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2
votes
1answer
65 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: for momentum space and for position space. How do we know that Ψ is not the Fourier transform of Φ but we suppose that its the other way around(Ψ would be ...
29
votes
5answers
1k views

In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?

A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential. Is there a one to one correspondence between the potential and its spectrum? If the ...
0
votes
0answers
42 views

Lowering operator and degenerate eigenvalue [duplicate]

In this video at 5:00: https://www.youtube.com/watch?v=xniXpD3bCYU David Miller derives how the lowering operator $\hat a$ produces some wave function that has the eigenvalue $(n-1)$; thus it ...
0
votes
1answer
68 views

spin independent observable

Let's consider a spin independent observable $O$ (the terms of the operator don't depend upon the spin operator). If we are interested to find an eigenfunctions' basis of the wave-functions' space, ...
2
votes
1answer
26 views

Proton spin independent fine structure “Hamiltonian” $W_f$

To find the perturbation correction (fine structure) in the case of a degenerate energy $E_n^0$, we can diagonalize the operator $W_f^n$, the restriction of $W_f$ to the eigen-space associated to ...
0
votes
3answers
94 views

Both Eigenvalues and Operators are “Observables”? [duplicate]

I am having a bit of difficulty wading through the what seems to be multiple usages for Observables in Quantum Mechanics. " Mathematically observables are postulated to be Hermitian operators.. " ...
1
vote
1answer
120 views

Quick question on sketching wavefunction in well

Usually for an infinite well, the sketch for n=3 level is this: Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than ...
0
votes
1answer
36 views

Problem with the Cooley-Numerov Method for Solving the Radial Nuclear Schodinger Equation in the Born-Oppenheimer Approximation

I have been trying to implement a solver for the radial nuclear Schodinger equation in the Born-Oppenheimer approximation using a similar method to R. J. Le Roy's LEVEL program[1]. I have as input a ...
1
vote
3answers
54 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
63 views
3
votes
2answers
929 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 ...
1
vote
1answer
34 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
73 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
votes
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
69 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
62 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 ...
5
votes
1answer
923 views

Can any physical rigid body be represented by an ellipsoid with the same angular dynamics?

According to wikipedia, the inertia tensor of an ellipsoid with semi-axes $a,b,c$ and mass $m$ is $$\left[\begin{array}{ccc} \frac{m}{5}(b^2+c^2)&0&0\\ 0&\frac{m}{5}(a^2+c^2)&0\\ ...
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
152 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 ...
8
votes
2answers
278 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 ...
1
vote
1answer
79 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 ...
0
votes
1answer
60 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
87 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 ...
3
votes
2answers
844 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 ...
2
votes
3answers
222 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
73 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 ...
2
votes
3answers
129 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 ...
0
votes
0answers
39 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 ...
0
votes
0answers
56 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 ...
15
votes
4answers
3k views

How to tackle 'dot' product for spin matrices

I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
1
vote
3answers
215 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
160 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 ...
2
votes
1answer
109 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}.$
0
votes
2answers
77 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
59 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 ...
-1
votes
2answers
116 views

How can we say that a wave function follows Schrödinger equation using operators?

If I have an operator which has an eigenfunction which satisfies Schrödinger's time-dependent equation, and I have another eigenfunction of this operator, can I say that the other eigenfunction will ...
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 ...
-2
votes
2answers
39 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
0answers
60 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
127 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
49 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 ...
2
votes
2answers
149 views

Eigenvalues being physical observables

I think I'm comfortable with the PDE solutions to the Schrodinger equation. But as soon as we start putting these values in a matrix (in dirac notation), I lose my understanding and everything ...
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
129 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 ...