# 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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### On the validity of energy eigenvalues obtained when solving the Schrödinger equation for a particle in a 1D box

I'm having trouble understanding the legitimacy of solving the Schrödinger equation for a particle confined in an infinite square well. Aren't we supposed to solve it for the whole space and not just ...
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### Property of the Hamiltonian's discrete spectrum

I have found a statement online saying that there must be an eigenvalue of the Hamiltonian inside the range $(E-\Delta H,E+\Delta H)$. Where the mean value and variance are defined for a random (...
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### Eigenvalues and Normal Modes in SHM

I'm reading Symmetries part from the textbook provided by MIT OCW Physics3 8.03SC course, but have a question about the condition to find normal modes of SHM. In the book they mentioned $S$ - symmetry ...
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### What kind of physical process would correspond to an operator that doesn’t result in an eigenvalue equation: $\hat{A}ψ=a ψ$?

I'm studying quantum mechanics and I'm trying to understand the concept of operators. They can be represented in general by the equation: $$\hat{A}ψ=ψ'.$$ Here the wavefunction is changed to $ψ'$ ...
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### Why is the spectrum of the momentum operator continuous for states containing 2 or more particles?

Consider a free theory and let $P^\mu = (H, \mathbf{P})$ be the 4-momentum operator. Since $P_\mu P^\mu = m^2$ is a Lorentz scalar, we get the relation $H^2 - |\mathbf{P}|^2 = m^2$. Here $H$ must be ...
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### What is the spectrum of a broken square drum?

Given a square drum with sides length equal to $L$, the squared raised frequencies are $(\pi m/L)^2 + (\pi n/L)^2$ with $m,n \in \mathbb{N}^*$. Here we have four boundary conditions (no vibration on ...
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### Eigenvalues of Hermitian Operators [duplicate]

In quantum mechanics, it's well-known that observables are associated as the eigenvalue of a Hermitian operator. My question is, is the converse also true? i.e. the eigenvalue of a Hermitian operator (...
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### Proof/Explanation for why the Hamiltonian operator's eigenvalues are the permitted energy values? [closed]

I'm looking for a proof as to why the Hamiltonian operator's eigenvalues in quantum mechanics are the permitted energies of a quantum particle. I am looking for an intuitive explanation as well as a ...
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### Eigenenergies shift of a two-level atom in a semi-classical electric field [closed]

Let's consider a two-level atom, with resonance frequency $\omega_a$, interacting with a semi-classical field, with frequency $\omega$ and Rabi-frequency $\Omega$. We could for example write our ...
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### Relation between eigenvalue equation for an operator and for its square

Consider a time indipendent Schrodinger problem: $$\hat{H}\psi_E(p) = E \psi_E(p)$$ with suitable boundary conditions. We know that $\psi_E$ are the eigenfunctions of $\hat{H}$. If we now consider 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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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 ...