The eigenvalue tag has no wiki summary.
2
votes
1answer
62 views
Eigenvectors of a 4D rotation, and their interpretation
Let us define a 4D rotation by using two unit quaternions: $$\mathring{q}_l=\frac{a+ib+jc+kd}{\left|a+ib+jc+kd\right|}$$ and $$\mathring{q}_r=\frac{e+ib+jc+kd}{\left|e+ib+jc+kd\right|}.$$ They differ ...
3
votes
0answers
59 views
Discreteness of set of energy eigenvalues
Given some potential $V$, we have the eigenvalue problem
$$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$
with the boundary condition
$$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$
If we ...
0
votes
1answer
40 views
Eigenvalue $a_n$
Q1:
In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
2
votes
1answer
63 views
NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms
From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other ...
2
votes
2answers
112 views
How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?
I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times:
$\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue
$W$.
...
0
votes
2answers
108 views
Geometrical interpretation of complex eigenvectors in a system of differential equations
Let's consider a system of differential equations in the form
$$\dot{X} = M X$$
in two dimensions ($X = (x(t), y(t))$).
In the case that $M$ has real values, it is easy to give a geometric ...
1
vote
0answers
26 views
Quantum graph theory: complex spectra
In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely ...
3
votes
1answer
65 views
Spectral properties of CFT
What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
1
vote
0answers
44 views
What class of theories/physical systems own finite/infinite complex eigenvalues?
What class of theories/physical systems own finite/infinite complex eigenvalues? I mean, do you know an updated list of physical systems or theories with complex eigenvalues-either finite or infinite? ...
1
vote
1answer
84 views
Mysterious spectra?
In my blog post Why riemannium? , I introduced the following idea. The infinite potential well in quantum mechanics, the harmonic oscillator and the Kepler (hygrogen-like) problem have energy spectra, ...
10
votes
3answers
330 views
How to tackle 'dot' product for spin matrices
I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
3
votes
1answer
133 views
How does a state in quantum mechanics evolve?
I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
$$
I am ...
3
votes
5answers
231 views
Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
1
vote
0answers
78 views
2D quantum well energy spectrum (analytical vs numerical)
I am trying to understand the energy spectrum difference between the analytical and the approximated solution for a quantum well.
The particle is inside a box with domain $\Omega=(0,0)$X$(1,1)$. For ...
3
votes
2answers
218 views
Quantum Mechanics Notation for BRA KET
I've been given this homework problem, but I do not understand its notation.
Please perform the following where the wavefunctions are the normalized eigenfunctions of the harmonic oscillator ...
2
votes
0answers
37 views
How to get from angular velocity to acquired phase for neutrino oscillations in matter?
I am reading Akhmedovs 2000 paper on parametric resonance, and I cannot figure out the math of this particular passage:
The difference of the neutrino eigenenergies in a matter of density $N_i$ is ...
2
votes
1answer
58 views
Schriffer Wolff Transformation - for first order change in eigenvalues
Step 1
Let me formulate the problem to convey my notation.
I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation
$$ U_A A\,\,U_A^{-1} = A_{diag}$$
Now the matrix is ...
2
votes
1answer
70 views
Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)
Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function ...
1
vote
0answers
15 views
could we obtain the potential (in one dimension) from the Gutzwiller trace?
to solve and obtain the potential of a 1-D Hamiltonian we must solve an integral equation
$$ N(E)= A \int_{0}^{E}\frac{V^{-1}(x)}{\sqrt{E-x}}$$
fro a some constant 'A' , then my question is since ...
4
votes
2answers
227 views
Eigenvalues of a quantum field?
Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t).
In quantum field theory, the classical field is promoted to an operator-valued ...
4
votes
0answers
105 views
Physical meaning of Laplace-Beltrami eigenfunctions?
The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of ...
7
votes
0answers
208 views
Lower bounds on spectral gaps of ferromagnetic spin-1/2 XXX Hamiltonians?
Question. Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamitlonians, on general interaction graphs, or tree-graphs?
I'm interested ...
1
vote
2answers
209 views
Angular Momentum Operators Non-Degenerate
Typically one writes simultaneous eigenstates of the angular momentum operators $J_3$ and $J^2$ as $|j,m\rangle$, where
$$J^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle$$
$$J_3 |j,m\rangle = \hbar ...
2
votes
2answers
359 views
Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]
Possible Duplicate:
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
2
votes
1answer
103 views
Perturbation method & eigenvalues
I have a problem but I don't understand the question. It says:
"Show that, to first order in energy, the eigenvalues are unchanged."
What does it mean?
It means that if the Hamiltonian has the ...
1
vote
1answer
42 views
growth condition for the potential
what growth conditions should the potential inside the Hamiltonian $ H=p^{2}+ V(x) $ has in order to get ALWAYS a discrete spectrum ??
for example how can we know for teh cases $ |x|^{a} $ , $ ...
2
votes
2answers
461 views
Using eigenvalues to determine the stability/behaviour of the system
first time I've been on physics.se but have used the math and cs before...
Anyway, here's my question:
If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the ...
1
vote
0answers
28 views
Random quantum systems with asymmetric Lifshitz tails?
For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
3
votes
1answer
81 views
Random Hankel matrix and eigenvalues distribution
I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices.
2
votes
1answer
81 views
When does the “norm of quasi-eigenvectors” matter in calculations? For which physical results are these even used?
Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the ...
2
votes
1answer
97 views
Are Quantum Physics and statistical theory always the same as semiclassical approximations?
Quantum Mechanics and Statistical physics is a bit hard , could we then study only the WKB approximation ?
In the form:
replace $ \sum_{n=0}^{\infty}exp(- \beta E_{n})=Z(\beta)\sim\iint ...
1
vote
1answer
149 views
Inverse of a sum of two easy matrices
Let $A$ be a symmetric positive semidefinite matrix and $I$ the identity matrix.
Given the linear equation
$$
y = A(A + \sigma^2I)^{-1} x
$$
Write $A$ in terms of its eigenvectors $|u_i\rangle$,
...
1
vote
2answers
110 views
eigenvalue staircase and hamiltonians
Let two Hamiltonians $H_{1}$ and $H_{2}$ be defined in such a manner that their eigenvalue staircases satisfy
$ N_{1} (E) = N_{2} (E)+A +O(E^{-1})$
What can we say about their potentials $ V_{1} ...
20
votes
4answers
758 views
In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?
A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential.
Is there a one to one correspondence between the potential and its spectrum?
If the ...
