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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Efficient ways of numerically calculating eigenenergies?

I want to calculate the eigenenergies of the ground state, first excited state and second excited state for the Hamiltonian: $$ H = (1-s)H_x + sH_z$$ for $s=0, 0.01, 0.02, ..., 1$. The Hamiltonian $...
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How the eigenvalue problem was solved?

In Gasiorowicz 3rd edition Chapter 3, I've tried to solve this problem I checked the solution's manual, When I tried to integrate it, the answer I got is $$ \psi(x)=Ce^{x^2/2\lambda} $$ Can you ...
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What is an eigensystem? Could you provide a simple example? [closed]

Also, what is the difference between an eigensystem and the eigenspace?
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Eigenstates meet the condition but resulting Linear Combination state doesn't

There is something I don't understand. When we solve the Time Independent S.E. for example for a particle on a circle $[0,L]$ we have our state $\Psi$ that writes as a Linear Combination of $\Psi_n$ (...
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How to get the weight of an eigenstate inside the state of the system without knowing the state?

Let us suppose we have a system in a state $\Psi$, with: $\Psi = \sum_m c_m \psi_m$ Let us further suppose that we don't know what $\Psi$ or the $c_m$ are, but that we know what the $\psi_m$ are since ...
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What is the physical meaning of the eigenstates of an operator in quantum mechanics?

Let us suppose that we have an Hamiltonian that describes a quantum system. If one would like to know all of the possible values that the energy of the system described by that hamiltonian, one has to ...
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Do operators always give a number after operating?

I am having some doubts regarding operators. In QM, when operators work on a wave function, will it always give a number times the wave function? Suppose I applied it on any normal function of x. Will ...
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Probability of possible results of measurement - hydrogen atom

I know that hydrogen atom's eigenfuction is : $\psi(\vec r)= \sqrt{ 1 \over 24 a_B^5} r e^{-r \over2a_B} Y_1^0 |+>$, where $Y_1^0 =\sqrt{3 \over 4 \pi} cos\theta$. So, since I have just one ...
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Periodicity of Wavefunction and Eigenvalues of Angular Momentum

I am currently looking through the following paper on circuit quantum electrodynamics in order to explore connections between a Cooper pair Hamiltonian and the quantum rotor model. Here is an excerpt ...
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What does eigenvalues of the Lorentz matrix represent physically speaking? [duplicate]

In special relativity, if we have a boozt in the x - direction, the relationship between the coordinates of the inertial frame of reference S, and the one of S' (moving with velocity v relative to S), ...
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Meaning of the Eigenvalues of the Hessian Matrix in Polar Coordinates in a Ferromagnetic System

I have the Hamiltonian for a magnetostatic system (exchange, dipole-dipole, zeeman) which is in polar coordinates since the spins are confined to being in-plane. If I calculate the Hessian Matrix of ...
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Why not imaginary eigenvalues in the eigenvalue equation $A\psi=\lambda\psi$?

in quantum mechanics we always see the eigenvalue equation $\hat A\psi=\lambda\psi$ and $\lambda$ is the probability amplitude meaning $\lambda^2$ is the actual probability of finding the system in ...
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Instantaneous eigenstate and time dependent Schrodinger equation

Instantaneous eigenstate $\psi(t)$ is defined as $\hat H(t)\psi(t)=E(t)\psi(t)\tag{1}$ But in the lecture notes of Quantum Physics III MIT (in the section of adiabatic approximation), it is written ...
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Clarification on the Premises of the Einstein-Podolsky-Rosen Argument

In their famous EPR paper, Einstein, Podolsky and Rosen argue that quantum mechanics does not provide a complete description of physical reality. To do this, they make two key assumptions: in a ...
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How to understand Heisenberg time in random matrix theory?

Recently, from few papers, I have encountered the word 'Heisenberg time' $t_{\text{H}}$ which is an inverse of a mean level spacing $\Delta(\hat{\mathcal{H}})$ of a finite system Hamiltonian $\hat{\...
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Why is the ground state of an atom never degenerate?

