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-1
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0answers
59 views

Solving for the resonant frequencies of a mass-spring system [closed]

How do I solve for the resonant frequencies of a mass-spring systems. Given 3 masses ($M_1, M_2, M_3$) connected linearly with 2 springs ($K_1,K_2$), let $X_1,X_2,X_3$ be the displacement of the ...
1
vote
3answers
51 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
votes
1answer
58 views

Why do electrons occupy in discrete energy states?

Why can't there be any continuous energy band in an atom?
1
vote
1answer
32 views

“Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit ...
4
votes
1answer
72 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 ...
0
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0answers
35 views

How to calculate the eigenenergies of a particle in a triangular billiard?

Suppose we take the Dirichlet boundary condition, namely the wave function must vanish on the boundary. How about a general n-polygon?
1
vote
0answers
59 views

Selecting physical solutions in numerical eigenvalue problems

I try to solve a certain time-independent Schrodinger equation numerically, using the method of finite differences. My boundary conditions are such that the finite difference method gives me an ...
0
votes
3answers
58 views

Do we get the same answer at any time if we measure a system's energy?

Schrödinger's equation says that the only allowed energy states of a system are the eigenvalues of the energy operator $H$. This means that if we measure the energy of the system at any time we ...
0
votes
0answers
30 views

Occurance and disappearance of degeneneracies in a periodic structure of (quantum) LC circuits

Introductory part I'm currently studying an analytical model of coupled LC circuits, in preparation for actually performing measurements on such structures. While the final goal will struggle with a ...
1
vote
1answer
71 views

How to act an operator on a two-particle spin state?

I'm doing an assignment for my quantum class at the moment and I'm having trouble figuring out how to act a Spin operator on a two-particle state - specifically in finding the eigenvalues - I've spent ...
8
votes
2answers
272 views

How to guarantee square integrable solutions to time-independent Schrödinger's equation?

Given the time-independent Schrödinger’s equation in one dimension $$H\psi = E\psi$$ what restrictions can we place on V(x) (inside the hamiltonian) and E to guarantee that the solutions won't have ...
0
votes
1answer
59 views

Why do we need the eigenvalue? [closed]

Now I'm trying to review about the image noise. http://people.csail.mit.edu/celiu/denoise/estnoise/and I have found an article about the eigenvalue. Why do we need the eigenvalue, what is its use?
3
votes
2answers
83 views

Units of eigenvectors

Consider for example a mass matrix $M$, $\lambda$ one eigenvalue and $X$ a corresponding eigenvector. Then $[M]=\text{mass}$ (the brackets indicate the "unit operator"), and $MX=\lambda X$ so ...
2
votes
3answers
220 views

Spin in magnetic field and eigenvalues

We have some arbitrary quantum state, lets say $$\vert\Psi\rangle=\alpha_{1}\vert\uparrow\rangle+\alpha_{2}\vert\downarrow\rangle= \begin{pmatrix} \alpha_{1} \\ \alpha_{2} \\ \end{pmatrix}$$. And ...
0
votes
0answers
29 views

Density of energy eigenstates in Heisenberg model

Consider the Heisenberg model for spin-$\frac{1}{2}$ particles in 1 dimension. Take the case described by the hamiltonian: $J\sum_{i,i+1}(\sigma^x_i\sigma^x_{i+1} + \sigma^y_i\sigma^y_{i+1} + ...
2
votes
1answer
39 views

Can someone clarify what should and should not be an operator in my verification of the 1D solution to the SE for a free particle?

I just worked out the 1D free particle solution to the Schrödinger equation. My wave function was \begin{equation} \psi(x,t) = Ae^{i(px-Et)/\hbar} \end{equation} So I plugged this into both sides ...
1
vote
0answers
67 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
0
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0answers
36 views

Eigenstates of operators on constituent systems in tensor product space

Suppose I have two entangled physical systems $\mathcal{A}$ and $\mathcal{B}$ with respective hilbert spaces $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$. If $A,B$ are operators on ...
3
votes
2answers
715 views

Numerical solution to Schrödinger equation - eigenvalues

This is my first question on here. I'm trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfunctions but I am confused about how ...
1
vote
1answer
151 views

Eigenvalues of the radial Schrödinger equation on a finite integration interval

There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations ...
0
votes
0answers
53 views

Why is the matrix representation in the same basis not same for a density operator?

