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2
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1answer
62 views

Eigenstates of a density matrix of continuous variables

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. The reduced density matrix of the second oscillator is given by: $$\rho_2 = ...
2
votes
1answer
204 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 ...
8
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0answers
277 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 ...
7
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0answers
95 views

Role of physics in the zeta function $\zeta$ and the Riemann hypothesis

Hilbert and Polya suggested a physical way to verify the Riemann hypotesis about $\zeta(x)$. If the Riemann hypotesis is true, we can state all eigenvalues of physical problems are real. What is the ...
5
votes
0answers
181 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 ...
2
votes
0answers
147 views

Adiabatic quantum evolution of single photon or biphoton system

The prerequisite for adiabatic quantum evolution of single photon or biphoton system is as follows. We have to prepare a single photon or biphoton quantum system which has a ground and a higher level ...
2
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0answers
40 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 ...
1
vote
0answers
33 views

Quick question on sketching wavefunction in well

Usually for an infinite well, the sketch for n=3 level is this: Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than ...
1
vote
0answers
233 views

Eigenvalues of the square of Pauli-Lubanski operator

Let's have Pauli-Lunanski 4-operator: $$ \hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma}, $$ which easy transforms to $$ \hat {W}^{\nu} = ...
1
vote
0answers
162 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 ...
1
vote
0answers
16 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 ...
1
vote
0answers
45 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 $\sigma(H)$ ...
0
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0answers
31 views

Nuclear Physics: Eigenvalues of nucleus angular moment

In the shell model of nuclei, when we talk about collective motions, we describe any nucleus deformation, expanding its radius on spherical harmonics base like this $R(\theta,\phi) = ...
0
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0answers
66 views

What is the eigenvalue of $J_z$?

In the calculation of the Zeeman Effect, the most important calculation is $$\langle J_z + S_z\rangle.$$ Suppose we want to find the Zeeman Effect for $(2p)^2$, meaning $l = 1$. In Sakurai's book, ...
0
votes
0answers
43 views

Reduced density matrix of coupled oscillators and entanglement entropy

This paper describes a way to find the entanglement entropy of $N$ entangled harmonic oscillators, after tracing out the first $n$. A few statement made within have royally confused me, and I haven't ...
0
votes
0answers
42 views

Showing that the maximum possible uncertainty for any observable is half the difference between its maximum and minimum eigenvalues

Show that the maximum possible uncertainty for any observable is $\frac{1}{2}|x_2 - x_1|$ where $x_1$ and $x_2$ are the extreme eigenvalues of X (Maximize $\Sigma_i p_ix_i^2 - (\Sigma_i p_ix_i)^2$) ...
0
votes
0answers
51 views

Meaning of Eigenvalues/Eigenvectors of a linear system of equations

I have a 41x41 system of linear equations (inhomogen) which I derived with Eureqa by describing the timecourse of fMRI haemodynamic data from a brain area as a function of the timecourses of 40 other ...
0
votes
0answers
205 views

Dirac Delta Potential and bound/scattered states

Why does the attractive Dirac Delta distribution (function) potential $V = \alpha\delta$(x) (for negative $\alpha$) yield both bound AND scattered states? Is this due to the definition of the Dirac ...