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Physics

Questions tagged [eigenvalue]

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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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Consider Kicked field Ising model Hamiltonian given as follows. (I am following the this paper, relevant calculations are in appendix A.)$$H_I=2 J \sum_k\left[\cos k\left(\hat{b}_k^{\dagger} \hat{b}...
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I am a Math student and I am following a course in Quantum Mechanics. I am having some trouble understanding the physical solution of some problems. For example, consider the simple problem of a “...
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I am familiar with the single quantum harmonic oscillator: using either the algebraic ladder-operator method or by solving the Schrodinger equation, one obtains the well-known energy spectrum\begin{...
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I am familiar with calculating the energy of a single quantum harmonic oscillator, where the Hamiltonian is\begin{equation}\hat{H} = \omega \hat{a}^\dagger \hat{a}\end{equation}and the energy ...
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I am studying Bloch's theorem on the wave function in a periodic potential. In the proof, the translational operator $T_R$ is found to commute with the Hamiltonian $H$. The proof then concludes that ...
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We know that the inner product of a basis vector of an observable or operator with itself should be 1 and should be 0 when inner producted with any other basis vector of the same observable is $0$.But ...
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$\newcommand{\ket}[1]{|#1\rangle}$I'm trying to figure out what the general form of the vector state (and wave function) look like in the case of a continuous spectrum with (for now) discrete ...
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Given an operator $A$ with continuous eigenbasis $\left\{ \vert a' \rangle \right\}$ ($a'$ is the eigenvalue), its expectation value for a state vector $\vert \Psi \rangle$ is given by:$$\langle A \...
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Suppose that we have two operators $A$ and $B$ which satisfy $[A,B]\neq0, A^{T}=A, B^{T}=B$. I'm going to keep these operators vague to be concise, but I have precise definitions of $A$ and $B$ in ...
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Suppose a particle is in the quantum state $\vert \Psi \rangle$. Then the expectation value for an observable $A$ is given by:$$\langle \Psi \vert A \vert \Psi \rangle.$$But why is this the case? ...
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I'm currently trying to equate two functions represented by unequal Fourier Besselseries within a specific region. The coefficients have to be independent of anyvariables, as their dependency would ...
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Not a physicist, apologies in case I lack rigor.Given the thermal average:$$\langle H\rangle_\beta = \frac{\sum_j \lambda_je^{-\beta \lambda_j}}{\sum_j e^{-\beta \lambda_j}}.$$Assuming to collect ...
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Not a physicist, apologies in case I lack rigor.Consider the following Hamiltonian:$$H=\sum_j \gamma_j\sigma^z_j\sigma^z_{j+1} + h\sum_j \sigma^x_j.$$I am looking for a lower-bound to the spectral ...
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I understand that in quantum mechanics we can represent an observable with a matrix that has certain vectors as eigenvectors, and these correspond to observable states. But we already have the ability ...
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Consider a single non-relativistic particle in a spherical box of radius $R$. I want to find the lowest energy level (or the ground state energy). In this case, the Schrödinger equation is simple, ...

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