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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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1answer
57 views

Finding the eigenfunctions of the $\hat{\vec l}.\hat{\vec s}$ operator for a single p-electron

I'm trying to calculate the SO-coupling for a single p-electron ($l=1$, $s=\frac{1}{2}$) in the uncoupled representation. This comes down to calculating these matrix elements: $$\left\langle ...
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1answer
466 views

What is the point of degenerate perturbation theory in quantum mechanics?

What is the point of degenerate perturbation theory in quantum mechanics? Let's disregard for a moment the issue of constructing the perturbed wave functions and assume that the 1st order correction ...
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585 views

Hamiltionian spectrum in unstable systems

I have heard that the eigenvalue of Hamiltonian in an unstable system can contain an imaginary part corresponding the tunneling. Is that true? If it is the case, then I am very confused about it. Let ...
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1answer
318 views

Solving the radial part of the Schrodinger equation for a central potential with two radial terms

a "hydrogen-like" atom has the following modified coulomb potential $$V(r)=\frac{-Ze^2}{r}+\frac{\alpha}{r^2}$$ Where Z is the number of positive charges and $\alpha$ is a positive energy constant. ...
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653 views

Why does the expectation value of a QM operator squared $\hat{Q}^2$ take the form $\big\langle \hat{Q}^2\big\rangle=\sum\limits_n |a_n|^2{q_n}^2$?

I understand the proof for the expectation value of $\langle\hat{Q}\rangle$, which is shown for reference. Note, if you are already familiar with this proof then there is no need to read the contents ...
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1answer
85 views

Eigenstates of Conical Potential in 3-dimensions?

If we take an ordinary time-invariant Schödinger equation: $$H|\psi\rangle = E|\psi\rangle,$$ and use a conical potential $V(r) = A r$ we get a differential equation: $$\left[-\left(\frac{\hbar^2}{2m}\...
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2answers
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Finding the ground state of a Hamiltonian Matrix

I have numerically constructed a Hamiltonian matrix. I am currently finding the ground state by full diagonalisation of the matrix (with the GSL library) and finding the most negative eigenvalue and ...
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2answers
454 views

Calculating eigenvalues from time independent Schrödinger equation [closed]

We calculate eigenstates by solving the time independent Schrödinger equation. The time independent Schrödinger equation is an energy eigenvalue problem, so we will get energy eigenstates as the ...
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1answer
98 views

Eigenvalue problems in finite volume

Does every eigenvalue problem of a selfadjoint operator which should be solved in a finite volume have a discrete set of eigenvalues (no matter it is treated classically or in a quantum mechanical ...
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1answer
567 views

Solving Majorana Hamiltonian

I am currently going through the Kitaev paper on quantum wires [1]. Given a Hamiltonian of the form $H = 1/2 i \epsilon b' b''$, where $b'$ and $b''$ are majorana modes, we can solve it by ...
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2answers
482 views

Eigenfunctions of angular momentum operators in momentum representation

I want to know the eigenfunctions of $L^2$ and $L_z$ in the momentum representation. Do I need to Fourier transform the spherical harmonics?
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2answers
178 views

The eigenfunctions of vanishing eigenvalues of Dirac operator

Consider the eigenfunctions to the operator $i\gamma^\mu D_\mu$: $$i\gamma^\mu D_\mu\Psi_i=\lambda_i\Psi_i.$$ Because $\{\gamma^5,\gamma^\mu\}=0$, we know $$i\gamma^\mu D_\mu\gamma^5\Psi_i=-\lambda_i\...
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293 views

Bra-ket notation average measured value

Let's consider an operator $A$ with eigenket $|a^\prime\rangle$. Then the average measured value according to 1.4.6 is $$\langle A\rangle=\sum_{a^\prime} \sum_{a^{\prime\prime}} \langle \alpha|a^{\...
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52 views

Which eigenvectors should I ignore?

