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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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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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What is the importance of eigenvalues? [closed]

I am wondering what is the meaning of eigenvalues applied to physics.
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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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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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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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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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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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157 views

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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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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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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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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93 views

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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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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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
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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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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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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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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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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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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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79 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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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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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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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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108 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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149 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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133 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
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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 ...
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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 ...
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Is the ground state energy always larger for the system with higher potential energy?

Say we have two Hamiltonians $\hat{H}_1$ and $\hat{H}_2$ that differ only in their potential energies and $$V_2(x) > V_1(x)$$ for all $x$. Is the energy of the ground state of system 2 necessarily ...
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163 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{...
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Am I correct to say that ladder operators have complex eigenvalues?

From the definition: $$\left. \begin{array} { l } { \hat { L } _ { + } = \hat { L } _ { x } + i \hat { L } _ { y } } \\ { \hat { L } _ { - } = \hat { L } _ { x } - i \hat { L } _ { y } } \end{array} \...
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1answer
148 views

How to diagonalise a hamiltonian which posesses symmetry?

I have a large hamiltonian but I know that it posseses some symmetries. How do you reduce the hamiltonian in order to find the eigenenergies?
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mutual coherence

I read Vertex-Frequency Analysis on Graphs (arxiv link) Shuman, David I, et al. “Vertex-Frequency Analysis on Graphs.” Applied and Computational Harmonic Analysis, vol. 40, no. 2, 2016, pp. 260–291....
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Quantum Spherical Pendulum [closed]

I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of ...
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2answers
744 views

Eigenvalues of momentum operator

I had a homework problem in my intro QM class, basically asking me to find which of a given set of functions were eigenfunctions of the momentum operator, $\hat{p_x}$. For example, $$ \psi_1 = Ae^{ik(...
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Significance of eigenvalues of an observable Of a wavefunction [closed]

What really is meant by eigenvalue of an observable? Does it mean that everytime we measure a value of an observable the result obtained is equal to the eigenvalue of the observable?
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Why the eigenvalues of hamiltonian are discrete in bounded systems (only discrete energy levels)? [duplicate]

In one dimensional motion the general potential is given as in the figure above When energy is between V min and V 1 the energy levels are discrete like in the potential barrier and well example or ...
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105 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/...
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1answer
124 views

Under what conditions is a wavefunction $\psi(x)$ equal to the probability amplitudes $a(x)$?

For context, consider a general expansion of a wavefunction into continuous eigenstates of position, $\phi(x_m,x)$, multiplied by continuous probability amplitudes, $a(x_m)$ $$\begin{align}\psi(x) &...
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1answer
104 views

A naive question on the eigenvalues of fermionic operators?

Let $A$ be a fermionic operator which is a product of odd number of fermion operators or a summation of them, say $A=C_{i_1}^{\dagger}\cdot \cdot\cdot C_{i_m}^{\dagger}C_{j_1}\cdot \cdot\cdot C_{j_n}...