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

learn more… | top users | synonyms

0
votes
1answer
33 views

Interaction Hamiltonian coupling question

System is composed of two qubits and the bath is one bath qubit. The interaction Hamiltonian is: $$\sigma_1^x\otimes B_1 + \sigma_2^x\otimes B_2$$ where $B_i$ is a 2 by 2 matrix. I try to interpret ...
1
vote
0answers
26 views

Meaning of Hamiltonian between two different states

If we have states $\left | 1 \right>$ and $\left | 2 \right>$, and the Hamiltonian operator $\hat{H}$, what is the meaning of the expression $$\left< 1 \right | \hat{H} \left | 2 \right ...
3
votes
1answer
36 views

Momentum operator in effective mass approximation

When we calculate the band structure of some solid then we often find that in the bottom of the conduction band the dispersion looks approximately quadratic with some new effective mass: $$E(k) = ...
0
votes
0answers
15 views

What is Loewdin downfolding method?

I am a student in solid state physics. I wonder if somebody could explain the mathematical background of downfolding method that is often used by Ole K. Anderson. Which restriction (conditions) to ...
2
votes
0answers
34 views

Is expectation value of the Hamiltonian always the energy? [duplicate]

There are time dependent & space dependent systems (magnetic fields) and time independent (particle in a box or harmonic oscillator). In the latter the expectation value is the 'average' energy ...
4
votes
2answers
101 views

Is the eigenvalue of Hamiltonian invariant under linear transformation of momentum operator?

It is given The dynamics of a particle moving one-dimensionally in a potential V(x) is governed by the Hamiltonian $H_0 = p^2 /2m + V(x) $, where $p = -i\hbar d/dx$ is the momentum operator. ...
1
vote
0answers
46 views

Bogoliubov transformation with two pairing terms

Let us assume that we have a Hamiltonian of the form: $$ H = \sum_{k,\sigma,s}\epsilon_{\sigma s}\left(k\right)c_{k\sigma s}^{\dagger}c_{k\sigma s} + \sum_{k,s}\Delta_{0}\left(k\right)c_{k\uparrow ...
0
votes
0answers
41 views

Build Hamiltonian function

Suppose we have three-point system Points A and B are connected with rod of fixed length $r_0$. Point C rotates around rod, vector R begins at rod's centre of mass. There is a potential of general ...
0
votes
0answers
14 views

Fermi-level of lattice model

I have a chain of $N$ lattice points with periodic boundary conditions. Every lattice point only has one orbital for an electron to occupy and spin is not included for simplicity. The lattice points ...
1
vote
2answers
42 views

Quantization of the Hamiltonian of a particle in a uniform magnetic field

If a particle of mass $m$ and charge $q$ is subject to a uniform magnetic field and if we have a vector potential $\mathbf{A}$ then we know that classically the dynamics of the particle will be ...
0
votes
1answer
55 views

Single particle tunneling Hamiltonian

In reference to Problem 9, Chapter 2 in Modern Quantum Mechanics by JJ Sakurai, For a single particle tunneling in a 1D double well potential, with position eigenkets $\mid R\rangle$, $\mid ...
1
vote
1answer
53 views

Going to the interaction picture in the Jaynes–Cummings model [closed]

In the Jaynes–Cummings model for a two level atom, the Hamiltonian for the atom is defined as (I let $\bar{h}=1$) $$H_a=\omega_a\frac{\sigma_z}{2}$$ and the field Hamiltonian is ...
5
votes
2answers
205 views

What is $\langle \phi | H | \psi \rangle$ in QM?

I know that $\langle \phi | \psi \rangle$ is the probability of going from the $\psi$-state to the $\phi$-state, and that $\langle \phi | H | \phi \rangle$ is the expectation value of the energy for ...
1
vote
0answers
20 views

Can you help me solve this using the current value Hamiltonian? [closed]

Okay, so I am getting a little stuck on this question, I will post it and then tell you how far I get. $$ max - \int_0^2 (x^2 + u^2)e^{-0.03t}dt\, $$ $$ x' = x-2u $$ $$ x(0) = 3 $$ $$ x(2)free $$ ...
3
votes
2answers
113 views

Why do we need both Hamiltonian and Hilbert Space to specify a Quantum System?