Solving an Euler-Bernoulli equation for a beam using Finite Element Method leads to the following equation $$M \ddot{\vec d}(t)+K \vec d(t)=\vec F$$ where $M$ is system mass matrix, $K$ system ...
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1answer
711 views

Matrix representation of operators

I have a question about the matrix representation of quantum operators, in one of the books I'm reading I found this: Let ${\{\psi_{n}\}}$ be a complete orthonormal system and $\bf A$ a operator. ...
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11k views

Energy eigenvalue of Hamiltonian operator

Hamiltonian is the total energy of the system. Then, is its eigenvalue $E$ also total energy of the system? What is the difference between them? Both of them are energy.
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1answer
805 views

Eigenvalues of Hamiltonian for a system w/ three interacting spin degrees of freedom with spin-1/2

I have three interacting spin-1/2 particles and I want to find the energy eigenvalues of H. The Hamiltonian for the system is $ H = \frac{J}{\hbar^2}(S_1\cdot S_2+S_2\cdot S_3+S_3\cdot S_1)$ (where J ...
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1answer
249 views

Eigenvalue of Hermitian Matrices [closed]

The way I have proved the Eigen values of Hermitian matrices are real like this: I considered $H$ is a hermitian matrix. Operator applied in ket space: $\left<\psi|H | \psi\right>= \lambda$ ...
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1answer
296 views

Multiplicity of degeneracy.

Let the Hamiltonian operator of a rigid dumbbell (such as $O_2$) be $\hat{H}=\hat{L}^2/(2I)$, whereas I is the moment of inertia and $L_i$ is the angular momentum of the molecule by rotation around ...
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1answer
800 views

How to prove bound state's spectrum must be discrete and scattering state's spectrum must be continuous?

Consider $d$-dim Schrodinger equation without internal degree of freedom, that is, we don't consider spin etc. How to prove that bound state's spectrum must be discrete and scattering state's ...
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0answers
61 views

Matching measured eigenvalue of a matrix [closed]

Given a Matrix of the form below, we have 8 variables, $\omega_{i}$ and $J_{ij}$. we want to diagonalise the Matrix to obtain values to match the observed Eigenvalue in an Experiment. i.e. $\bar{\...
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1answer
108 views

Does dividing a box full of gas at equilibrium change its entropy? [duplicate]

Imagine you have a box full of a gas. It is at thermodynamic equilibrium. The entropy of the gas in the box is proportional to the logarithm of the number of available microstates. The number of ...
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899 views

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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2answers
58 views

Trouble understanding math in expectation value definition

This is the part of the equation I do understand: $$ \langle A \rangle_\alpha = \langle \alpha | A | \alpha \rangle = \sum_{a}\sum_{b} \langle \alpha | b\rangle \langle b | A | a\rangle \langle a | \...
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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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3answers
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Physical interpretation of Neumann boundary conditions for wave equation on a disk?

If you take consider the wave equation on a disk $D$, then if we use Dirichlet boundary conditions, it means the wave function is fixed at $0$ on the boundary of the disk, and if we consider the ...
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1answer
772 views

To prove that an operator commutes with any function of it

(This is a homework question.) The question is to prove that a general operator $\hat{A}$ commutes with any function $\hat{B} = f(\hat{A})$. $$ \newcommand{\ket}[1]{\left| #1 \right\rangle} \...
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1answer
106 views

Why are eigen-things better/more useful than non-eigen-things?

An eigenvector is a vector that doesn't change direction when a transformation is applied. So in the case of, say, an energy (Hamiltonian) eigenstate, it's a state that doesn't change 'direction' (...
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1answer
171 views

An apparent inconsistency in the form of Spin-z operator for spin 1/2 system [closed]

Problem: This is problem 7 from the first chapter of Modern Quantum Mechanics by Sakurai (page 59). Consider a ket space spanned by the eigenkets $\{ \mid a'\rangle \}$ of some Hermitian operator $...
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1answer
91 views

Name of $H|a\rangle=n|a\rangle$

I was wondering if the form $$H\vert a\rangle=n\vert a\rangle$$ has a proper name. I am familiar with each part like the Hermitian matrix, eigenvalue and eigenstate, but is there a word to classify ...
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2answers
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Eigenvalues of Hermitian operators are real and the dependence/independence of boundary conditions

Without reproducing proofs: Eigenvalues of a Hermitian operator are real (proof does not rely on the boundary conditions). The momentum operator is Hermitian (proof does not rely on the boundary ...
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1answer
47 views

Oscillator in Energy Basis Lowering and Raising Operators

On page 205 of Shankar's Intro to Quantum Mechanics, equation 7.4.12 does not make sense to me. I understand why a|e> is an eigenvector and why e-1 is its Eigenvalue, but I don't understand how that ...
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1answer
834 views

Eigenvalue equation for kinetic and potential energy

In Boas' Mathematical Methods there is a section on linear algebra in which it is stated that we can write the eigenvalue equation for a set of springs using the kinetic energy and the potential ...
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3answers
142 views

In quantum mechanics how can eigenfunction, eigenvalues, matrix methods give us values of real physical quantities? [closed]

Eigenfunction, eigenvalues, eigenstates & matrix methods used in quantum mechanics seems purely mathematical.How can they give us values of real physical quantities in quantum mechanics?
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1answer
185 views

Neutron in a magnetic field (Schrödinger Equation, Eigenstates, Eigenvalues).

