The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.

learn more… | top users | synonyms

2
votes
1answer
175 views

Perturbation method & eigenvalues

I have a problem but I don't understand the question. It says: "Show that, to first order in energy, the eigenvalues ​​are unchanged." What does it mean? It means that if the Hamiltonian has the ...
2
votes
1answer
535 views

Solving time dependent Schrodinger equation in matrix form

If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$ With Hamiltonian $H$ given by $$H=\hbar\omega ...
2
votes
1answer
342 views

Canonical transformation generated by hamiltonian?

Someone told me that, in a hamiltonian system, the hamilonian function is the generating function of the canonical transformation given by time translation. However, this statement doesn't make any ...
2
votes
1answer
303 views

Computing a density of states of Hamiltonian $ H=xp$

How could I compute the integral $$ N(E)~=~ \int dx \int dp~ H(E-xp) $$ the 'Area' inside the Phase space is taken for $ x \ge 0 $ and $ p\ge 0 $? The result should be $$ N(E)~=~ ...
2
votes
1answer
63 views

Diagonalizing Van der Waals Hamiltonian

In Kittel's Solid State Physics, he attempts to find the energy exchange due to the van der Waals interaction. He starts by writing the hamiltonian: two oscillators with coordinates $x_1$ and $x_2$ ...
2
votes
1answer
143 views

How do I show that the eigenstates of a Hamiltonian can be made orthonormal?

I've been tearing my hair out over this all evening. It should be simple but I must be missing something somewhere. Can someone show me how to prove that the eigenstates of a Hamiltonian can be made ...
2
votes
1answer
96 views

Hamiltonian form of Noether's Theorem

I understand that Noether's Theorem has a Hamiltonian form, whereby {X, H} = 0 iff {H, X} = 0. The proof of this is trivial, as it follows from the antisymmetry of the Poisson Brackets. First ...
2
votes
1answer
124 views

Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
2
votes
2answers
2k views

Two expressions for expectation value of energy

I was looking up expectation value of energy for a free particle on the following webpage: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/expect.html It says that $E=\frac{p^2}{2m}$ and ...
2
votes
1answer
250 views

Hamiltonian matrix off diagonal elements?

I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I ...
2
votes
1answer
240 views

The relation between Hamiltonian and Energy

I know Hamiltonian can be energy and be a constant of motion if and only if: Lagrangian be time-independent, potential be independent of velocity, coordinate be time independent. Otherwise ...
2
votes
0answers
40 views

The proof that Dirac's hamiltonian commutes with inversion operator

I tried to check the statement that Dirac free Hamiltonian commutes with inversion operator. For $$ \hat {P}\Psi(\mathbf r , t) = i\hat {\gamma}_{0}\Psi (-\mathbf r , t), \quad \hat {H} = (\hat ...
2
votes
1answer
62 views

Eigenenergies and eigenkets given the Hamiltonian

For a two level system the Hamiltonian is: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|) $$ where $a$ is a number with the dimension of an energy. I need to ...
2
votes
0answers
139 views

Adiabatic quantum evolution of single photon or biphoton system

The prerequisite for adiabatic quantum evolution of single photon or biphoton system is as follows. We have to prepare a single photon or biphoton quantum system which has a ground and a higher level ...
2
votes
0answers
47 views

Spin Transition Energies

I am reading a paper: http://arxiv.org/ftp/arxiv/papers/1305/1305.2445.pdf On p. 22, the following Hamiltonian is given: $$ H = \mu_B g \mathbf{B} \cdot \mathbf{S} + D(S_Z^2+\frac{1}{3}S(S+1)) + ...
2
votes
0answers
107 views

Number theoretical function applied in physics? [closed]

I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does ...
2
votes
0answers
66 views

Boundary condition Hamiltonian with point tinteractions

I`m studying the Hamiltonian with point interaction centered in $y$ in three dimensions. I know that the elements in the domain of the Hamiltonian are of the form $$\psi=\phi+qG^z(\cdot-y)$$ where ...
2
votes
0answers
144 views

Hubbard Model Hamitonian

$H = -\sum\limits_{i,j} A_{ij} c_i^{\dagger} c_j + \frac{U}{2} \sum\limits_i(c_i^\dagger c_i)(c_i^\dagger c_i -1)$ is defined to be a Hamiltonian for modeling quantum random walk of identical ...
2
votes
1answer
82 views

What does $\psi_j(r_i)$ mean?

