Eigenvalues Definition and 820 Threads

Hi PF! Here's an ODE (for now let's not worry about the solutions, as A LOT of preceding work went into reducing the PDEs and BCs to this BVP): $$\lambda^2\phi-0.1 i\lambda\phi''-\phi'''=0$$ which admits analytic eigenvalues $$\lambda =-2.47433 + 0.17337 I, 2.47433 + 0.17337 I, -10.5087 +...

I know the basis I should use is |m_1,m_2> and that each m can be 1,0,-1 but how do I get the eigenvalues from this?

I'm fairly stuck, I can't figure out how to start. I called the matrix ##\mathbf{A}## so then it gives us that ##\mathbf{A}\mathbf{A}^\intercal = \mathbf{I}## from the orthogonal bit. I tried 'determining' both sides... $$(\det(\mathbf{A}))^{2} = 1 \implies \det{\mathbf{A}} = \pm 1$$I don't...

Is the spiral I drew here clockwise or counterclockwise ? What’s a trick to know whether it’s going CCW or CW. Thanks!

The eigenvalue equation is $$\frac{d^2}{d \phi^2} f(\phi) = q f(\phi)$$ This is a second order linear homogeneous differential equation. The second order polynomial associated to it is $$\lambda ^2 - q = 0 \rightarrow \lambda = \pm \sqrt{q}$$ As both roots are real and distinct, the...

From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...

Given a wave function \Psi which is an eigenstate of a Hermitian operator \hat{Q}, we can measure a definite value of the observable corresponding to \hat{Q}, and the value of this observable is the eigenvalue Q of the eigenstate $$ \hat{Q}\Psi = Q\Psi $$ My question is whether it's a postulate...

I am wondering what's the best option to compute the eigenvalues for such a determinant$$\begin{vmatrix} \sin \Big( n \frac{\omega}{v_1} \theta \Big) & \cos \Big( n \frac{\omega}{v_1} \theta \Big) & 0 & 0 \\ 0 & 0 & \sin \Big( n \frac{\omega}{v_2} (2 \pi - \theta) \Big) & \cos \Big( n...

Anyone know what result this article is talking about? https://www.theatlantic.com/science/archive/2019/11/neutrino-oscillations-lead-striking-mathematical-discovery/602128/

A link to an interesting article I found is below: https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/

Hi PF! I'm trying to find the eigenvalues of this ODE $$y''(x) + \lambda y = 0 : u(0)=u(1)=0$$ by using the basis functions ##\phi_i = (1-x)x^i : i=1,2,3...n## and taking inner products to formulate the matrix equation $$A_{ij} = \int_0^1 \phi_i'' \phi_j \, dx\\ B_{ij} = \int_0^1...

I have a question relates to a 3 levels system. I have the Hamiltonian given by: H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|) I have been asked to find that H has an eigenvector with zero eigenvalues at any time t

Hi all- I am trying to obtain eigenvalues for an equation that has a very simple second order linear differential operator L acting on function y - so it looks like : L[y(n)] = Lambda (n) * y(n) Where y(n) can be written as a sum of terms in powers of x up to x^n but I find L is non self...

The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...

Hello, I would like to start with an assumption. Suppose a system is in the state: $$|\psi\rangle=\frac{1}{\sqrt{6}}|0\rangle+\sqrt{\frac{5}{6}}|1\rangle$$ The question is now: A measurement is made with respect to the observable Y. The expectation or average value is to calculate. My first...

I know the eigen value of energy in a Morse potential is Evib= ħωo(v+ 1/2) - ħωoxe(v+ 1/2)2 but is this the same for every Morse potential, given that the masses μ of the diatomic molecules are the same? The two potentials are these:

hey :) assume I have an operator A with |ai> eigenstates and matching ai eigenvalues, and assume my system is in state |Ψ> = Σci|ai> I know that applying the measurement that corresponds to A will collapse the system into one of the |ai>'s with probability |<Ψ|ai>|2. with that being...

Eigenvalues ##\lambda## for some operator ##A## satisfy ##A f(x) = \lambda f(x)##. Then $$ Af(x) = \lambda f(x) \implies\\ xf(x) = \lambda f(x)\implies\\ (\lambda-x)f(x) = 0.$$ How do I then show that no eigenvalues exist? Seems obvious one doesn't exists since ##\lambda-x \neq 0## for all...