I have a $\rho : V \to V$ density operator of a $n$ dimensional space $V$ and $\{i\}=\{i_1,i_2..i_n\}$ is an orthonormal basis of this space. The density operator is defined as $$\rho=\sum ...
1
vote
3answers
208 views

Quantum Mechanincs - Dirac notation and solving time dependant schrodinger [closed]

The $\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}$ obviously correlate to $x,y,z$ components of the operators. Consider the Hamiltonian: $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$ where $C$ is a ...
1
vote
2answers
157 views

How can I prove following density matrices have same eigenvalues?

I have the following two density operators, the paper I am reading says that these two operators have same eigenvalues $$\rho^i = \frac{1}{3} ( |0\rangle \langle 0 | +|1\rangle \langle 1 |+|2\rangle ...
1
vote
0answers
59 views

WKB formula and Langer correction [duplicate]

The general WKB approximation formula states that $$ \int_a^b\sqrt{E_k-V(x)} = (k+1)\pi \text{ with } x \in [a,b] $$ for a regular Schrödinger equation (without the $\hbar$ and such). However, in the ...
0
votes
2answers
74 views

When generalizing from discrete (but infinite) eigenstates to continuous eigenstates, Why do we change the definition?

The propagator function for discrete eigenstates is $$u(t)=\sum_{n=1}^{\infty}|E_n\rangle\langle E_n|e^{-iE_nt/ \hbar } \tag{1}\ .$$ But when we have continuous eigenstates, (like for the case of ...
0
votes
2answers
58 views

What is the purpose of knowing the value of ground state energy of a potential well?

Using the formula $$E ~=~ \frac{\pi^2\hbar^2}{2 m a^2}$$ where $a$ is the length of an infinite potential well. It is apparent that as $a$ get smaller i.e. from a metal to the size of an atom, the ...
-2
votes
2answers
38 views

Find the eigenvalues of the operator [closed]

A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator. In this question is it necessary to define what the projection operator is? And ...
0
votes
2answers
73 views

Calulate the eigenvalues and the possible states after measurement [closed]

An observable is given by $$\sum\limits_{n= 1}^N a_n|a_n\rangle\langle a_n | $$ Here $\langle a_n |a_m\rangle = \delta_{nm}$. What are the possible measurement results corresponding to the operator ...
0
votes
0answers
59 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
0
votes
1answer
113 views

Eigenvalues of a nearest-neighbour tight-binding Hamiltonian in (Mahan, 2003)

In this paper by G. D. Mahan, he obtains the following electron Hamiltonian in a nearest-neighbour tight binding scheme: (page 2 of the paper, top of the right column) \begin{align} H_0 &= J_0 ...
0
votes
1answer
44 views

Eigenvalue for interacting Hamiltonian [closed]

Consider the Hamiltonian $$H=\omega_{1} a_{1}^\dagger a_{1}+\omega_{2}a_{2}^\dagger a_{2}+\alpha a_{3}^\dagger a_{3}(a_{1}^\dagger a_{2}+a_{2}^\dagger a_{1})$$ with $$ ...
0
votes
1answer
48 views

Scaling of an eigenvalue with the coupling constant

Consider the Hamiltonian $H = - \frac{d^2}{dx^2}+gx^{2N}$. Scaling out the coupling constant $g$, the eigenvalues scale as $\lambda \propto g^{\frac{2}{N+2}}$. So, we can drop the g dependence and ...
0
votes
1answer
40 views

A series of bound states covering an interval

Generally, the bound states (normalizable eigenvectors) of a Hamiltonian have discrete eigenvalues. Is it possible for the eigenvalues to cover an interval? Say, $(a,b)$? That is, for each $E \in ...
1
vote
1answer
54 views

How can you tell if a GTO function is an eigenfunction of hamiltionan H?