I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by $V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$ I ...
1
vote
1answer
580 views

Cyclic Coordinates in Hamiltonian Mechanics

I was reading up on Hamiltonian Mechanics and came across the following: If a generalized coordinate $q_j$ doesn't explicitly occur in the Hamiltonian, then $p_j$ is a constant of motion ...
1
vote
2answers
150 views

Lagrangian and hamiltonian of interaction

How to prove that lagrangian of interaction is equal to hamiltonian of interaction with minus sign? For example, I can't prove it for special case - quantum electrodynamics.
1
vote
2answers
140 views

Sign in the time-independent Schrödinger's equation

In the time-independent Schrödinger's equation: $$ -\frac{\hbar^2}{2m} \frac{d^2} {dx^2} u + Vu ~= Eu, $$ why there is a minus sign before the first term?
1
vote
3answers
422 views

Propagators and Probabilities in the Heisenberg Picture

I'm trying to understand why $$\Bigl|\langle0|\phi(x)\phi(y)|0\rangle\Bigr|^2$$ is the probability for a particle created at $y$ to propagate to $x$ where $\phi$ is the Klein-Gordon field. What's ...
1
vote
3answers
145 views

The notion of bounded states in quantum mechanics and their characterization with operators

Is there any case of potential $V$, such that the continuity of the operator $H=c\ \Delta+V$ is not spoiled? And I don't know any non-differnetial operator examples for continous spectra. I ...
1
vote
1answer
155 views

Quantum Stat-Mech Proof of an Inequality for the Partition Function

I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been ...
1
vote
3answers
756 views

Energy Spectrum of pair of spin-1/2 particles with general Hamiltonian

I found this problem, and so far I am stumped. I was wondering if anyone wanted to solve it with me, or help me calculate eigenvectors, or just give insight on my questions. Consider a system of ...
1
vote
1answer
64 views

The Molecular Hamiltonian and the avoidance of Overcounting

Whenever I see the total non-relativistic molecular Hamiltonian, $\hat{H}_{molecular} = \hat{T}_{e} + \hat{T}_{n} + \hat{V}_{ee} + \hat{V}_{nn} + \hat{V}_{en}$ I always notice that the sums ...
1
vote
3answers
241 views

Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field

I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows: $$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
1
vote
1answer
224 views

Symmetry and overlapping of ground states

In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$: $$E_0 = ...
1
vote
1answer
562 views

The Hermiticity of the Laplacian (and other operators)

Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator? Alternatively: is the matrix representation of the Laplacian Hermitian? i.e. $$\langle \nabla^{2} x | y \rangle = \langle x | ...
1
vote
3answers
115 views

Bogoliubov transformation with a slight twist

Given a Hamiltonian of the form $H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix} \begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \omega_0\end{pmatrix} ...
1
vote
1answer
104 views

Change of operator in the Hamiltonian [closed]

We are told that the particle has mass m and charge e and is moving in 2 dimensions. The position operator $\mathbf{X}=(X_{1},X_{2})$ and momentum operator $\mathbf{P}=(P_{1},P_{2})$ We are given ...
1
vote
2answers
161 views

Representation of Hamiltonian in terms of “creation” and “destruction” operators

Let's have Schrodinger equation or Dirac equation in Schrodinger form: $$ i \partial_{0}\Psi = \hat {H}\Psi . $$ Sometimes we can introduce some operators $\hat {A}, \hat {B}$ (the second is not ...
1
vote
2answers
446 views

Getting Energies and Probabilities from the Hamiltonian

So I need to find the possible energies and the probabilities of these using the eigenvalues of a Hamiltonian. Once I obtain the eigenvalues, are those the energies E_n in and of themselves? Or do ...
1
vote
1answer
120 views

