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1
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3answers
56 views

How does one describe a state with a density matrix after measuring position?

My question is about position measurement in non relativistic quantum mechanics. I've been taught that when you measure the value of an observable for some state of a system described by ...
4
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1answer
76 views

Constructing differential equation from arbitrary Hamiltonian

Suppose I begin with the time-independent Schrodinger equation $$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$ ordinarily we specify the function $V$ and then solve for a ...
2
votes
1answer
65 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: for momentum space and for position space. How do we know that Ψ is not the Fourier transform of Φ but we suppose that its the other way around(Ψ would be ...
2
votes
1answer
26 views

Proton spin independent fine structure “Hamiltonian” $W_f$

To find the perturbation correction (fine structure) in the case of a degenerate energy $E_n^0$, we can diagonalize the operator $W_f^n$, the restriction of $W_f$ to the eigen-space associated to ...
1
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1answer
35 views

“Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit ...
1
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1answer
120 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 ...
0
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1answer
63 views

Why do electrons occupy in discrete energy states?

Why can't there be any continuous energy band in an atom?
8
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0answers
339 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
165 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
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0answers
234 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 ...
3
votes
0answers
113 views

The conjugate representation in $su(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $su(2)$ and that for the diagonal operator $T_3$, \begin{align} T_3\psi_1 ...
2
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0answers
50 views

Making An Energy Momentum Plot For A Rashba Model (Using Discretization)

I want to make a plot of the Energy versus the Momentum of the Rashba model, using discrete matrices. First Ill show how I did this for the free particle. Subsequently I will show what goes wrong for ...
2
votes
0answers
311 views

What is the physical meaning of complex eigenvalues?

I understand the mathematical origin of complex eigenvalues, and that complex eigenvalues come in pairs. But what is the meaning of the imaginary part? In particular I refer to an acoustic problem ...
2
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0answers
155 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
votes
0answers
44 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
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0answers
70 views

Selecting physical solutions in numerical eigenvalue problems

I try to solve a certain time-independent Schrodinger equation numerically, using the method of finite differences. My boundary conditions are such that the finite difference method gives me an ...
1
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0answers
75 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
1
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0answers
24 views

Can an axisymmetric solution produce antisymmetric eigenfunctions?

I'm solving a vibrating membrane. In order to simplify my calculations, it's tempting to assume axisymmetric behaviour. If I solve an axisymmetric problem, am I going to lose all the antisymmetric ...
1
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0answers
250 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
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0answers
18 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
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0answers
49 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
35 views

How to calculate the eigenenergies of a particle in a triangular billiard?

Suppose we take the Dirichlet boundary condition, namely the wave function must vanish on the boundary. How about a general n-polygon?
0
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0answers
30 views

Occurance and disappearance of degeneneracies in a periodic structure of (quantum) LC circuits

Introductory part I'm currently studying an analytical model of coupled LC circuits, in preparation for actually performing measurements on such structures. While the final goal will struggle with a ...
0
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0answers
29 views

Density of energy eigenstates in Heisenberg model

Consider the Heisenberg model for spin-$\frac{1}{2}$ particles in 1 dimension. Take the case described by the hamiltonian: $J\sum_{i,i+1}(\sigma^x_i\sigma^x_{i+1} + \sigma^y_i\sigma^y_{i+1} + ...
0
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0answers
39 views

Eigenstates of operators on constituent systems in tensor product space

Suppose I have two entangled physical systems $\mathcal{A}$ and $\mathcal{B}$ with respective hilbert spaces $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$. If $A,B$ are operators on ...
0
votes
0answers
56 views

Why is the matrix representation in the same basis not same for a density operator?

I have a $\rho : V \to V$ density operator of a $n$ dimensional space $V$ and $\{i\}=\{i_1,i_2..i_n\}$ is an orthonormal basis of this space. The density operator is defined as $$\rho=\sum ...
0
votes
0answers
60 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
0
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0answers
24 views

distance random matrix

In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by ...
0
votes
0answers
31 views

Energy eigen value for a perturbed free particle system

Suppose we have a one-dimensional free particle system and we introduce a perturbation like $V(x)=V_{0} \cos(Gx)$, where $G$ is the reciprocal lattice vector (it's a periodic perturbation, I think) ...
0
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0answers
36 views

What equation can be used to solve an ideal string/membrane in a non-vacuum medium?

I'm interested in the eigenmodes of the membrane for various mediums, such as vacuum, air, water, etc., which impose a damping effect on the membrane. This cannot be done by merely changing the value ...
0
votes
0answers
70 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
113 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
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0answers
350 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 ...