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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3answers
89 views

Is $X\otimes X$ not the simultaneous position operator?

I had thought that $X\otimes X$ would be the operator on $H_1\otimes H_2$ to simultaneously measure the x-positions of two particles. But there seems to be something wrong with this -- for a given ...
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Chua's Circuit: an inequality ensuring that the equilibrium is not stable

According to Kennedy's Robust op-amp realization of Chua's circuit(1992), the differential equations satisfied by several physical quantities in Chua's circuit are $$\begin{aligned} C_{1} \frac{d v_{...
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1answer
78 views

Different formula to find $2\times 2$ Hamiltonian's eigenvalues [closed]

Consider the Hamiltonian $$ \left[ \begin{matrix} E_1 & -A\\ -A& E_2\\ \end{matrix} \right] $$ where $A$, $E_1,E_2$ are real numbers. I have seen a different formula to ...
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1answer
71 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 ...
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64 views

How to solve these coupled differential equations?

I am trying to solve for wavefunctions of 2D tilted Dirac systems, the Hamiltonian for which is: $$\hat H = v_{x}\sigma_{x}\hat p_{x}+v_{y}\sigma_{y}\hat p_{y}+I_{2}(v_{t}^{x}\hat p_{x}+v_{t}^{y}\hat ...
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What are the Eigenstates and Eigenvalues? [closed]

In quantum mechanics I keep hearing about them. Kindly tell about them...not at a very very high level but simple enough to understand completely
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44 views

Eigenvalues of Unitary Matrices

I am considering the standard equation for a unitary transformation $\alpha^* = U \alpha U^{-1}$, where $\alpha$ is an arbitrary linear operator and $U$ is a unitary matrix. Since in quantum ...
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1answer
26 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: $$T = \frac{1}{2} \dot q^T M \dot q \tag{1}$$ ...
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31 views

How to do Weierstrass-transform in MATLAB? [closed]

I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform?) on them. So I have the wave functions ($\Psi$, $1\times N$ vectors), ...
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164 views

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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1answer
56 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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1answer
60 views

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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2answers
85 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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2answers
125 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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110 views

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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169 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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1answer
90 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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68 views

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
102 views

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
27 views

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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1answer
111 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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1answer
151 views

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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1answer
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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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52 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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23 views

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
122 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
40 views

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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171 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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43 views

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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28 views

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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1answer
34 views

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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1answer
151 views

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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61 views

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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1answer
93 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
85 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
47 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
582 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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1answer
73 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^...
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1answer
140 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
163 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
157 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
168 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 ...
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1answer
98 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 ...
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1answer
44 views

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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1answer
192 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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3answers
180 views

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} \...