Transform hamiltonian

I have got the following Quantum Hamiltonian: $$H=\frac{p^{2}}{2m}+k_{1}x^{2}+k_{2}x+k_{3}$$ Which transformation can I use to change this Hamiltonian into an harmonic oscillator hamiltonian? ...
1
vote
1answer
412 views

Commutation relation with Hamiltonian

How do we get $[\beta , L] = 0$ , where $L$= orbital angular momentum and $\beta$= matrix from Dirac equation?
1
vote
2answers
463 views

Confusion about Free Energy and the Hamiltonian

I'm probably making a relatively basic mistake here, but I'm a bit confused about the relation between the Hamiltonian and Helmholtz free energy. From what I can see, the free energy can be written ...
1
vote
2answers
523 views

Canonical transformations and conservation of energy

I have an important doubt about the nature of canonical transformations in hamiltonian mechanics. Suppose I have a one-degree-of-freedom lagrangian system, whose hamiltonian depends explicitly on ...
1
vote
2answers
112 views

Molecular Hamiltonian

I was reading some material on the Molecular Hamiltonian on Wiki. It said that, Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first ...
1
vote
1answer
100 views

Where can I find hamiltonians + lagrangians?

Where would you say I can start learning about Hamiltonians, Lagrangians ... Jacobians? and the like? I was trying to read Ibach and Luth - Solid State Physics, and suddenly (suddenly a Hamiltonian ...
1
vote
0answers
70 views

Second quantization Hamiltonian Matrix for an aggregate

I am working on the matrix form of the Hamiltonian in the second quantization. I haven't taken any course on second quantization and I'm learning it on my own. I'm a little bit confused about the ...
1
vote
0answers
81 views

Exotic coupling

I have encountered the minimal coupling between a field and charges before $$H = \frac{1}{2m}(p-qA)^2,$$ whereby I am considering the classical case. The description minimal leads me to ask if ...
1
vote
0answers
233 views

Anharmonic oscillator solution function

I am solving a CLASSICAL an-harmonic oscillator problem with Hamiltonian given by $H= (1/2)\dot{x}^2+(1/2)x^2-(1/2)x^4$ with all the constants (k's) and mass being taken as 1 (one). I find that $x= ...
1
vote
0answers
65 views

What is the difference between broken and unbroken sypersymmetry?

In a physics article I read recently, the author introduces the notion of supersymmetry by saying basically that the system described by the Hamiltonian $H$ is supersymmetric if $H$ can be decomposed ...
1
vote
0answers
72 views

Equivalence of Bogoliubov-de Gennes Hamiltonian for nanowire

I'm trying to show that the Hamiltonian for a nanowire with proximity-induced superconductivity $$ \hat{H} = \int dx \text{ } ...
1
vote
0answers
98 views

What is the energy scale of a Hamiltonian?

On the second page of this paper a term 'fundamental energy scale' is used while talking about a Hamiltonian. The context is implementing Deutsch's algorithm using Adiabatic Quantum Computation. What ...
1
vote
0answers
35 views

Delivered/Reflected Power by Drive on a Hamiltonian System

Imagine a SHO with a drive F(t). (or in general a Hamiltonian system) What is the power delivered to the system and can we talk about the power reflected? is i am imagining say a MW oscillator ...
1
vote
0answers
108 views

Measurement in Quantum mechanics

I have got a quantum conservative system whose Hamiltonian is $H$. I consider an selfandjoint operator $O$ whose eigenvalues and eigenvectors are: $$O|\psi _{n}\rangle = \lambda _{n}|\psi ...
1
vote
0answers
55 views

Length of the orbit (semiclassical orbits)

The Gutzwiller trace is about $$ d(E)=d_{0} (E)+ \sum_{p.o}A_{p}Cos(S(E)\ell_{p}) $$ and $ \ell_{p} $ are the length of the orbit. However my question is, how can one derive the length of the orbit ...
0
votes
3answers
165 views

Equation $H(q,p)=E$ is the equation of motion or energy-conservation law?

I do not completely understand, why do we consider Hamilton–Jacobi equation $H(q,p)=E$ as equation of motion, whereas it is looks like energy-conservation law?