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.
845 questions
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Eigenvalues of integrable Floquet system
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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Interpretation of a solution in Quantum Mechanics [closed]
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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What is the energy spectrum of two coupled quantum harmonic oscillators? [duplicate]
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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How to calculate the energy of two coupled bosonic cavity modes?
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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Question on proof of Bloch's theorem
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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Magnitude of basisvectors [duplicate]
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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How to obtain the general explicit form of the vector state and wavefunction in the case of a continuous degenerate spectrum
$\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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Finding the expectation value of an operator in the eigenbasis of a different operator
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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Computing the trace of powers of a transfer matrix when we can diagonalize its components
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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Why is the expectation value given by $\langle\Psi\!\! |A\,|\Psi\rangle$ in quantum mechanics?
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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How do I solve the 3D magnetic field of a Halbach Rotor?
I'm currently trying to equate two functions represented by unequal Fourier Bessel
series within a specific region. The coefficients have to be independent of any
variables, as their dependency would ...
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Spectral gap lower bound - according to thermal average
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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Spectral gap lower-bound for transverse Ising model?
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 ...
- 261
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Meaning of operating on non-eigenvectors? [closed]
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 ...
- 799
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Quantum particle in the ground state of a spherical box
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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