Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
1,086
questions
22
votes
2answers
698 views
Is there a physical interpretation to invariant random matrix ensembles?
Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
5
votes
0answers
186 views
CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates
Statement of the problem:
I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a ...
4
votes
1answer
675 views
Quantum mechanics with non-cartesian coordinates
Let say we have the classical hamiltonian of a harmonic oscillator: $$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$
and we want to find the hamiltonian operator in quantum mechanics, ...
8
votes
1answer
560 views
Role of physics in the zeta function $\zeta$ and the Riemann hypothesis
Hilbert and Polya suggested a physical way to verify the Riemann hypotesis about $\zeta(x)$. If the Riemann hypotesis is true, we can state all eigenvalues of physical problems are real. What is the ...
3
votes
2answers
5k views
What is the energy operator and from where do we get it?
I am trying to learn Quantum mechanics from MIT OCW Videos
about quantum mechanics. I have reached the 5th lecture. Please help me in understanding this:
In the middle (At 32:08), the professor wrote ...
4
votes
2answers
4k views
What is an “Interaction Hamiltonian”
I'm an undergraduate reading up on some quantum physics so that I can help out more in the lab that I'm working in this summer. In the book I'm reading (Shankar's "Principles of Quantum Mechanics") I ...
2
votes
3answers
5k views
Hamiltonian mechanics and conservation of energy?
Can anyone explain to me Hamiltonian mechanics relation to conservation of energy? I'm not very good at mathematics, and I know it's important into understanding Hamiltonian mechanics. However, can ...
2
votes
1answer
846 views
Vanishing diagonal matrix elements of pertubation
In time-dependent pertubation theory we can denote the Schrödinger equation by a set of two equations
$$\dot{c_a} = -\frac{i}{\hbar}\Big[c_aH'_{aa}+c_bH'_{ab}e^{-i(E_b-E_a)t/\hbar}\Big] \\
\dot{c_b} =...
3
votes
1answer
1k views
Quick question on perturbation theory
Suppose we have a particle in an infinite potential well, with $V(x) = 0,\space 0< x < a $ and infinity everywhere else.
Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, ...
0
votes
1answer
217 views
Hamiltonian Operator for Harmonic Oscillator
I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both ...
3
votes
2answers
845 views
Feynman's $i \epsilon$ prescription in loop expansion
I have some questions about the $i\epsilon$ factor in Feynman diagrams. First, what is the physical meaning of $i\epsilon$ in loop amplitudes. Second, how does it ensures unitarity?
And third, Dyson ...
3
votes
1answer
174 views
$\gamma^5$ factor in Quantum Field Theory
I have a problem with interpretation of $\gamma^5$ factor in the interaction Hamiltonian. I know that $\frac{1\pm\gamma^5}{2}$ is a helicity projection and it requires helicity conservation in ...
2
votes
1answer
279 views
On use of Hamiltonians for Helium
The Hamiltonian of helium can be expressed as the sum of two hydrogen Hamiltonians and that of the Coulomb interaction of two electrons.
$$\hat H = \hat H_1 + \hat H_2 + \hat H_{1,2}.$$
The wave ...
2
votes
1answer
397 views
Expectation value of Hamiltonian in different pictures of quantum mechanics
We start with the familiar Schrodinger equation:
$$
i\hbar \frac{\partial \left|\psi_S\right\rangle}{\partial t} = \hat{H}_S \left|\psi_S\right\rangle
$$
As we switch to a different picture than ...
1
vote
2answers
1k views
Showing that Hamiltonian expectation value is time independent
I want to check that I am getting the concept right here, and my question is: if the expectation value of a Hamiltonian is the same whether you use the time dependent version or not. I thought I had ...
2
votes
1answer
26k views
Calculating the expectation value of a Hamiltonian
I want to calculate the expectation value of a Hamiltonian. I have a wave function that is $$\psi = \frac{1}{\sqrt{5}}(1\phi_1 + 2\phi_2).$$
I want to know if I set this up properly. The Hamiltonian ...
0
votes
1answer
577 views
Expectation value of Hamiltonian on number state [closed]
Hamiltonian is defined by $H_I = \hbar \omega (\hat{a}^+ \hat{a} + 1/2)$
What is the expectation value of the energy on the number state
$$\vert \psi \rangle = \frac{1}{\sqrt{2}} ( \vert 1 \rangle +...