Consider the spin of a neutron in a magnetic field $\vec{B}$. A neutron is a neutral particle with the mass of a proton and the spin $\frac{1}{2}$. The Hamiltonian is $H=\mu_n\vec{S}\cdot\vec{B}$...
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1answer
546 views

Eigenvalues of spherical harmonics in $d$ dimensions

I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...
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1answer
233 views

Zeeman effect - eigenstates and its degeneracy - with and without magnetic field

Consider a hydrogen atom in a homogeneous magnetic field $\vec{B}=B\vec{e_z}$. Using the coulomb gauge ($\nabla \vec{A}=0$) we can take $\vec{A}=\frac{1}{2}\vec{B}\times \vec{r}$ as a vector potential....
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1answer
179 views

Random operator in heisenberg/schrödinger picture (heisenberg's equation of motion) [closed]

Consider a system whose hamiltonian isn't explicitly dependent on time. Let A be the operator for the eigenvalue a in the Schrödinger picture and $A_H=U^\dagger A U$ the corresponding operator in the ...
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1answer
77 views

How to understand Preskill's argument for degeneration of eigenstates?

In his notes on topological quantum computation on page 18, Preskill uses the "commutator" $T_2^{-1}T_1^{-1}T_2T_1 = e^{-2 i \vartheta}$ to show that the eigenstates of $T_1$ are degenerate. But I don'...
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1answer
346 views

Reasoning behind taking the Fourier transform of the fermionic operators for a circular $1$D spin chain [closed]

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
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1answer
118 views

State and measurement after a sequence of measurements

Hi I just want to confirm my interpretation of the following question: Let the quantum state be given as $$|\psi_0 \rangle = [\sqrt{2}|\phi_1 \rangle + \sqrt{3}|\phi_2 \rangle + | \phi_3 \rangle + |\...
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1answer
417 views

Verification of proof of complete set of commuting operators

Hi I am interested in the validity of the following proof. I am interested in the validity of this particular proof as I am aware of how to prove this result in a different way. Theorem: If two ...
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0answers
312 views

Eigenstates of 2D harmonic oscillator in a constant magnetic field

I want to find the eigenstates of the 2D harmonic oscillator in a constant magnetic field $\vec B = \vec B(x,y)$. My Hamiltonian reads $H_0 = H_{xy} + H_z$ where $H_{xy}$, is the hamiltonian of the ...
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1answer
2k views

Eigenstates of position and momentum operators in QM

In Griffiths pages 103-105 "Introduction to Quantum Mechanics" 2nd editiion he states that the eigenfunctions of the position and momentum operators are $$g_y(x) = \delta(x-y)$$ where the eigenvalue ...
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1answer
484 views

probability of finding the system in the ground state [closed]

$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ Assume that a quantum mechanical system is described by two orthonormal states $\ket{+}$ and $\ket{-}$, defined by the property of being the ...
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1answer
541 views

How to check if a Hamiltonian is PT symmetric or not?

Consider the Hamiltonian $$H=p^2+ix^3+ix.$$ This paper by Carl M bender claims this is a $PT$ symmetric Hamiltonian. In this he describes $PT$ symmetry as parity $P$, whose effect is to make ...
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74 views

Physical Meaning of Over- / Underdetermined Acoustic Eigenvalue Problem

I am performing an eigenmode study on a system of acoustic ducts. The system consists of two large cylindrical volumes connected by several smaller cylindrical volumes (modeling a combustion chamber ...
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1answer
253 views

Diagonalisation: Schmidt vs eigenvalue - when to use which?

In physics we encounter diagonalisation of matrices or operators in a variety of areas. But there are different kinds, the main two being Schmidt decomposition and eigenvalue diagonalisation. The two ...
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3answers
1k views

What does a zero eigenvalue mean to its eigenstate?

Assume that initial wave function had the form of $\psi(x)= u_1(x) + u_2(x)$ where $u_1$ and $u_2$ are eigenfunctions of $\psi(x)$ to an observable operator $S$. The eigenvalues of $u_1$ and $u_2$ are ...
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1answer
55 views

Taking Measurements of Quantities in QM

I have a quick question relating to Annihilation and Creation operators, and in taking observables in general. Let's say, for instance, that I prepare a particle so that I consider the projection of ...