In this paper https://link.springer.com/article/10.1007/BF01391720 , Kellner argues that the ground state of the helium atom must be spherically symmetric because "it is known that the ground ...
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Commutation relation confusion of ladder operators in Quantum Mechanics

Suppose that $X$ and $N$ are operators such that they follow the commutation relation $$[N,X]=cX$$ for some scalar c. In this Wikipedia article it is shown that if $|n \rangle$ is some eigenstate of ...
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Valid Intuition? - Why observables are represented by eigenstates/eigenvalues

So I've been frustrated with the usual presentation of the operator formalism being presented as an axiom, and have been after a more intuitive explanation. Would the following intuition be considered ...
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Wavefunction vs state vector

When I was first introduced in quantum mechanics I learned that the wavefunction $\psi (x, t)$ (one dimension for simplicity) contains all the information about the system like $x(t)$ in classical ...
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Physical Interpretation of eigenvalues/eigenvectors of the density matrices of a given order

I was reading this paper by Per-Olov Löwdin and it discusses how density matrices can be used to represent/interpret the wavefunction. And, I had a question regarding how the eigenvalues and ...
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Is an Hamiltonian like this non-degenerate? [closed]

If I have a system of only two energy levels $E_0$ and $E_1$ and an Hamiltonian $$H=\begin{pmatrix} E_0 & 0 \\ 0 & E_1 \\ \end{pmatrix}$$ can the Hamiltonian be both degenerate and non-...
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Dirac notation notation equality

I'm going trough my quantum mechanics notes but I don't understand why: $$H|\phi_{m}\rangle\langle\phi_{n}|-|\phi_{m}\rangle\langle\phi_{n}|H=a_{m}\langle\phi_{n}|-a_{n}|\phi_{m}\rangle.$$ What is ...
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Form of the wave function in the TDSE

I was reading the derivation of Expectation value of energy for an eigen-state $\psi_n(\overrightarrow{r,}t)$ There $<E>$ is expressed as: $$\int_V{\psi_n^*}(\overrightarrow{r})e^{iE_nt/ℏ}\hat{E}...
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How to obtain the eigenvalues of Hamiltonian $H=p^2+x^2 (i x)^{\epsilon}$?

For quantum harmonic oscillator Hamiltonian $H=p^2 + x^2$, one can calculate the eigenvalues $(2n+1)\omega/2~$ ( with $\hbar =1).$ Now I came across this Hamiltonian $$H=p^2+x^2 (i x)^{\epsilon},\tag{...
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Orthogonality of eigenfunctions

I came across a derivation in my book which proves that any two eigenfunctions with different eigenvalues are orthogonal. The derivation goes like this: $$\hat{A}u_1(x)=a_1u_1(x)\tag{i}$$ $$\hat{A}u_2(...
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Stark Effect in Hydrogen Degenerate Perturbation Theory

I am going though this example of degenerate perturbation theory. We are examining the Stark effect in hydrogen for $n=2$. After finding the 4 degenerate cases; $|0, 0⟩, |1,0⟩, |1,1⟩, |1,-1⟩$, we ...
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Infinite circular well and Bessel functions

When we study one particle in the infinite circular well we have a potential $$ V(r) = \begin{cases} 0 & r\le a\\ \infty & r>a \end{cases} $$ so that TISE is: $$\left(-\frac{\hbar^2}{2m}\...
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Obtaining Dirac spectrum on unorientable manifold ($RP^n$) from orientable manifold

The Dirac spectrum for $S^n$ is well known along with its multiplicities. In Appendix D of https://arxiv.org/abs/1510.05663 author computes dirac spectrum of $RP^4$ from that of $S^4$. The argument ...
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What is the difference between an eigenfunction and a wavefunction?

This question is an additional point of clarification to my previous question about adding position and momentum eigenstates. For simplicity, suppose I had a particle in an eigenstate of momentum, $|p\...
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Doubt on Pauli matrices eigenvectors

I want to write the $\sigma_{z}$ eigenvectors $\{|+\rangle,|-\rangle\}$, in terms of the eigenvectors of $\sigma_{x}$, $\{|+\rangle _{x},|-\rangle _{x}\}$. But I simply didn't reach the right result: $...
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Time-independent Schrodinger equation from energy eigenvalue equation [duplicate]

$$ \left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right]\langle x \mid E\rangle=E\langle x \mid E\rangle $$ is often referred to as the time-independent Schrödinger equation in ...
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Why is radius of gyration for an ellipsoid derived from a tensor different than the formula?

The first method one can use to solve for radius of gyration, $R_g$, of an ellipsoid with semi-axes a, b, and c involves using the formula: $$ R_g = \sqrt{\frac{a^2+b^2+c^2}{5}}$$ However, it appears ...
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What is the physical meaning of the eigenvalues of a state-space representation of a physical system?

