# 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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### Why is the ground state of an atom never degenerate?

In this paper https://link.springer.com/article/10.1007/BF01391720 , Kellner argues that the ground state of the helium atom must be spherically symmetric because "it is known that the ground ...
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### Commutation relation confusion of ladder operators in Quantum Mechanics

Suppose that $X$ and $N$ are operators such that they follow the commutation relation $$[N,X]=cX$$ for some scalar c. In this Wikipedia article it is shown that if $|n \rangle$ is some eigenstate of ...
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### Valid Intuition? - Why observables are represented by eigenstates/eigenvalues

So I've been frustrated with the usual presentation of the operator formalism being presented as an axiom, and have been after a more intuitive explanation. Would the following intuition be considered ...
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### Wavefunction vs state vector

When I was first introduced in quantum mechanics I learned that the wavefunction $\psi (x, t)$ (one dimension for simplicity) contains all the information about the system like $x(t)$ in classical ...
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### Physical Interpretation of eigenvalues/eigenvectors of the density matrices of a given order

I was reading this paper by Per-Olov Löwdin and it discusses how density matrices can be used to represent/interpret the wavefunction. And, I had a question regarding how the eigenvalues and ...
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### Is an Hamiltonian like this non-degenerate? [closed]

If I have a system of only two energy levels $E_0$ and $E_1$ and an Hamiltonian $$H=\begin{pmatrix} E_0 & 0 \\ 0 & E_1 \\ \end{pmatrix}$$ can the Hamiltonian be both degenerate and non-...
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### Dirac notation notation equality

I'm going trough my quantum mechanics notes but I don't understand why: $$H|\phi_{m}\rangle\langle\phi_{n}|-|\phi_{m}\rangle\langle\phi_{n}|H=a_{m}\langle\phi_{n}|-a_{n}|\phi_{m}\rangle.$$ What is ...
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### Obtaining Dirac spectrum on unorientable manifold ($RP^n$) from orientable manifold

The Dirac spectrum for $S^n$ is well known along with its multiplicities. In Appendix D of https://arxiv.org/abs/1510.05663 author computes dirac spectrum of $RP^4$ from that of $S^4$. The argument ...
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### Time-independent Schrodinger equation from energy eigenvalue equation [duplicate]

$$\left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right]\langle x \mid E\rangle=E\langle x \mid E\rangle$$ is often referred to as the time-independent Schrödinger equation in ...
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### Why is radius of gyration for an ellipsoid derived from a tensor different than the formula?

The first method one can use to solve for radius of gyration, $R_g$, of an ellipsoid with semi-axes a, b, and c involves using the formula: $$R_g = \sqrt{\frac{a^2+b^2+c^2}{5}}$$ However, it appears ...
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