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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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47 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 ...
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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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307 views

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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2answers
82 views

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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107 views

Solving the free particle problem in momentum space

$\newcommand{\ket}[1]{|#1\rangle}$$\newcommand{\bra}[1]{\langle#1|}$(Note: this question was asked before here but I didn't follow the answer.) For the free particle, Schrödinger's equation is given ...
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2answers
139 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 ...
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149 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 ...
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Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
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181 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: ...
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Superposition principle forbids quantisation?

Apparently bound states in quantum mechanics require energy states to be discrete. That means energy in such systems is quantized, right? However, say that we have a superposition of energy ...
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234 views

Bloch's Theorem without periodic boundary condition - mathematically rigorous way

I am looking for a proof of Bloch's Theorem which does not use periodic boundary conditions. Sometimes one happens to see non-rigorous demonstrations of Bloch's Theorem without the use of periodic ...
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89 views

What is energy in quantum mechanics?

Is it wrong to say energy is the expectation value of Hamiltonian? Or should I say energy is the eigenvalue of Hamiltonian?
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Arriving at the Quantum Mechanial Potential From The Energy Eigenvalues [duplicate]

In Quantum Mechanics, we know that given a potential we can solve the eigen value problem to find out the energy eigen values and eigen functions. Now suppose in an experiment we have information only ...
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Numerical exact diagonalization of tight binding Hamiltonian

I want to exactly diagonalize the following Hamiltonian for $10$ number of sites and $4$ number of spinless fermions $$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{...
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2answers
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Creation operator acting on a coherent state. Occupation number operator

For a coherent state $$|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}(a^{\dagger})^n}{n!}|0\rangle$$ I want to find a simplified expression for $a^{\dagger}|\alpha\...
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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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1answer
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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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1d Ising model: Transfer matrices

we came across a peculiarity when calculating the partition function of $N$ spins $s_i=\pm1$ with Hamiltonian $$H=-J\sum_{i=1}^Ns_is_{i+1}$$ where we impose periodic boundary conditions such that $s_{...
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Eigenvalues of the thermal state density operator

We define the thermal density operator as $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ where $H$ is the systems Hamiltonian. Today I was told that the eigenvalues of the ...
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Coherent state being the eigenstate of the annihilation operator

From what I understand, the physical relevance and interest of a coherent state is that its dynamics closely resembles the one of its classical analogue. For example, for a quantum SHO $\langle x \...
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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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7answers
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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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29 views

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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0answers
150 views

Expansion of the infinite square well [closed]

I was studying the expectation value of the energy of a particle in the groud state of the infinite square well after its expansion in terms of width (from $a$ to $2a$), which is: $$\langle H\rangle= ...
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51 views

Applying Sylvester's theorem in quantum mechanics

A $2d$ system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is an unitary Hermitian $2$x$2$ matrix with eigenvalues $\exp(±i\theta)$. I need to use ...
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Quantum Potential - how to find its eigenstates?

I am studying the Pöschl-Teller potential in spherical coordinates and doing a change of variable is enough to put it in another sort of differential equation and thus, obtain the solution. The point ...
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2answers
119 views

Conceptual understanding of Schrödinger equation

So I followed this lecture: https://www.youtube.com/watch?v=qu-jyrwW6hw which starts of with the statement: If you have a Schrödinger equation for an energy eigenstate you have $$-\frac{\hbar}{2m}...
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1answer
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Does the eigenbasis associated with an observable changes after measuring a different observable?

Suppose a system is initially in a superposition: $$\psi(x) = \sum\limits_{i}|c_i\phi_i(x)\rangle$$ After a position measurement, the wave function collapses to one of the position eigenfunctions,$\...
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Why is this function an eigenfunction of $\hat{L}_{z}$?

$$\Psi(\varphi)=\frac{1}{\sqrt{2\pi}}(\sin\varphi-\cos\varphi)$$ I am not able to see why the above function is an eigenfunction of $\hat{L}_{z}$ and which is its eigenvalue. I've been trying with ...
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Applications of Hamiltonians proportional to nth order momentum

Has anyone come across any physical applications of a linear theory where Hamiltonians are proportional to an arbitrary order of momentum? This paper https://michaelberryphysics.files.wordpress.com/...
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If $L_z$ has a $0$ eigenfunction, since $[L_x, L_y] = i\hbar L_z,$, then can $L_x, L_y$ have a simultaneous eigenfunctions?

In the lecture Quantum Mechanics by Dr. Adams in ocw.mit.edu, in the 16th lecture at 7:11, it is stated that since $$[L_x, L_y] = i\hbar L_z,$$ there is no state s.t it is eigenfunction of both $L_x, ...
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1answer
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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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Woods-Saxon energy levels

I'm trying to calculate the bound state energy levels for a Woods-Saxon potential, given by: $$V(r) = -\frac{V_{0}}{1+\exp(\frac{r-R}{a})}$$ I'm using Numerov's matrix method, which led me to: $$H\...
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3answers
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Eigenfunctions of the Runge-Lenz vector

The hamiltonian for the hydrogen atom, $$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r} $$ is spherically symmetric and it therefore commutes with the angular momentum $\mathbf{L}$; this causes all its ...
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1answer
80 views

Can a general many-body Hamiltonian with quadratic and biquadratic terms be diagonalized?

Can an arbitrary many-body hamiltonian in second quantization form with quadratic and biquadratic terms $$H=\sum_{v_1,v_2} \alpha_{v_1 v_2}\ c_{v_1}^{\dagger}c_{v_2}+ \sum_{v_1,v_2,v_3,v_4}\beta_{v_1 ...
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1answer
82 views

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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1answer
45 views

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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1answer
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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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1answer
417 views

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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1answer
111 views

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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1answer
150 views

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\...
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1answer
134 views

Eigenvalues and functions in quantum mechanics [closed]

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).
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1answer
142 views

Finding measurements in non-Hermitian operators

I know how the measurement postulate in quantum mechanics works, in regard to hermitian operators, but what if an operator is non-hermitian? Can i apply the following reasoning? If an operator is ...