Here's the problem along with the solution. The correct answer listed in the book for the eigenvectors are the expressions to the right (inside the blue box). To find the eigenvectors, I tried using a trick, which I don't remember where I saw, but said that one can quickly find eigenvectors (at...

Homework Statement Diagonalize the matrix $$ \mathbf {M} = \begin{pmatrix} 1 & -\varphi /N\\ \varphi /N & 1\\ \end{pmatrix} $$ to obtain the matrix $$ \mathbf{M^{'}= SMS^{-1} }$$ Homework Equations First find the eigenvalues and eigenvectors of ##\mathbf{M}##, and then normalize the...

Homework Statement Show that for $$W^\mu = -\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma},$$ where ##M^{\mu\nu}## satisfies the commutation relations of the Lorentz group and ##\Psi## is a bispinor that transforms according to the ##(\frac{1}{2},0)\oplus(0,\frac{1}{2})##...

Hey! :o We have the matrix $$A=\begin{pmatrix}2 & 0.4 & -0.1 & 0.3 \\ 0.3 & 3 & -0.1 & 0.2 \\ 0 & 0.7 & 3 & 1 \\ 0.2 & 0.1 & 0 & 4\end{pmatrix}$$ We get the row Gershgorin circles: $$K_1=\{z\in \mathbb{C} : |z-2|\leq 0.8 \} \\ K_2=\{z\in \mathbb{C} : |z-3|\leq 0.6 \} \\ K_3=\{z\in \mathbb{C} ...

Homework Statement Find the eigenvalues and eigenvectors of the following matrix: $$ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -1 & 0 \end{bmatrix} $$ Homework Equations Characteristic polynomial: $$ \Delta (t) = t^3 - Tr(A) t^2 + (A_{11}+A_{22} +A_{33})t - det(A) .$$ The Attempt at...

Homework Statement Find the eigenvalues and eigenvectors fro the matrix: $$ A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$. Homework Equations Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## . The Attempt at a Solution I've found...

Hello everyone, For weeks I have been struggling with this quantum mechanics homework involving writing a code to determine the energy spectrum and eigenvalues for the stationary Schrodinger equation for the harmonic oscillator. I can't find any resources anywhere. If anyone could help me get...

Hi PF! Given the ODE $$f'' = -\lambda f : f(0)=f(1)=0$$ we know ##f_n = \sin (n\pi x), \lambda_n = (n\pi)^2##. Estimating eigenvalues via Rayleigh quotient implies $$\lambda_n \leq R_n \equiv -\frac{(\phi''_n,\phi_n)}{(\phi_n,\phi_n)}$$ where ##\phi_n## are the trial functions. Does the...

[Note from mentor: This thread was originally posted in a non-homework forum, so it lacks the homework template. Even though the solution was resolved there, the thread has been moved here for future reference.]So I'm given Φ = N(φ1+2*φ2 + 3*φ3) and the operator A with eigen values λ1 = 1, λ2...

Homework Statement Following from \hat{b}^\dagger_j\hat{b}_j(\hat{b}_j \mid \Psi \rangle )=(|B_-^j|^2-1)\hat{b}_j \mid \Psi \rangle , I want to prove that if I keep applying ##\hat{b}_j##, ## n_j##times, I'll get: (|B_-^j|^2-n_j)\hat{b}_j\hat{b}_j\hat{b}_j ... \mid \Psi \rangle . Homework...

Homework Statement A particle with spin s=1/2 moves under the influence of a magnetic field given by: $$\vec{A}=B(-y,0,0)$$ Find the eigenvalues of the corresponding Pauli hamiltonian. Repeat the same process for: $$\vec{A}=\frac{B}{2}(-y,x,0)$$ Explain your result by relating the...

Homework Statement Again, consider the two-dimensional vector space, with an orthonormal basis consisting of kets |1> and |2>, i.e. <1|2> = <2|1> = 0, and <1|1> = <2|2> = 1. Any ket in this space is a linear combination of |1> and |2>. a) [2pt] The operator A acts on the basis kets as A|1> =...

Homework Statement I got a solution for finding eigen values. It evaluates to: (Lambda)^3 -12(lambda) -16=0, Then it that the list of possible integer solutions is: +-1, +-2, +-4, _-8, +-16 (i.e. plus minus 1, plus minus 2 and so on). I can't understand, why he says list of possible solution...

