calculating eigenvalues from time independent Schrödinger equation [closed]

We calculate eigenstates by solving the time independent Schrödinger equation. The time independent Schrödinger equation is an energy eigenvalue problem, so we will get energy eigenstates as the ...
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### Eigenvalue problems in finite volume

Does every eigenvalue problem of a selfadjoint operator which should be solved in a finite volume have a discrete set of eigenvalues (no matter it is treated classically or in a quantum mechanical ...
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### Solving Majorana Hamiltonian

I am currently going through the Kitaev paper on quantum wires [1]. Given a Hamiltonian of the form $H = 1/2 i \epsilon b' b''$, where $b'$ and $b''$ are majorana modes, we can solve it by ...
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### Eigenfunctions of angular momentum operators in momentum representation

I want to know the eigenfunctions of $L^2$ and $L_z$ in the momentum representation. Do I need to Fourier transform the spherical harmonics?
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### Which eigenvectors should I ignore?

Solving an Euler-Bernoulli equation for a beam using Finite Element Method leads to the following equation $$M \ddot{\vec d}(t)+K \vec d(t)=\vec F$$ where $M$ is system mass matrix, $K$ system ...
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### Matrix representation of operators

I have a question about the matrix representation of quantum operators, in one of the books I'm reading I found this: Let ${\{\psi_{n}\}}$ be a complete orthonormal system and $\bf A$ a operator. ...
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### Energy eigenvalue of Hamiltonian operator

Hamiltonian is the total energy of the system. Then, is its eigenvalue $E$ also total energy of the system? What is the difference between them? Both of them are energy.
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### Eigenvalues of Hamiltonian for a system w/ three interacting spin degrees of freedom with spin-1/2

I have three interacting spin-1/2 particles and I want to find the energy eigenvalues of H. The Hamiltonian for the system is $H = \frac{J}{\hbar^2}(S_1\cdot S_2+S_2\cdot S_3+S_3\cdot S_1)$ (where J ...
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### Eigenvalue of Hermitian Matrices [closed]

The way I have proved the Eigen values of Hermitian matrices are real like this: I considered $H$ is a hermitian matrix. Operator applied in ket space: $\left<\psi|H | \psi\right>= \lambda$ ...
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### Multiplicity of degeneracy.

Let the Hamiltonian operator of a rigid dumbbell (such as $O_2$) be $\hat{H}=\hat{L}^2/(2I)$, whereas I is the moment of inertia and $L_i$ is the angular momentum of the molecule by rotation around ...
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### How to prove bound state's spectrum must be discrete and scattering state's spectrum must be continuous?

Consider $d$-dim Schrodinger equation without internal degree of freedom, that is, we don't consider spin etc. How to prove that bound state's spectrum must be discrete and scattering state's ...
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### Why are eigen-things better/more useful than non-eigen-things?

An eigenvector is a vector that doesn't change direction when a transformation is applied. So in the case of, say, an energy (Hamiltonian) eigenstate, it's a state that doesn't change 'direction' (...