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, ...

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0
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1answer
75 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 ...
7
votes
2answers
292 views

Determinant and adjunct of $k-\omega^2m$ in terms of natural frequencies

Given is a mechanical multiple degree of freedom system described by the following matrices and equation: mass matrix ${\bf{m}} = \left[\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 ...
0
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1answer
42 views

probability of finding the system in the ground state [on hold]

$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ Assume that a quantum mechanical system is described by two orthonormal states $\ket{+}$ and $\ket{-}$, defined by the property of being ...
-2
votes
1answer
57 views

Eigenvalues and states of hamiltonian [closed]

A quantum mechanical system is described by a two dimensional Hilbert space of states, spanned by an orthonormal basis {|1>, | − 1>}, with the following Hamiltonian: $ H | 1> = | ...
4
votes
4answers
229 views

How does one describe a state with a density matrix after measuring position?

My question is about position measurement in non relativistic quantum mechanics. I've been taught that when you measure the value of an observable for some state of a system described by ...
0
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1answer
33 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 ...
2
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3answers
190 views
3
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0answers
34 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 ...
1
vote
2answers
94 views

Pauli Matrices & 2D Rotation Operators?

I was doing a strange calculation with my teacher the other day: find the eigenvalues and eigenvectors of the 2D rotation operator. Intuitively, there should be no solution to this problem in ...
7
votes
2answers
1k views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
0
votes
0answers
30 views

Generalized Eigenvalue Problem from linear stability analysis

I am trying to solve a generalized eigenvalue problem raised by linear stability analysis $$AV=\lambda BV.$$ $A$ and $B$ are non-symmetric complex valued matrices. The set of equations I am trying to ...
0
votes
0answers
19 views

How strong is the HCN Union when modelling with springs

I'm modeling the HCN Molecule with springs, giving the bounds between H and C the name k1 and between C and N k2. Is there any information of how strong is the bound? We were asked to get the ...
2
votes
1answer
68 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 ...
12
votes
3answers
5k views

Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
3
votes
1answer
388 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: $$\phi(\vec{k}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{r\text{ space}} \psi(\vec{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^3r$$ for momentum space and ...
0
votes
1answer
74 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 ...
-1
votes
1answer
46 views

Deriving eigen values of $\hat{N}$

So let's say we have an operator $\hat{a}$ (ladder operator), where $\left[\hat{a},\hat{a}^\dagger\right] = 1$, and $\hat{a}^2 |\phi\rangle = 0$. How do I show that the eigenvalues of ...
1
vote
1answer
27 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 ...
0
votes
1answer
467 views

Dirac Delta Potential and bound/scattered states

Why does the attractive Dirac Delta distribution (function) potential $V = \alpha\delta$(x) (for negative $\alpha$) yield both bound AND scattered states? Is this due to the definition of the Dirac ...
1
vote
0answers
22 views

Particle in a $V(\rho)$ potential in cylindrical coordinates

Consider cylindrical coordinates $(\rho,\phi,z)$ and consider a particle with a potential energy $V(\rho)$. If we write the Hamiltonian operator in these coordinates we find that $$H = ...
3
votes
1answer
103 views

Quantum mechanics - measuring position

I am watching Susskind's Stanford Lectures on quantum mechanics. The eigenvectors (eigenfunctions) of the position operator are of the form $\delta(x-k)$. But $$\int\delta^{*}(x-k)\delta(x-k)\, ...
3
votes
1answer
1k views

How does a state in quantum mechanics evolve?

I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as $$ i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle $$ I am ...
3
votes
4answers
169 views

Help understanding proof in simultaneous diagonalization

The proof is from Principles of Quantum Mechanics by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
1
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2answers
49 views

What is localization length of eigenvectors?

Apology if this question is not appropriate. I was looking to associate entropy to eigenvectors for some of my work and I found the link http://chaos.if.uj.edu.pl/~karol/pdf/ZK94.pdf . This leads to ...
1
vote
1answer
341 views

Diagonalize a dot product with Pauli matrices

How can I diagonalize the following operator? $$\lambda \hat{\vec{\sigma}}\cdot\vec{r}$$ where $\lambda$ is a real constant, $\hat{\vec{\sigma}}=(\hat{\sigma_{x}},\hat{\sigma_{y}},\hat{\sigma_{z}}) ...
1
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0answers
30 views

How to estimate the ground state of a potential well when a confinement dimension is added

I have a finite harmonic potential where I trap an electron. The confinement length changes in size. Now, I'm interested in the ground state energy, so I have this 1D Poisson solver which gives me the ...
-1
votes
1answer
146 views

Meaning of Eigenvalues/Eigenvectors of a linear system of equations

I have a 41x41 system of linear equations (inhomogen) which I derived with Eureqa by describing the timecourse of fMRI haemodynamic data from a brain area as a function of the timecourses of 40 other ...
0
votes
1answer
94 views

eigenvalues and eigenvectors for a generalized Pauli matrix in spherical coordinates [closed]

If we have $nz=\cosθ$, $nx=\sinθ\cosφ$ and $ny = \sinθ\sinφ$, how we can compute the eigenvectors of the Linear Operator of the spin ?
1
vote
0answers
47 views

Eigenvalues of Hamiltonian with on-diagonal coordinate

A bit abstract, but if I take the standard graphene Hamiltonian (around the Dirac point) and introduce an on-diagonal term proportional to the coordinate $\hat{y}$, how would I find the eigenstates ...
3
votes
0answers
95 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
1
vote
1answer
81 views

what is eigenvalue of $P^{1/n}$ operator if we know eigenvalue equation of $P$ ? [closed]

If $P$ is an operator and $PΨ=pΨ$ ( $p$ as the eigenvalue ) then is it true to say $P^{1/n}Ψ=p^{1/n}Ψ$ ( n is an integer and positive number )
5
votes
1answer
82 views

Can I *always* decompose a normalizable function into the discrete Hydrogen spectrum?