Consider the following state-space model of a physical system. $\begin{bmatrix} \dot{{x_1}} \\ \dot{{x_2}} \end{bmatrix} = \begin{bmatrix}\ 0 & 1 \\ 0 & -c/m \end{bmatrix}\begin{bmatrix} \ ...
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Eigenvalues of Hamiltonian in Another Basis

I am taking a quantum mechanics class and was assigned this problem: Among other things, I am asked to find the eigenvalues of $H$ in terms of $a$, $b$ and $\sigma$. I'm sort of lost of even how to ...
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Quantum numbers which are (not) eigenvalues of the symmetry operators

I noticed that if a QM Hamiltonian is translationally&rotationally invariant (typically it is a single-particle version of some QFT Hamiltonian, like Klein-Gordon, Dirac or free Schrödinger in the ...
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Why do the boundary conditions change the eigenvalues between unitarily equivalent Hamiltonians?

Let's say we have a particle in a circle (let e.g. $x = x+1$) with Hamiltonian $$ H = \frac{1}{2} p^2, $$ Let's also set $\hbar = 1$. Now the solution of the Schrödinger equation $H \psi(x) = E \psi(x)...
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Symmetries in QM

I have the following question; if we an operator corresponding to a spacetime translation: $$ \hat{\Omega}$$ and a hermitian operator, $$\hat{A}$$ commutes with this translation: $$[\hat{\Omega},\hat{...
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Eigenvalue Equation : $ A |\psi\rangle = 0 |\phi\rangle$?

A typical eigenvalue equation goes like: $ A |\psi\rangle = e\: |\psi\rangle$, where $|\psi\rangle$ is an eigenstate for operator $A$ with eigenvalue $e$. Suppose that $e=0$ in the above equation, ...
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Contradiction in TISE of two particle system $V(x_1-x_2)$

For this Hamiltonian, we have the classical Poisson bracket $\{H,p_1+p_2\}=0$. Quantising gives $$[H,P_1+P_2]=0.$$ Eigenvectors of $P_1+P_2$ are the functions $e^{i(p_1x_1+p_2x_2)}$. I say these are ...
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Dirac's Principles of QM the eigenstate existence problem [duplicate]

In the second chapter (pp32-34) PAM states that should a real linear operator obey a certain expression (the characteristic equation, from what I understand) of degree $n$ then there are 2 statements ...
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How come the spin operator has one-half or integer eigenvalues? [duplicate]

The generator of spin corresponds to the 90 degree rotation 2x2 matrix, right? A block-diagonal matrix composed of these 2x2 matrices will generate the rotations of the vectors in a wave function of ...
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Discreteness of spectrum of Hamiltonian operator on a bounded interval

Consider the following Schroedinger equation on a bounded interval $[-a,a]\in \mathbb{R}$: \begin{equation} -\frac{d^2}{dx^2}\Psi(x)+V(x)\Psi(x)=0 \end{equation} with boundary condition \begin{...
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Need help understanding product of Kronecker delta function with an outer product

I need help solving the the below expression which is from the book Modern Quantum Mechanics by J. J. Sakurai and Jim Napolitano. $$A=\sum_{a''}\sum_{a'}|a''\rangle a'\delta_{a'a''}\langle a'| =~?$$ I ...
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2 votes
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Time evolution Operator: write expression for $t>0$

I have a question related to time evolution operator. I was analyzing my teacher solution after solving the problem myself, but there is a detail I dont get. I have a hamiltonian that is represented ...
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2 votes
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Is the eigenvalue of an eigenstate the same as its (global) phase?

I'm trying to understand Shor's Algorithm by reading this Qiskit textbook. At some part the following equation comes up: \begin{equation} U |u_0\rangle = \frac{1}{\sqrt{12}} \begin{pmatrix} |3\rangle +...
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Eigenfunction of Momentum operator

I've been having trouble solving this question given by one of my professors. If eigenfunction of momentum operator is $e^{-x^3}$, then calculate its eigenvalue. So far, if $p = i(h/2\pi)(d/dx)$, if ...
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2 votes
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Observables in QM, are they the eigenvalues or the elements of the spectrum?

The question is rather simple. We all hear all the time that an observable of a quantum mechanical system (for some observable that is given by a some self-adjoint operator, let's call it $H$) is ...
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Why does a Hermitian operator have a basis of its own eigenvectors?

Suppose I have a hermitian operator $\Omega$. The proof of the existence of a orthonormal eigenbasis as given in Shankar is given. What I don't understand is why the second eigenvector $\left| \...
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Number Operator Eigenvectors of Amplifier

For the number operator $\hat{N}$, it's eigenvectors are the Fock basis vectors $|n\rangle$, as $\hat{N}|n\rangle = n|n\rangle$. Let us suppose we have a bipartite set of basis vectors $\{|n,n\rangle\...
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Why aren't the eigenvectors of a tight-binding Hamiltonian periodic?

I try to calculate the Berry connection for a simple graphene model and stumbled across the following question. Suppose I have a tight binding Hamiltonian (further details here or here): $$H = \begin{...
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