From my understanding, when we have the Hamiltonian, in principle we can know the eigenstates for our system of interest. Then, we can calculate everything we want. In addition, these eigenstates ...
6
votes
1answer
100 views

Hamiltonian and Energy of a charged particle in an Electromagnetic field

The Lagrangian of a charged particle of charge $e$ moving in an electromagnetic field is given by $$L=\frac{1}{2}m\dot{\textbf{r}}^2-e\phi-e\textbf{A}\cdot \textbf{v}$$ where $\phi(\textbf{r},t)$ is ...
1
vote
2answers
55 views

Fourier transform of Hamiltonian for scalar field

In the Srednicki notes (http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) page 36 he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int ...
0
votes
1answer
54 views

What is the deBroglie Wavelength in a region with potential energy?

This question was on a physics GRE: A free particle with initial kinetic energy $T$ and deBroglie wavelength $\lambda$ enters a region in which it has potential energy $V$. what is the particle's new ...
1
vote
2answers
100 views

Mathematical Proof the angular momentum and Hamiltonian commute?

I'm in a quantum mechanics class, and it is given in the book that the operators $\hat{L^{2}}$ and $\hat{H}$ commute for the 3D Harmonic Oscillator, but no definite mathematical proof is given, and ...
9
votes
4answers
426 views

Where does the $i$ come from in the Schrödinger equation?

I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow ...
1
vote
0answers
46 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 ...
2
votes
1answer
37 views

How to scale variables in a classical Hamiltonian?

So I looked at some research articles where one has a classical Hamiltonian $H(p,q,t) = p^{2}/2 + V(q,t)$. If one introduces the scaling transformation $$t \mapsto t/\sqrt{s}, \quad H \mapsto Hs, ...
0
votes
3answers
64 views

How does commutation between the hamiltonian and angular momentum operator (squared) imply conservation of Angular momentum?

So we are looking at central potentials in QM; The lecturer poses the question, when is $\textbf{L}$ conserved? He then considers the commutator of $\hat{H}$ and $\hat{L^2}$. We have; ...
1
vote
1answer
58 views

Conservation of energy in quantum mechanics

In Griffiths' book Introduction to quantum mechanics (second edition, page 37) it states: The time-independent Schrödinger equation says $$\hat{H} \psi_{n} = E_{n}\psi_{n}$$ so $$\langle H ...
3
votes
1answer
49 views

What is meant by taking the partial derivative of the Hamiltonian in this situation?

I'm doing a computation involving the quantum mechanical harmonic oscillator, and I have an expression of the form $\frac{\partial}{\partial \omega} \hat{H}$ where $$\hat{H} = \frac{1}{2m} \left( - ...
3
votes
1answer
149 views

How I can prove the Commutation between hamiltonian and Runge-Lenz vector? [closed]

I am a undergraduate student in physics. I found this page that shows a way to prove the commutator between Runge-Lenz vector and Hamiltonian .$\left [\hat{A}_{i},\hat{H}\right]=0$ I believe he did a ...
2
votes
0answers
69 views

Transfer from Heisenberg to Ising model

It is well know, that ferromagnets can be described using Hamiltonian $$ H = -\sum\limits_{i<j}J_{ij}\, (\mathbf{s}_i \cdot \mathbf{s}_j). $$ where (three dimensional) spins $\mathbf{s}_i$ ...
1
vote
0answers
56 views

Expectation value of the Hamiltonian [closed]

How to calculate expectation value of the Hamiltonian for hydrogen atom? $$\langle H \rangle_{\alpha l} \equiv \frac{\langle \psi_{\alpha l m}|H(r)| \psi_{\alpha l m}\rangle} {\langle \psi_{\alpha l ...
0
votes
1answer
32 views

Energy conservation Hamiltonian dependency

Suppose the a system has a Hamiltonian $H = H(q,p)$, and suppose $H$ does not depend explicitly on time. If $H\neq E$ the total energy of the system, does this necessarily say that $E$ is not ...
16
votes
2answers
428 views

Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
0
votes
1answer
59 views

Energy of hydrogen atom - Schrodinger equation [closed]

The wavefunction of the electron in the hydrogen atom is $ k* exp(-r/a)$ (k is the normalization constant), but it does not take n into account, whereas the solution of Schrödinger's equation ...
0
votes
1answer
71 views

Hamiltonian operator in spherical coordinates

I'm studying the hydrogen atom from a quantum mechanics perspective, but I'm having troubles understanding a step. Consider the stationary Schroedinger equation: $$\hat H \psi = E\psi$$ Let $M$ be ...
1
vote
1answer
51 views

Can we measure the energy of one of several identical particles?

