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.

Filter by
Sorted by
Tagged with
0
votes
1answer
138 views

Energy of Free-electron Gas - Landau Levels in 3D

so i am looking into Landau Diamagnetism and am reading Dupre's paper. I am slightly confused at where he has got a term in his value of E from. He states that: $$ E=(n+1/2)\hbar\omega+\hbar^2k_z^...
0
votes
1answer
254 views

Eigenvalues of the Klein-Gordon operator

If I've understood what I've read correctly, the eigenvalues of the Klein-Gordon (KG) operator $\Box+m^{2}$ are $-p^{2}+m^{2}$, but how does one show this? Naively I assumed that the eigenfunctions ...
0
votes
0answers
56 views

Quantum mechanics problem set doubt [closed]

Let {|+⟩, |−⟩} be an orthonormal basis for a two-dimensional complex vector space. The basis vectors are the eigenvectors of the operator σ3: σ3|+⟩ = |+⟩; σ3|−⟩ = −|−⟩, Define two linear operators σ+ ...
0
votes
1answer
323 views

Graphical determination of energy eigenvalues (symmetrical potential well)

It is about a particle with mass $m$ in a potential $V(x)$: I want to do a graphical determination(at first only the symmetrical case) of the energy eigenvalues. I will show you my previous work: ...
1
vote
1answer
38 views

Momentum of Eigenstate of Spin Operator

For the Hamiltonian $H = \dfrac{p^2}{2m} - \mu S_z$, the eigenstates are the vectors $\begin{pmatrix} 1 \\ 0 \end{pmatrix} $ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix} $. I understand these are ...
2
votes
0answers
78 views

Eliminating an eigenvalue from the Hamiltonian

I have a momentum space Hamiltonian $H(\vec k)$ for a Kagome lattice and I want to find its eigenvalues which may be dependent on $\vec k$. Now, I'm told that one of the eigenvalues for such ...
1
vote
2answers
52 views

Can local projections on different parties give the same reduced state?

Suppose I have a bipartite pure state $\vert\psi\rangle_{AB}$. By the Schmidt decomposition, we know that the reduced states $\rho_A$ and $\rho_B$ have the same eigenvalues. I am now interested in ...
-1
votes
0answers
26 views

A Projection Operator generated by the Eigenspaces of an Observable?

What does it mean for a projection operator on to a subspace of Hilbert space for a system $S$, $H_{S}$, to be generated by the eigenspaces of an observable $A$ that correspond to a certain set $\...
2
votes
1answer
65 views

Why are time derivatives of states in QFT equal to zero?

In equation 6-38 on page 176 of the book "Student Friendly QFT" by Robert D. Klauber it is said that the partial derivative w.r.t. time of a multi-particle state is equal to zero since we ...
0
votes
1answer
51 views

Understanding bases in quantum mechanics

For $l = 1$ the angular momentum operator $L_z$ has the eigenvalues $\hbar,0,-\hbar$ and the eigenstates are then $|1,1\rangle, |1,0\rangle, |1,-1\rangle$. Now, we can calculate the matrix elements of ...
0
votes
1answer
100 views

Physical significance of this operator in quantum mechanics

I have stumbled across this question and cannot seem to find an answer to it. Consider an operator $\textbf{A}$ with eigenkets $|{a_{i}\rangle}$ and distinct eigenvalues $a_{i}$ . One can check that ...
3
votes
2answers
131 views

How come the eigenvalues of the Hamiltonian represent energy levels when the Hamilton doesn't represent the energy of the system?

Like in the Hamiltonian for a particle in an electromagnetic field. This is not a conservative field so the Hamiltonian doesn't represent the energy of the system. And yet the time independent ...
2
votes
1answer
23 views

How are the saddle-point equations derived in the single random matrix model?

In this question, I'm referring to a specific step in https://arxiv.org/abs/hep-th/9306153. I want to reproduce equation (2.4) on page 15. I think I lack the experience required for dealing with ...
0
votes
1answer
45 views

Measuring a state in a basis other than eigenbasis

Suppose I have a state expressed in its eigenbasis as follows. $\rho = \sum_i\lambda_i\vert i\rangle\langle i\vert$. It is now measured in some other basis $\{\vert x\rangle\}$ that is distinct from ...
1
vote
2answers
78 views

Is symmetrization $xp-px$ required for commutation $[H,x]=0$?

Given a Quantum Hamiltonian: $$\hat{H}=ax^2+bp^2$$ It does not commute with either $x$ or $p$. Suppose we have a Hamiltonian :$$H = k \hat{p}\hat{x}$$ why do we need it to be: $$H = k (\hat{p}\hat{x} -...
2
votes
2answers
74 views

(How) can you tell that a given operator's eigenspectrum will feature degeneracy?

