The eigenvalue tag has no wiki summary.
13
votes
6answers
1k views
Why are Only Real Things Measurable?
Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
1
vote
1answer
51 views
Weird Behaviour of the act of measurement to a quantum system
I and my friend were disputing about some weird behaviour of the act of measuring some observables quantities e.g. Energy, position.
But I still don't think what he said is strictly true.
He said" ...
2
votes
1answer
50 views
Relationship between two eigenfunctions of the time-independent Schrödinger Equation in one dimension?
What is the relationship between two eigenfunctions of the time-independent Schrödinger Equation (in one spatial dimension) if they both have the same eigenvalue?
2
votes
2answers
79 views
Angular momentum of quantum system
Problem:
A physical system is in the common eigenstate of $\hat{L^2}$ and $\hat{L_z}$. Calculate the following quantities: $\langle L_x\rangle,\langle L_y\rangle,\langle L_z\rangle,\langle L_x L_y + ...
0
votes
2answers
171 views
Expectation Values in Quantum Mechanics
Why is the expectation value what it is? Why don't you apply the operator, then multiply that by it's conjugate?
1
vote
2answers
71 views
Complex Versus Real Wave Velocities in Quantum Mechanics
There's a fantastic quote in Schrodinger's second 1926 paper1 that apparently provides some motivation for the discrete energy levels (I think) that I'm having trouble interpreting:
I would not ...
2
votes
1answer
56 views
Is continuous evolution from one eigenstate of operator $O$ to another $O$-eigenstate possible?
Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics.
Does this entail that a quantum system cannot continuously evolve from one eigenstate ...
1
vote
1answer
64 views
Total angular momentum - single electron
I have been dealing with total angular momentum of the single electron which is outside the closed shells in which sum of the angular momentums is zero.
My book says that total atomic angular ...
1
vote
0answers
42 views
Arbitrarily using Sin and Cos as eigenfunctions of a Hamiltonian? [closed]
In the context of quantum optics, the rotating wave Hamiltonian can be written:
$\hbar\begin{pmatrix}
-\Delta & \Omega/2\\
\Omega/2 & 0
\end{pmatrix}$
The eigenvalues can then be calculated ...
1
vote
0answers
127 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
1answer
107 views
Diagonalization of a hamiltonian for a quantum wire with proximity-induced superconductivity
I'm trying to diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor it's all fine. However, with a superconductor I don't get the ...
0
votes
0answers
91 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 ...
8
votes
2answers
236 views
Bounded and Unbounded (Scattering) States in Quantum Mechanics
I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state ...
0
votes
2answers
220 views
Finite potential well problem - calculating the ground state
1. The problem statement, all variables and given/known data
Electron of is in a 1-D potential well of depth $20eV$ width $d=0.2 nm$ in his ground
state $N=1$. What is the energy of the ground ...
1
vote
2answers
179 views
The Energy Eigenvalue of a Wavefunction
I've been reading an introduction to quantum mechanics online, and while constructing the Schrodinger equation for a free particle, the equation $i\hbar \frac{d \Psi}{dt}=\hbar\omega\Psi$ is obtained. ...
2
votes
0answers
128 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 ...
-1
votes
1answer
82 views
Wavefunction operators and the observable [closed]
So I got this from the exam I had yesterday. I couldn't really answer it other and it played on my mind through the night
Show that if a wave function $\psi$ , is an eigenfunction of an operator [Q], ...
-2
votes
2answers
91 views
Determine whether the ground state is an eigenfunction of [p] and of [p^2] [closed]
Consider a particle confined in an infinite square well potential of width L,
$$V(x)=\left\{ \begin{array}{ll}\infty, &{\rm for}\ (x \le 0)\vee (x \ge L) \\0, &{\rm for} \ 0 < x < L ...
2
votes
1answer
204 views
Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?
These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
4
votes
3answers
224 views
Momentum of particle in a box
Take a unit box, the energy eigenfunctions are $\sin(n\pi x)$ (ignoring normalization constant) inside the box and 0 outside. I have read that there is no momentum operator for a particle in a box, ...
2
votes
1answer
202 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 ...
13
votes
1answer
311 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
56 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
109 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
186 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
$W$.
...
1
vote
2answers
213 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
1answer
74 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
79 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
1answer
88 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
590 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
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
3
votes
1answer
200 views
How does a state in quantum mechanics evolve?
I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
$$
I am ...
4
votes
5answers
282 views
Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
1
vote
0answers
111 views
2D quantum well energy spectrum (analytical vs numerical)
I am trying to understand the energy spectrum difference between the analytical and the approximated solution for a quantum well.
The particle is inside a box with domain $\Omega=(0,0)$X$(1,1)$. For ...
4
votes
2answers
424 views
Quantum Mechanics Notation for BRA KET
I've been given this homework problem, but I do not understand its notation.
Please perform the following where the wavefunctions are the normalized eigenfunctions of the harmonic oscillator ...
2
votes
0answers
40 views
How to get from angular velocity to acquired phase for neutrino oscillations in matter?
I am reading Akhmedovs 2000 paper on parametric resonance, and I cannot figure out the math of this particular passage:
The difference of the neutrino eigenenergies in a matter of density $N_i$ is ...
2
votes
1answer
63 views
Schriffer Wolff Transformation - for first order change in eigenvalues
Step 1
Let me formulate the problem to convey my notation.
I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation
$$ U_A A\,\,U_A^{-1} = A_{diag}$$
Now the matrix is ...
2
votes
1answer
95 views
Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)
Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function ...
1
vote
0answers
16 views
could we obtain the potential (in one dimension) from the Gutzwiller trace?
to solve and obtain the potential of a 1-D Hamiltonian we must solve an integral equation
$$ N(E)= A \int_{0}^{E}\frac{V^{-1}(x)}{\sqrt{E-x}}$$
fro a some constant 'A' , then my question is since ...
4
votes
2answers
304 views
Eigenvalues of a quantum field?
Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t).
In quantum field theory, the classical field is promoted to an operator-valued ...
5
votes
0answers
130 views
Physical meaning of Laplace-Beltrami eigenfunctions?
The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of ...
9
votes
0answers
232 views
Lower bounds on spectral gaps of ferromagnetic spin-1/2 XXX Hamiltonians?
Question. Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamitlonians, on general interaction graphs, or tree-graphs?
I'm interested ...
1
vote
2answers
257 views
Angular Momentum Operators Non-Degenerate
Typically one writes simultaneous eigenstates of the angular momentum operators $J_3$ and $J^2$ as $|j,m\rangle$, where
$$J^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle$$
$$J_3 |j,m\rangle = \hbar ...
2
votes
2answers
424 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
127 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
43 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
755 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
36 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
110 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.
8
votes
4answers
325 views
Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?
I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
2
votes
1answer
92 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 ...