How can you tell if a Gauss-type orbital is an eigenfunction of Hamiltionan $H$? For example: $$GTO = N z^2 \exp\left(-\alpha r^2\right)$$ I know it is and eigenfunction of $L_z$ and not $L_x$ and ...
1
vote
1answer
122 views

numerical diagonalization of tight-binding hamiltonian

I would like to find the exact eigenvalues of the following tight-binding Hamiltonian, written here in second quatization: \begin{eqnarray} \hspace{-0.25in}{\mathcal{H}} &=& \mathcal{H}_0+ ...
0
votes
0answers
24 views

distance random matrix

In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by ...
2
votes
1answer
167 views

Is there a simple way of finding the eigenstates of the creation and annihilation operator in QM?

How can I find the eigenstates of creation and annihilation operator in QM? My attempt: Such eigenstate will obey: $$ a^{\dagger} |\psi \rangle = \alpha |\psi \rangle. $$ We can expand $|\psi ...
0
votes
1answer
86 views

Where does this commutator relation come from?

What is the origin of this relation: $$ [H,a_n^\dagger] = \epsilon_n a_n^\dagger $$ for Hamiltonian $H$, creation operator $ a_n^\dagger $, and eigenvalue $ \epsilon_n $. The square brackets denote ...
0
votes
0answers
28 views

Energy eigen value for a perturbed free particle system

Suppose we have a one-dimensional free particle system and we introduce a perturbation like $V(x)=V_{0} \cos(Gx)$, where $G$ is the reciprocal lattice vector (it's a periodic perturbation, I think) ...
2
votes
1answer
108 views

Eigenfunctions of Schrödinger equation

Why are solutions of the Schrödinger equation called eigenfunctions? For an electron moving in one dimensional lattice the eigenfunctions are given by$$\psi(x)=u_k(x)e^{ikx}.$
6
votes
2answers
200 views

Does the Hermitian operator $H=-\frac{d^2}{dx^2}$ have imaginary eigenvalues?

In quantum mechanics, Hermitian operators play a very important role because they possess real eigenvalues. Considering $-\frac{d^2}{dx^2}$, it is a Hermitian operator (Actually it's the simplest ...
0
votes
1answer
34 views

Minimizing a damping constant in order to minimize the amplitude of oscillations

How can I determine the damping coefficient that minimizes the amplitude of vibrations? This is an extension of Coupled ODEs that model a quad rotor \begin{align} ...
0
votes
1answer
47 views

Physical observables and hermiticity

Is it necessary for an operator to be Hermitian in order to be a physical observable or is it just sufficient that the operator obeys the eigenvalue equation? If I were to check whether an operator is ...
1
vote
1answer
208 views

Eigenvalues of hamiltonian [closed]

Q: THe hamiltonian which describes the motion of a particle in an one dimensional potential V(x) is $H_0=\frac{p^2}{2m}+V(x)$ , where $p=-i\hbar \frac{d}{dx}$ is the momentum operator. $E_n^0$ , ...
2
votes
0answers
47 views

Making An Energy Momentum Plot For A Rashba Model (Using Discretization)

I want to make a plot of the Energy versus the Momentum of the Rashba model, using discrete matrices. First Ill show how I did this for the free particle. Subsequently I will show what goes wrong for ...
2
votes
3answers
142 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 $A$.
4
votes
2answers
849 views

Understanding the Jacobian Matrix

Taking the example of a two dimensional system, desribred by the following ODE's: \begin{align} \frac{dx_1}{dt}&=f_1(x_1,x_2)\\ \frac{dx_2}{dt}&=f_2(x_1,x_2) \end{align} The Jacobian Matrix ...
3
votes
1answer
437 views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
0
votes
0answers
36 views

What equation can be used to solve an ideal string/membrane in a non-vacuum medium?

I'm interested in the eigenmodes of the membrane for various mediums, such as vacuum, air, water, etc., which impose a damping effect on the membrane. This cannot be done by merely changing the value ...
1
vote
2answers
129 views

What is the meaning of pre-tension for a stiff membrane?

On one hand, I know that the tighter a drum head is stretched, the higher its natural frequencies. This relation is given by: $$f_{ij}=\frac{k_{ij}}{2\pi R}\sqrt{\frac{T_0}{h\rho}}$$ where $k_{ij}$ ...