I am speaking about operators representing physical observables and am not interested in purely mathematical objects (if that's relevant to answering the question). I know that a degenerate ...
1
vote
0answers
18 views

Electromagnetic Variational Principle and Concentration of Optical energy

While reading a book on photonic crystals, I came across the Electromagnetic Variational Principle. If the electric and magnetic fields are represented by the real part of $$\mathbf{H}(\mathbf{r}, t)...
0
votes
3answers
83 views

Collapse of wavefunction to its eigenfunction upon measurement

In quantum mechanics, it is postulated that to every observable, we have an associated operator. It is further postulated that when we do a measurement on a system, the measured value is one of the ...
2
votes
1answer
46 views

Translation operator eigenvalues can be real and arbitrary?

Consider the translation in space operator in $1D$: $$D(a)=e^{-ia\hat{p}/\hbar}$$ It is unitary - $D(-a)=D^{\dagger}(a)=D^{-1}(a)$ - which implies that $D(a)$ has eigenvalues on the unit circle like ...
1
vote
1answer
35 views

Numerical way of finding energy spectrum of $N$-body Schrodinger equation

For a single particle trapped in a potential, one can discretize the Time Independent Schrodinger Equation and hence find the eigenvalues of the corresponding Hamiltonian by diagonalising numerically. ...
0
votes
0answers
33 views

How to choose boundary conditions for numerical solution of Schrodinger's equation whose solutions are expected to die out “at infinity”?

I am using the "Shooting method" for solving the TISE with a "reasonably arbitrary" potential in 1D,with boundary conditions such that the eigenfunctions $\psi_n\to0$ as $x\to\infty$(And another ...
3
votes
1answer
59 views

Floquet bandstructure calculation

In this paper "Photonic Floquet Topological Insulators" the authors calculate the bandstructure of a time-periodic Hamiltonian. They create a time-dependent tight-binding Hamiltonian via the ...
0
votes
1answer
120 views

Symmetry-breaking matrix/operator deformations: uniquely splitting eigenspaces into smaller ones?

Operators used in quantum mechanics, like Hamiltonian or angular momentum operator, usually have huge degeneracy of eigenspaces (symmetry inside them) - bringing a question of possibility to uniquely ...
5
votes
1answer
867 views

What is the problem of having an inertia tensor not satisfying the triangle inequality?

It is well known that rigid body inertia tensors are 3 by 3 positive semidefinite matrices, which is the same as saying that their eigenvalues are all non-negative. A little less known is the fact ...
0
votes
1answer
35 views

Number Operator on the Product State of Identical Bosons

Suppose that we have a single photon (or any elementary boson) with the state $$\Phi_{1} = |n\rangle.$$ Suppose also that there is a two-particle system whose state is given by $$\Phi_{2} = |n\rangle_{...
3
votes
4answers
351 views

What is a physical example of an observable with degenerate eigenvalues? [closed]

If eigenvalues of an observable have the physical meaning of a possible result after a measurement, what's the interpretation of degenerate eigenvalues, and what is an example of such an observable?
4
votes
3answers
71 views

For $[A,B]=0$, if an eigenfunction of $A$ not an eigenfunction of $B$, does that imply degeneracy of one operator?

When two operators $A$ and $B$ commute, there can be functions which are eigenfunctions of $A$ but not that of $B$. For example, in case of the one-dimensional harmonic oscillator, any linear ...
1
vote
1answer
71 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) = \...
2
votes
1answer
58 views

If two operators commute, does it mean that every eigenfuction of one is also an eigenfunction of the other?

I have trouble interpreting the result of a problem. If we have a function $$\psi ( \theta , \phi) = e^{-3i\phi}cos \theta $$ and two operators $$A=\frac{\partial}{\partial \phi} $$ $$B=\frac{\partial}...
1
vote
1answer
115 views

Reconcile a pair of two-qubit boundary-state separability probability analyses

It is now clearly well-established--though formalized proofs are still largely lacking—that the probability, with respect to Hilbert-Schmidt measure, that a generic two-qubit state is separable/...
2
votes
1answer
40 views

Eigenvalues of a two particle system in a coupled vs. uncoupled basis

Consider a system of two distinguishable spin-1/2 particles with Hamiltonian \begin{align} H &= \frac{\alpha}{4} \vec{\sigma}_1 \cdot\vec{\sigma}_2.\\ \end{align} where $\vec{\sigma}_1 = (\sigma_x\...
2
votes
2answers
116 views

Is it wrong to assume that $|{-\vec r}⟩ = - |{\vec r}⟩$?