Hello! (Wave) Let $A$ be a $n \times n$ complex unitary matrix. I want to show that the eigenvalues $\lambda$ of the matrix $A+A^{\star}$ are real numbers that satisfy the relation $-2 \leq \lambda \leq 2$. I have looked up the definitions and I read that a unitary matrix is a square matrix...

defind ## \hat{A}f(x)=f(-x) ## find eigenfunction and eigenvalue I think ## \frac{d}{dx} ( \hat{A}f(x) ) = \frac{d}{dx} f(-x) ## ## \hat{A} \frac{d}{dx}f(x) + f(x) \frac{d}{dx} \hat{A} = -\frac{d}{dx} f(x)## ## \hat{A} \frac{d}{dx}f(x) + \frac{d}{dx} f(x) = -f(x) \frac{d}{dx} \hat{A}## ##...

I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it. 1. Homework Statement Given an infinite base of orthonormal states in the Hilbert space...

When one wants to represent a general ket in a basis consisting of eigenkets each attributed to an eigenvalue in a range, say from a to b, why does one take the integral of said kets from a to b w.r.t. the eigenvalues? I understand that the integral here plays a role analogous to a sum in the...

If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?

Homework Statement Find the normalised eigenspinors and eigenvalues of the spin operator Sy for a spin 1⁄2 particle If X+ and X- represent the normalised eigenspinors of the operator Sy, show that X+ and X- are orthogonal. Homework Equations det | Sy - λI | = 0 Sy = ## ħ/2 \begin{bmatrix} 0...

Find a basis for eigenspace corresponding to the listed eigenvalue: just seeing if these first steps are correct $$\begin{align*} A_{15}&=\left[ \begin{array}{rrr} -4&1&1\\ 2&-3&2\\ 3&3&-2 \end{array} \right],\lambda=-5&(1)\\ A-(-5)i&=\left[ \begin{array}{rrr} -4&1&1\\ 2&-3&2\\ 3&3&-2...

Hello, I have begun to teach myself Control Theory. I am looking for a book that is focused for mechanical engineers. I do not mind examples in electrical engineering, but they bore me (no offense). Also, I find some books begin with Laplace Transforms. Yet I found this online lecture...

In non-relativistic QM, given a wave function that has a degenerate eigenvalue for some observable, say energy. There is a whole subspace of eigenvectors associated with that single degenerate eigenvalue. How is the measurement probability for that degenerate eigenvalue computed from the...

Homework Statement The Hamiltonian of the positronium atom in the ##1S## state in a magnetic field ##B## along the ##z##-axis is to good approximation, $$H=AS_1\cdot S_2+\frac{eB}{mc}(S_{1z}-S_{2z}).$$ Using the coupled representation in which ##S^2=(S_1+S_2)^2##, and ##S_z=S_{1z}+S_{2z}## are...

In non-relativistic QM, say we are given some observable M and some wave function Ψ. For each unique eigenvalue of M there is at least one corresponding eigenvector. Actually, there can be a multiple (subspace) eigenvectors corresponding to the one eigenvalue. But if we are given a set of...

Homework Statement Find the eigenvalues and eigenvectors of the matrix ##A=\matrix{{2, 0, -1}\\{0, 2, -1}\\{-1, -1, 3} }## What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is the identity matrix?Homework EquationsThe Attempt at a Solution So I've found the...

Hi, one more question! How do I prove that A has eigenvalues equal to its singular values iff it is symmetric positive definite? I think I have the positive definite down but I can't figure out the symmetric part. Thanks!

Homework Statement Homework EquationsThe Attempt at a Solution I solved it by calculating the eigen values by ##| A- \lambda |= 0 ##. This gave me ## \lambda _1 = 6.42, \lambda _2 = 0.387, \lambda_3 = -0.806##. So, the required answer is 42.02 , option (b). Is this correct? The matrix is...

Homework Statement The Hamiltonian of a certain two-level system is: $$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$ Where ##|1 \rangle, |2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...

Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...

Homework Statement Coupled Harmonic Oscillators. In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 6.2 to the case where the oscillators are coupled. Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure...

Hi PF! I am looping through a linear system and each time I do I generate a new matrix, call this matrix ##A##. When finding the eigenvalues of ##A## in Matlab is use [a,sigma2M] = eig(A);% a eigenvector and sigma2M matrix of eigenvalues sigma2(:,ii) = sum(sigma2M);% create matrix with rows of...

Hi, I am studying about circulant matrices, and I have seen that one of the properties of such matrices is the eigenvalues which some combinations of roots of unity. I am trying to understand why it is like that. In all the places I have searched they just show that it is true, but I would like...