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.
800
questions
1
vote
1
answer
94
views
Confusion Regarding the Derivation of Graphene Dispersion Using Annihilation and Creation Operators
I am going through a text which derives the energy bands in graphene (https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/3/67057/files/2018/09/graphene_tight-binding_model-1ny95f1.pdf) and am stuck on a ...
- 121
1
vote
1
answer
422
views
Physical meaning of eigenvalues in the heat equation problem
Let's consider the heat equation on a $\Omega \subset \mathbb{R}^2$ manifold with a boundary $\Gamma$, with initial and boundary conditions
\begin{align}
\dot{u}(\mathbf{r}, t) &= \Delta u(\mathbf{...
- 43
1
vote
2
answers
391
views
Eigenvalues in Floquet theory
After calculating Floquet Hamiltonian and then it's eigenvalues I stumbled upon a problem with ordering of eigenvalues. I am using eigen library for c++ and for every Floquet Hamiltonian for given ...
- 11
1
vote
1
answer
279
views
Finding the probability of measuring a particular eigenvalue of an operator for a system after time evolution
Consider a quantum system with Hamiltonian H and consider the measurement
of an observable $a_n$ associated with a different operator A.
Initially the system is an eigenstate $|\phi_n \rangle$ with ...
- 978
1
vote
1
answer
203
views
Spectrum of the angular momentum and angular momentum squared
I am trying to understand how to build the spectrum of the angular momentum; of course since different components of the angular momentum do not commute with each other we must chose only one ...
- 4,474
1
vote
1
answer
231
views
Doppler as an eigenvalue of the Lorentz transformation
It is a known fact that
$$
\gamma (1\pm\beta) = \sqrt\frac{1\pm\beta}{1\mp\beta}
$$
is an eigenvalue of the Lorentz transformation (which is a linear transformation).
This is also (as stated in the ...
- 147
1
vote
1
answer
549
views
How the matrix representation of a Hamiltonian affects the eigenvalues?
Suppose we're given the following Hamiltonian: $$\hat{H}=\frac{\omega}{\hbar} \left(\hat{S}_+^2+\hat{S}_-^2\right)$$ Suppose also that we measure $\vec{S}^2$ and get $6\hbar^2$, i.e. reduced to the $s=...
- 685
1
vote
1
answer
133
views
Discreteness of the general angular momentum in quantum mechanics [duplicate]
When studying the general angular momentum $\textbf{J}$, which is defined as a vector operator with its components being Hermitian operators satisfying the commutation relations
\begin{align*}
\textbf{...
1
vote
1
answer
363
views
Wavefunction of a particle on a ring ($E > V$) using WKB method
For a particle on a ring (with radius $R$ and changing angle $\theta$) with only kinetic energy ($V=0$) we get the expressions for the wavefunction (normalized) and eigenvalues $$\Psi_n (\theta) = \...
- 43
1
vote
1
answer
104
views
Completeness relation, commuting operators
I have a question about some formulars our professor wrote on the black board.
Let $\hat{Q}_{1},...,\hat{Q}_{N}$ be operators, which are a CSCO. We know now that there exists a set of eigenvectors $\{...
- 854
1
vote
1
answer
174
views
Eigenstructure of the Dicke Model
I am beginning a study of the Dicke model and found a very interesting publication: "The Dicke model in quantum optics: Dicke model revisited" by Barry M Garraway in Phil. Trans. R. Soc. A (2011).
I ...
1
vote
1
answer
255
views
Eigenvectors of the BCS Hamiltonian
In introductory superconductivity one often studies the BCS Hamiltonian
$$H=
\begin{pmatrix}
\xi & -\Delta \\
-\Delta & -\xi
\end{pmatrix}
$$
I can find the Eigenvalues and Eigenvectors by ...
- 441
1
vote
1
answer
371
views
Confusion on kinetic energy quadratic forms and eigenfrequencies
I am new to the idea of expressing kinetic energy in terms of the quadratic form. I noticed that online, people often express the kinetic energy as:
$$T = \frac{1}{2} \dot q^T M \dot q \tag{1}$$
...
- 694
1
vote
1
answer
1k
views
Eigenvectors of spin-spin coupling Hamiltonian
We want to find the eigenvectors and eigenvalues of the Hamiltonian, $H = \vec{\sigma_1}.\vec{\sigma_2}$ , where the subscript indicates the particle number.
The usual way to go about it is to find ...
- 668
1
vote
1
answer
1k
views
Eigenstates of a Hamiltonian [closed]
For a particle with a spin of 1/2, which was exposed to both magnetic fields
$B_{0}=B_{z}e_z$
and
$B_1=B_xe_x$
I already found the eigenvalues of its Hamiltonian which is given by
\begin{...
- 11
1
vote
1
answer
198
views
Use of operators in a time-dependent Hamiltonian quantum system
I am given the following Hamiltonian, $$H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$$ for $t<0$ and $$H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$$ for $t\geq0$. For some time $t_1(<0)$, ...