I found a problem that says the following: The 'even' operator is defined as: $$\Pi|{\vec r}\rangle = |{-\vec r}\rangle$$ Show that $\Pi$ is Hermitian and find $\Pi^2$ and its eigenvalues. All ...
1
vote
2answers
53 views

Moment of inertia tensor and symmetry of the object

What information does the moment of inertia tensor give on the structure of an item. I was told that its eigenvectors give the principal axes of the object. Do you know more about this?
2
votes
1answer
43 views

Ground state of hydrogen molecule using Heitler–London method (H-L)

I am reading The Theory of Magnetism I, by Mattis. In Chapter 2, a hydrogen molecule is studied in the following way: We have a Hamiltonian of a hydrogen molecule: $$H = H^0_1 + H^0_2 + H^\lambda \...
1
vote
0answers
32 views

Angular Momentum and Spherical Top

The Schrödinger equation for 2 nuclei at fixed distance $R$ can be transformed to relative and centre of mass coordinates and gives rise to the Eigenvalue problem $$\frac{{J}^2 \Psi_{jm}}{2\mu R^2} = ...
2
votes
0answers
66 views

Does no-level-crossing theorem (aka avoided crossing) always hold in perturbation theory?

In perturbation, J.J. Sakurai Modern Quantum Mechanics Second Edition page 310 stated a no-level-crossing theorem stated that "a pair of energy levels connected by perturbation do not cross as ...
1
vote
1answer
57 views

Guessing eigenvalue solution

I am reading The Theory of Magnetism I, by Mattis. In Chapter 2, he proposes the following eigenproblem: $$ \left ( \begin{matrix} V & U \\ U^\dagger& V \end{matrix} \right ) \left ( \begin{...
0
votes
2answers
67 views

Determining What Eigenfunction Occurs With Wavefunction Collapse

Suppose an operator $O$ has eigenfunction normalized $f$ corresponding to eigenvalue $n.$ Of course, any function $cf$, with $c$ on the unit circle, is also a normalized eigenfunction. Thus, if a ...
0
votes
1answer
100 views

How do you find the Triangle Inequality from an Inertia Matrix?

If you have an inertia matrix of the form $$\begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}=I$$ If the matrix ...
1
vote
0answers
36 views

Quantum observables in nonstandard Hilbert space

Consider a Hermitian $(n \times n)$-matrix $A$, and a Hilbert space $\mathbb{C}^n$, foreseen with a nonstandard inner product. (An inner product $s(\cdot,\cdot)$ is standard if for any two vectors $x =...
12
votes
2answers
2k views

Schrödinger wavefunctional quantum-field eigenstates

The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time. The problem is to construct the eigenstates of a quantum ...
0
votes
1answer
130 views

The energy eigenstates $\psi_n(x)$ are eigenstates of parity operator?

For one dimensional system system described by symmetric potential energies with the property $V(x)=V(-x)$, the energy eigenstates $\psi_n(x)$ are eigenstates of parity operator? Is the above ...
0
votes
0answers
28 views

Looking for a freeware software/app that can solve eigenvalue problems symbolically

I'm taking a quantum mechanics course and my homework involves extremely tedious algebra to solve symbolic eigenvalue problems. I'm looking for a software that I can give matrices with symbolic ...
1
vote
0answers
83 views

Confusion about calculating first order correction to energy eigenstate / state vector

I am trying to determine the first order correction to the ground state for a particle in an infinite square well with a given perturbation, $$V'(x) = \frac{2\pi^2 h^2}{mL^3} (\epsilon x- \frac{\...
0
votes
0answers
47 views

4×4 Cofactor Transpose Matrix calculation gone wrong, in Shankar's Principles of Quantum Mechanics

In Appendix A$.1$, Shankar, R; Principles of Quantum Mechanics, the cofactor transpose of a $3\times3$ matrix $M$ is given as (to be referred to as the first procedure by me) $$\overline{M}=\begin{...
0
votes
0answers
27 views

Spin-Spin Interaction: Ground State Degeneracy

I'm given the hamiltonian $$\hat{H}=\sum_{i=1}^{L-1}\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}_{i+1}$$ (reminiscent of a para- or ferromagnetism situation?) where $\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}...
2
votes
1answer
77 views

Are eigenfunctions always normalizable?

If $y$ is an eigenfunction which corresponds to the eigenvalue $a$ of the operator $A$: $$A\langle y|=a\langle y|$$ Can we assume that $$\int y^*y=1$$ ?
1
vote
1answer
78 views

Is the mass an eigenvalue of Dirac equation?

Writing the Dirac equation as: $$(i \hbar\gamma^{\mu}\frac{\partial}{\partial x^{\mu}})\psi = m \psi$$ it seems that $m$ is an eigenvalue of the operator of the left side, and we need to find the ...
0
votes
2answers
288 views

Proof that coherent states are eigenstates of annihilation operator [closed]

My goal is to prove that, for $|\lambda\rangle=N\exp(\lambda\hat{a}^\dagger)|0\rangle$ is an eigenvector of the operator $\hat a$. I took 2 approaches, but both make sense to me and I get different ...
1
vote
1answer
88 views

Why is $|\alpha\rangle$ not eigenstate of $a^{\dagger}$ for $\alpha^*$

I know that even if we have: $$a |\alpha \rangle = \alpha |\alpha\rangle$$ We don't have: $$a^{\dagger} |\alpha \rangle = \alpha^* |\alpha\rangle$$ Actually as explained in the second answer here ...

1
2 3 4 5
10