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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1answer
50 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 ...
0
votes
1answer
21 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....
1
vote
1answer
24 views

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

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 ...
1
vote
1answer
33 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'...
0
votes
1answer
33 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 ...
0
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1answer
44 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 + |\...
0
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1answer
46 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 ...
0
votes
0answers
23 views

Are there explicit formulas for the eigenvalues and eigenvectors of a generic 4x4 density matrix? [migrated]

I have a 4x4 density matrix all of whose elements are nonzero. Its form is $$\begin{pmatrix} a & b & c & d \\ b^* & e & f & g \\ c^* & f^* & h & j \\ d^*...
2
votes
1answer
51 views

Different postulates and statistical interpreations of quantum mechanics

Hi I have a query about the difference of two aspects of the statistical interpretation of quantum mechanics given in the popular introductory quantum mechanics books "Introduction to Quantum ...
0
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0answers
44 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 ...
0
votes
1answer
81 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 ...
0
votes
1answer
44 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 ...
0
votes
1answer
37 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 ...
3
votes
0answers
38 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 ...
0
votes
0answers
34 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
73 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 ...
0
votes
1answer
82 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
29 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 ...
-1
votes
1answer
47 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 $\hat{N}=\hat{...
1
vote
0answers
27 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 = -\dfrac{\hbar^...
3
votes
1answer
116 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)\, \...
1
vote
2answers
58 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 ...
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vote
0answers
31 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 ...
0
votes
1answer
104 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 ?
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0answers
48 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 )
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vote
2answers
208 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 ...
6
votes
1answer
89 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
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??
0
votes
1answer
71 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
112 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
140 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 ...
1
vote
2answers
100 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 $\mathbb{...
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 ...
2
votes
1answer
41 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 ...
0
votes
1answer
84 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
45 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
57 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
102 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
53 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 ...
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 ...
1
vote
1answer
82 views

What is the meaning of definite total energy of the wave function?

David J. Griffiths in Introduction to Quantum Mechanics asked: What's so great about separable solutions of time independent Schroedinger equation? His answer was They are states of ...
1
vote
2answers
344 views

Operator vs. Matrix in quantum formalism

We use in Dirac formalism of QM the tool of operators and kets in spatial and spin spaces to obtain eigenvalues and eigenkets. But the operation here is simply that of a matrix multiplication. Now ...
2
votes
0answers
76 views

Determine the number of bound states admitted in Schrodinger system

Is there a general method for determining the number of bound states admitted by a potential in the Schrodinger equation? Certainly the number of dimensions must factor in somehow: the delta ...
0
votes
1answer
122 views

Physical meaning of eigenvectors of mass matrix

What is the physical meaning of the eigenvectors of the mass matrix? If I consider a 2-dof system with one mass linked to two orthogonal springs and I write the equations in any orthogonal system of ...