Suppose we have a many-particle system described via a many-particle wavefunction that involves single-particle states $\lvert\lambda_{a}\rangle$, $\lvert\lambda_{b}\rangle$, ...
1
vote
1answer
60 views

Broadband light term in a Hamiltonian

In atomic systems, for a two-level system, the Hamiltonian can be written in the form: $$H=\left( \begin{array}{cc} E_1 & C_{12} \\ C_{21} & E_2 \\ \end{array} \right)$$ where $E_1$ and ...
3
votes
0answers
63 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = ...
1
vote
1answer
46 views

Hamiltonian average of energy of two stationary states

In quantum mechanics, the description of the infinite square well is given with the potential energy defined as $$V(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq a,\\ \infty & ...
0
votes
1answer
67 views

Constructing a Hamiltonian from a mass matrix?

I was solving some questions regarding the Hamiltonian, which required a lot of algebra, but as I finished and looked professor answer I saw that he constructed a matrix from the kinetic energy and ...
0
votes
1answer
42 views

Energy Expectation Value

I had an assignment question in which I was asked to calculate the expectation value of energy, $\langle E\rangle (t),$ and in the solution to it, the following was stated: \begin{align*} \langle ...
0
votes
1answer
37 views

Interaction Hamiltonian and shifts

When we quantize a free field theory, we set $\phi(x)$ to be the operators and we take the Fourier transform to determine the creation and annihilation operators $a_\omega,a^\dagger_\omega$ such that ...
1
vote
0answers
72 views

A mysterious conserved quantity for a central potential

In teaching a course in classical mechanics and I have come across (from my predecessor) a to me mysterious conserved quantity. We are considering a gravitational (or electric) potential with the ...
0
votes
1answer
47 views

Dispersion relation from Hamiltonian

Note: This is obviously for homework so I'm not asking for the answer to be spoon fed, I'm just not understanding the steps I have to take. I have a fairly simple Hamiltonian for a ring tight binding ...
1
vote
0answers
125 views

Hamiltonian of a 1D Linear Harmonic Oscillator [closed]

Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is: $$H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar$$ $$=\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} ...
1
vote
0answers
68 views

Commutation between angular momentum and Hamiltonian

Consider the following Hamiltonian of a 3-dimensional system: $$H=\frac{p^2}{2m}+V(r)$$ If the components of the angular momentum, $L_i$, commute with $H$, then: $$[H,L_i]=0$$ This condition can ...
2
votes
1answer
77 views

What happens to the wave function of a particle immediately after measuring its energy?

For this question, I will be adhering to the Copenhagen interpretation (since that's what I've learned in university so far). For the sake of brevity/clarity, also, assume the Hamiltonian here has ...
1
vote
0answers
50 views

Hamiltonian in commutator contradiction [duplicate]

Consider the following: $$[ \hat H, \hat x]=\left[-\frac{\hbar^2 \hat p^2}{2m}+V,\hat x\right]\ne0 \text{ in general}$$ But $$[ \hat H, \hat x]=\left[i\hbar \frac{\partial }{\partial t},\hat ...
1
vote
0answers
40 views

What's the point of hamiltonian mathematical formalism of classical mechanics? [duplicate]

Just what the title asks. What are the applications of it?
1
vote
2answers
76 views

Operators for a Perturbed Hamiltonian: Heisenberg Picture ($\hat{x}$, $\hat{p}$)

Problem I am trying to calculate the Equations of Motion in the Heisenberg picture for $\hat{x}$ and $\hat{p}$ in a perturbed Hamiltonian, $$ \tag{1} \hat{H} = \hat{H}_0 + \hat{H}' $$ Assume ...
0
votes
0answers
23 views

how to reduce the order of the hamiltonian equation for electrical problem given below?

I am having my FYP in this Hamiltonian project to analysis the integrator for Hamiltonian system. Can anyone please guide me how to reduce the equations to first order using substitution method?
0
votes
0answers
21 views

How to reduce the order of Hamiltonian equation for electrical problem [duplicate]

I want to reduce the order of this Hamiltonian but I don't know how to proceed. The equations are given below: $$H(p,q) = \frac{1}{2} (kq^2) + \frac{p^2}{2m} $$ This is the Hamiltonian for a simple ...
1
vote
0answers
19 views

ray tracing through a stack of flat plates

Say I want to trace rays in 3 dimensions through a stack of flat plates of various refractive indices. My rays have canonical coordinates {Q,P}. The plates are normal to the z axis, all the rays start ...