- 1,202
1
vote
1
answer
385
views
Measurement of position in quantum mechanics
I know that when you perform a measurement of position in quantum mechanics, the wave function collapses to something proportional to it, but in a small range of values of positions, depending on the ...
1
vote
1
answer
163
views
is it possible to have different states other than just one in Dirac Delta Potential?
Let's say, initially the state is in first excited state of finite well potential and then I change the width & depth of the well, eventually to Dirac delta potential, then what happens to the ...
1
vote
1
answer
71
views
Solve eigenvalue problem with known constraint on one of the Eigenvalues
I have the following problem and would appreciate any help.
I have a real, symmetric matrix M given by
$$M=\begin{pmatrix}
m_{11} & m_{12} & m_{13} & m_{14} \\
m_{12} & m_{...
- 337
1
vote
2
answers
764
views
Need help with clarification of eigenstates and wavefunctions
The book "Introduction to quantum mechanics" by Griffiths starts by introducing the wave function. The squared of the integral of the wave function gives you the probability of measuring the position ...
- 11
1
vote
2
answers
1k
views
Significance of using Eigenvalues / Vectors in QM?
A fundamental idea in Quantum Mechanics is that observable quantities are represented by linear, hermitian operators. Why is it that we represent distinguishable states as the eigenvectors of ...
user154080
1
vote
1
answer
242
views
How the useful, unitary "Folding Transform" is applied to a Hamiltonian
Summary
Given a unitary transformation $U_F(t)=\sum_n e^{in\omega t}\sum_{\lambda\in n}\left| \lambda, n\right\rangle \left\langle \lambda,n \right|$ applied to a Hamiltonian $H_0$ (with $H_0 \left|\...
- 106
1
vote
1
answer
131
views
Finding the eigenfunctions of the $\hat{\vec l}.\hat{\vec s}$ operator for a single p-electron
I'm trying to calculate the SO-coupling for a single p-electron ($l=1$, $s=\frac{1}{2}$) in the uncoupled representation.
This comes down to calculating these matrix elements:
$$\left\langle ...
- 1,373
1
vote
1
answer
142
views
Eigenstates of Conical Potential in 3-dimensions?
If we take an ordinary time-invariant Schödinger equation:
$$H|\psi\rangle = E|\psi\rangle,$$
and use a conical potential $V(r) = A r$ we get a differential equation:
$$\left[-\left(\frac{\hbar^2}{2m}\...
- 22.5k
1
vote
1
answer
485
views
Neutron in a magnetic field (Schrödinger Equation, Eigenstates, Eigenvalues).
Consider the spin of a neutron in a magnetic field $\vec{B}$. A neutron is a neutral particle with the mass of a proton and the spin $\frac{1}{2}$. The Hamiltonian is
$H=\mu_n\vec{S}\cdot\vec{B}$ ,
...
- 73
1
vote
1
answer
269
views
Random operator in heisenberg/schrödinger picture (heisenberg's equation of motion) [closed]
Consider a system whose hamiltonian isn't explicitly dependent on time. Let A be the operator for the eigenvalue a in the Schrödinger picture and $A_H=U^\dagger A U$ the corresponding operator in the ...
- 277
1
vote
1
answer
97
views
How to understand Preskill's argument for degeneration of eigenstates?
In his notes on topological quantum computation on page 18, Preskill uses the "commutator"
$T_2^{-1}T_1^{-1}T_2T_1 = e^{-2 i \vartheta}$ to show that the eigenstates of $T_1$ are degenerate. But I don'...
- 254
1
vote
1
answer
683
views
Reasoning behind taking the Fourier transform of the fermionic operators for a circular $1$D spin chain [closed]
In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
- 402
1
vote
1
answer
5k
views
Eigenstates of position and momentum operators in QM
In Griffiths pages 103-105 "Introduction to Quantum Mechanics" 2nd editiion he states that the eigenfunctions of the position and momentum operators are $$g_y(x) = \delta(x-y)$$ where the eigenvalue ...
user100411
1
vote
1
answer
92
views
Taking Measurements of Quantities in QM
I have a quick question relating to Annihilation and Creation operators, and in taking observables in general.
Let's say, for instance, that I prepare a particle so that I consider the projection of ...
- 344
1
vote
2
answers
347
views
Difference between operators used to represent quantum gates vs that to represent physical observables?
I have learnt that informations about a physical observable property is buried in the state vector of a quantum system. To get the possible value of a property all we need to do is multiply the state ...
1
vote
1
answer
241
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 ...
user84302
1
vote
1
answer
465
views
Eigenvalues of the radial Schrödinger equation on a finite integration interval
There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations ...
- 63
1
vote
1
answer
92
views
Eigenvalue for interacting Hamiltonian [closed]
Consider the Hamiltonian
$$H=\omega_{1} a_{1}^\dagger a_{1}+\omega_{2}a_{2}^\dagger a_{2}+\alpha a_{3}^\dagger a_{3}(a_{1}^\dagger a_{2}+a_{2}^\dagger a_{1})$$
with
$$ [a_\alpha^\dagger,a_\...
