# 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.

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### How Do We Define Integration over Bra and Ket Vectors?

I'm having trouble understanding the completeness condition for bra and ket vectors in Hilbert space, especially in the continuous case. The discrete case makes a fair amount of sense; given any ...
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### 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 ...
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### Is it possible to derive $2\times 2$ Lorentz transformation matrix from only eigenvectors?

As a preface, I am somewhat familiar with year 1 linear algebra but not too familiar with how one makes the connection to Lorentz transformation matrices so I apologize if the answer is obvious. One ...
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### How can I meaningfully diagonalize the eigenvector subspace of a degenerate phonon mode?

It often occurs to find phonon modes which are degenerate by symmetry. In such occasions the eigenvector is usually not physically insightful, as is is a linear combination of the n degenerate ...
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### 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 ...
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### Negativity of the real part of eigenvalues of Lindblad operators

I'm looking for a proof of the fact that the real part of eigenvalues of Lindblad operators is always negative. So far I have only found handwavy arguments such as "things should not blow up at ...
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### Eigenvalues, Hermitian operators and observables in quantum mechanics

Consider a hermitian operator. So a) in a space of infinite dimension its eigenvectors are a base. b) in a finite-dimensional space the matrix that represents the hermitian operator is always ...
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### Probability of a specific energy state

We consider the normalized wave function: $$\psi(x,t) = \sqrt{\frac{2}{3}}\psi_0(x)\exp\left(\frac{-iE_0t}{\hbar}\right) + \sqrt{\frac{1}{3}}\psi_1(x)\exp\left(\frac{-iE_1t}{\hbar}\right)$$ To ...
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### 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 ...
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### Finding the eigenstates of an operator [closed]

I am currently taking a course in QM and can't see how the eigenstates have been found for examples like this one: Question Let $\phi _1$ and $\phi _2$ be two normalised wavefunctions orthogonal onto ...
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### Some questions on coherent states and corresponding Hilbert spaces. Reproducing kernal

I have a few questions related to coherent states. I use this source https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf. Using standart inner product $\langle\cdot|\cdot\rangle$ ...
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### Confusion of measuring two quantities on a quantum system

Let's say there are two observables corresponding to two operators A and B, and let's say my system is in a state Phi where with probability 1 if I measure A I get 3 (let's say 3 Joules), If I ...
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### What does $\left|x,t\right>$ actually mean (Heisenberg picture)?

I am pretty much confused with this notation I believe. The Heisenberg states are denoted by $\left|x,t\right>$ and the Schrodinger states are given by $\left|x(t)\right>$. It seems like both of ...
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### 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 ...
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### What is the term for a particle spin's uncollapsed position? What is the orbiting “thing”?

I'm not sure if I have the correct visual model, but I imagine that a particle spin can be represented by a single point on the orbit, or by a superposition state (like a random plane through a corner ...
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### What exactly are control functions (used for parametrization)?

Let us consider a system in state $\rho$ with an internal hamiltonian $H_0$ on which we apply a cyclic, unitary evolution $H_t = H_0 + V(t)$ Where $V(t)$ is a time dependent external potential for ...
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### Wigner proof of the non-existence of finite unitary representation of the Lorentz group

I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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### What do I get by multiplying a 0 operator on a 0 eigenvector?

I don't know how to write the equation form. Assuming my notation as Dirac notation, what do I get from $$( 0 | 0 | 0 ) ~?$$
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### Problem 7-1 from the book “Quantum Mechanics in Simple Matrix Form” by T. Jordan [closed]

Introduction I'm reading the book "Quantum Mechanics in Simple Matrix Form" by T. Jordan. I try to solve the problem sets. With problem 7-1 i have a solution, i think, but i'm not sure if the writer ...
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### What is eigenvalue and eigenfunction in quantum mechanics?

What is the use of eigenvalue and eigenfunction in quantum mechanics specially Schrodinger equation? What is the physical meaning of having an eigenvalue and eigenfunction in Schrodinger equation?
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### 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^...
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### How to calculate the ground state of Ising model at non-zero temperature

I'm studying the quantum Ising model, i.e. with Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$. I know conceptually how to compute the ground state of the Ising model at zero ...
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### Diagonalization of a matrix with operators as elements

How to diagonalize a hamiltonian matrix that has differential operators as elements? My matrix looks something like: \begin{bmatrix} A \frac{d^{2}}{{d\theta}^{2}}+ B_{1} & a\cos{(b\theta +c)}\\ a\...
How do I determine if $\psi(x)$ is a eigenfunction of some operator and find the corresponding eigenvalues, where $\psi(x)$ is the wave function of free particle (potential = zero).