This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the discrete set of Hydrogen wavefunctions ...
1
vote
2answers
179 views

Eigenvalue physical meaning [closed]

What is the physical significance of eigenvalues or eigenvectors?? Please try to explain in very simple language simple harmonic oscillator , potential well could you support your answer by ...
1
vote
0answers
25 views

Expectation value of a state eigenvalue [closed]

Why is it such that Expectation value of an Observable quantity is always equal to one of the eigenvalue of the its operator?? an it be proved mathematically??
11
votes
2answers
489 views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
0
votes
1answer
64 views

Combination of quantum numbers for a particle in a 3D box

For a second excited state, the three combination of quantum number corresponds to $$n_{1}=2,n_{2}=2,n_{3}=1$$ or $$n_{1}=2,n_{2}=1,n_{3}=2$$ or $$n_{1}=1,n_{2}=2,n_{3}=2.$$ This is from the text ...
4
votes
2answers
110 views

Shifting momentum by a constant in the Schrodinger Equation

My book states that if we perturb a given Hamiltonian for the Schrödinger Equation $$ H = \frac{p^2}{2m} +V(x) $$ to $$ H' = \frac{p^2}{2m} + V(x) + \frac{\lambda p}{m} $$ then we can rewrite ...
1
vote
2answers
139 views

Are the eigenstates of an operator time independent?

In the Schrodinger picture, are the eigenstates of an operator time independent? Is it their expectation values that evolve in time rather than the actual eigenstates? For example, say I have an ...
8
votes
5answers
378 views

Why does the measurement of some observable $A$, the measured value is always an eigenvalue of the operator?

Explain why when we make a measurement of some observable $A$ in QM, the measured value is always an eigenvalue of the operator $A$.
2
votes
1answer
37 views

Optimizing the second, third,… eigenvalues - applications

I'm working on some topics related to spectral optimization as a function of the domain. For example it is known for almost a century (lord Rayleigh and Faber, Krahn) that the shape which minimizes ...
1
vote
0answers
43 views

Showing two wavefunctions are proportional to one another [duplicate]

I am struggling to answer the following question: Let ψ₁(x) and ψ₂(x) be normalisable energy eigenfunctions for a particle of mass m in one dimension moving in a potential V(x). Suppose that ψ₁ and ...
0
votes
1answer
80 views

Confused on how to interpret the energy eigenfunction of Hydrogen

So here is an image of the third lowest energy eigenfunction of an electron in a hydrogen atom: Image from http://imgur.com/Lu4MocL I understand well the eigenfunctions given by Schrodinger's ...
0
votes
1answer
41 views

quantum mechanics probability of +1 spin between arbitrary directions

So there are two unit vectors $\hat{m}$ and $\hat{n}$ with arbitrary directions in 3D space. There is a spin operator along a particular direction in space, say that of $\hat{r}$, is: $\sigma_r= ...
0
votes
1answer
56 views

Calculate mean number of particles of time evolution coherent state [closed]

I seem to be missing some identities. I know you need to calculate P_n = |<n|alpha_t>|^2 and mean number of particles is the infinite sum of nP_n. However I ...
4
votes
1answer
97 views

Closure relation for degenerate eigenkets

Consider an observable in quantum mechanics, with a degenerate eigenvalue in a continuous spectrum. Is it possible for such an eigenvalue to have a finite degeneracy? If the degeneracy is infinite, ...
1
vote
0answers
52 views

Why Hamiltonian is Hermitian? [duplicate]

Everyone knows that this is needed to make eigenvalues real, but still why we enforcing such a structure at first place? An arbitrary operator can have as complex as real eigenvalues, we can simply ...
7
votes
1answer
117 views

Constructing differential equation from arbitrary Hamiltonian

Suppose I begin with the time-independent Schrodinger equation $$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$ ordinarily we specify the function $V$ and then solve for a ...
1
vote
0answers
25 views

LinAlg based physics textbooks [duplicate]

I'm in my second year studying physics, and ever since I took LinAlg, I've been noticing LinAlg-related concepts pop up all over the place, but it has never been presented directly as matrices, bases, ...
0
votes
1answer
44 views

Distinguishing degenerate states physically

Suppose there is a free particle on a circle with radius r. The energy spectrum is then $$E_n = \frac{n^2\hbar^2}{2mr^2} \,.$$ Thus, when $n \neq 0$, then the spectrum of energies is degenerate ...
1
vote
1answer
48 views

What is the main difference between a free particle on a line and a free particle on a circle?

The energy spectrum for a free particle in a circle with radius $r$ is $$E_n=\frac{n^2\hbar^2}{2mr^2}.$$ The energy spectrum for a free particle on an infinite line is similar. If so, what is the ...