- 11
1
vote
1
answer
145
views
How can you tell if a GTO function is an eigenfunction of hamiltionan H?
How can you tell if a Gauss-type orbital is an eigenfunction of Hamiltionan $H$? For example:
$$GTO = N z^2 \exp\left(-\alpha r^2\right)$$
I know it is and eigenfunction of $L_z$ and not $L_x$ and $...
- 11
1
vote
1
answer
1k
views
numerical diagonalization of tight-binding hamiltonian
I would like to find the exact eigenvalues of the following tight-binding Hamiltonian, written here in second quatization:
\begin{eqnarray}
\hspace{-0.25in}{\mathcal{H}} &=& \mathcal{H}_0+ \...
- 317
1
vote
1
answer
955
views
Eigenvalues of hamiltonian [closed]
Q: THe hamiltonian which describes the motion of a particle in an one dimensional potential V(x) is $H_0=\frac{p^2}{2m}+V(x)$ , where $p=-i\hbar \frac{d}{dx}$ is the momentum operator.
$E_n^0$ , $n=...
- 110
1
vote
2
answers
960
views
What is the meaning of pre-tension for a stiff membrane?
On one hand, I know that the tighter a drum head is stretched, the higher its natural frequencies. This relation is given by:
$$f_{ij}=\frac{k_{ij}}{2\pi R}\sqrt{\frac{T_0}{h\rho}}$$
where $k_{ij}$ ...
- 3,194
1
vote
1
answer
405
views
Which textbooks contain info on Bessel functions & their use as basis functions?
As an exercise my research mentor assigned me to solve the following set of equations for the constants $a$, $b$, and $c$ at the bottom. The function $f(r)$ should be a basis function for a ...
- 189
1
vote
1
answer
97
views
How to write QM operator if I know all of it's eigenfunctions?
Suppose I have selected enough orthogonal functions in representation of operator A and I want to derive operator B which has ...
- 1,645
1
vote
1
answer
199
views
The Eigenstate Existence Problem in Dirac's book 'Principles of Quantum Mechanics'
In Chapter II p. 32 of Dirac's book Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/...
- 149
1
vote
1
answer
243
views
Clarify formula in quantum perturbation theory
I'm studying perturbation theory in the context of quantum mechanics. My lecture notes say that in order to calculate the first-order correction of eigenfunction $\psi_n$, that is $\psi_n^{(1)}$, I ...
- 367
1
vote
1
answer
684
views
Matrix operations on Quantum States in a composite quantum system
Intro (you may skip this if you're an expert, I'm including this for completeness):
Say I have two bases for two systems,
The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c}
1\\0
\...
- 3,447
1
vote
0
answers
60
views
Eigenvalues of Hermitian Operators [duplicate]
In quantum mechanics, it's well-known that observables are associated as the eigenvalue of a Hermitian operator.
My question is, is the converse also true? i.e. the eigenvalue of a Hermitian operator (...
1
vote
1
answer
123
views
Finding the zero-order wave-functions of the perturbed state
If I have an unperturbed Hamiltonian $\hat{H}_0$ which is four-fold degenerate corresponding to orthonormal eigenfunctions $\phi_1$, $\phi_2$, $\phi_3$, $\phi_4$ and I have some perturbing Hamiltonian ...
- 135
1
vote
1
answer
43
views
Non-Hermitian PT-symmetric Interacting Hamiltonian with Real Spectra
The following hamiltonian is $\mathcal{PT}$-symmetric $$\mathcal{H} = -J \sum_{j = 1}^{2N} [ 1 + (-1)^j \delta ] [ c^{\dagger}_{j} c_{j+1} + h.c. ] + \imath \gamma \sum_{j = 1}^{2N} (-1)^j c^{\dagger}...
- 199
1
vote
0
answers
44
views
Why energy eigenstates are extrema of the energy functional? [duplicate]
We have the energy functional of a system:
$$E[\psi] = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}$$
and over numerous textbooks it is said that the eigenstates of the ...
1
vote
0
answers
62
views
Appearance of Sturm-Liouville Problem in Quantum Mechanics
I am considering the below ODE (ordinary differential equation), which can be studied using techniques from Sturm-Liouville theory.
The context is mathematical, but I was wondering if anyone ...
- 1,220
1
vote
0
answers
40
views
The eigenvectors obtained in the diagonalization in the paper "Two Soluble Models of an Antiferromagnetic Chain" by Lieb, Schultz and Mattis [closed]
The diagonalization involves few transformations, that transforms the anisotropic XY model in a matrix eigenvalue equation as $(A-B)(A+B)\phi_k= \lambda^2_k \phi_k$ where the matrix $(A-B)(A+B)$ has a ...
1
vote
0
answers
86
views
Green Function associated to a periodic Schrodinger operator
If $V:\mathbb R\to \mathbb R$ is an $L$ periodic function in $\operatorname L^{\infty}$ we can always find two independent solutions for $$\psi''(x)+V(x)\psi(x)=E\psi(x)$$
$\psi^{\pm}(x)=e^{\pm ipx}\...
- 147