-1
votes
1answer
214 views
Apply the Heisenberg Equation to the Hamiltonian [closed]
$\frac{d}{dt}$$\hat{H}$ = $\frac{i}{\hbar}$$[\hat{H},\hat{H}]$ +$\frac{\partial{\hat{H}}}{\partial{t}}$
That's as far as I've got. I do not know much about the Heisenberg equation or even what it ...
4
votes
1answer
260 views
How would Hamiltonian for several fermions with spin look?
All discussions of Pauli exclusion principle I read usually talked about antisymmetric wavefunctions, from which the princinple appears. But I would like to see a Hamiltonian for multiple fermions, ...
17
votes
3answers
3k views
Correct way to write the eigenvector of a diagonalized hamiltonian in second quantization
I am studying diagonalization of a quadratic bosonic Hamiltonian of the type:
$$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j + \frac{1}{2}\displaystyle\sum_{<i,j>} [B_{ij} a_i^\...
6
votes
1answer
276 views
What are the restrictions on the Hamiltonian in QM?
In quantum mechanics, we usually write the Hamiltonian as:
$$\hat{H}=\hat{T}+\hat{V}$$
But in classical mechanics, there are several reasons why it would not have this form:
We've chosen some ...
27
votes
2answers
3k views
The formal solution of the time-dependent Schrödinger equation
Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as
$$
\tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi .
$$
Usually, one likes to write that it has a ...
7
votes
4answers
2k views
Bogoliubov transformation with a slight twist
Given a Hamiltonian of the form
$$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} \begin{...
6
votes
0answers
307 views
Topological Quantum Field Theories
I've asked this on Math.SE, but with no avail. So, I decided to ask it here.
I was wondering about the following after reading the Wikipedia article on TQFTs.
It is said that TQFTs have vanishing ...
3
votes
1answer
437 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
$$
\hat {P}\Psi(\mathbf r , t) = i\hat {\gamma}_{0}\Psi (-\mathbf r , t), \quad \hat {H} = (\hat {\...
3
votes
2answers
2k views
Eigenenergies and eigenkets given the Hamiltonian
For a two level system the Hamiltonian is:
$$
H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|)
$$
where $a$ is a number with the dimension of an energy.
I need to ...
1
vote
0answers
325 views
Second quantization Hamiltonian Matrix for an aggregate
I am working on the matrix form of the Hamiltonian in the second quantization. I haven't taken any course on second quantization and I'm learning it on my own. I'm a little bit confused about the ...
7
votes
2answers
7k views
Gauge Invariance of the Hamiltonian of the electromagnetic field
The Hamiltonian for an electron of mass $m$ and charge $e$ in an exterior electromagnetic field is
$$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$
The corresponding (via canonical quantization) quantum ...
0
votes
1answer
2k views
Proof of the time-independent Schrödinger equation
I have a question regarding the proof of the time-independent Schrödinger equation.
So if we have a time-Independent Hamiltonian, we can solve the Schrödinger equation by adopting separation of ...
4
votes
1answer
1k views
Diagonalizing Van der Waals Hamiltonian
In Kittel's Solid State Physics, he attempts to find the energy exchange due to the van der Waals interaction. He starts by writing the hamiltonian: two oscillators with coordinates $x_1$ and $x_2$
$$...
1
vote
1answer
224 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
0answers
113 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 ...
2
votes
1answer
5k 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 ...
11
votes
2answers
3k views
Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?
The classical Lagrangian for the electromagnetic field is
$$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} - J^\mu A_\mu.$$
Is there also a Hamiltonian? If so, how to derive it? I know how ...
9
votes
0answers
1k views
Exact diagonalization by Bogoliubov transformation
I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by,
$$
H =
\begin{pmatrix}
\xi_\mathbf{k} ...
5
votes
1answer
3k views
Conservation of Hamiltonian vs Conservation of Energy
What is the difference between conservation of the Hamiltonian and conservation of energy?
5
votes
1answer
6k views
When does the total time derivative of the Hamiltonian equal its partial time derivative?
When does the total time derivative of the Hamiltonian equal the partial time derivative of the Hamiltonian? In symbols, when does $\frac{dH}{dt} = \frac{\partial H}{\partial t}$ hold?
In Thornton &...
10
votes
1answer
308 views
Do systems with level crossings have unstable eigenbases?
It's folklore dating back to von Neumann and Wigner that time-dependent Hamiltonian systems tend not to have level crossings of their energy eigenvalues.
However, we can of course consider smoothly ...
2
votes
2answers
1k views
Lagrangian and hamiltonian of interaction
How to prove that lagrangian of interaction is equal to hamiltonian of interaction with minus sign? For example, I can't prove it for special case - quantum electrodynamics.
1
vote
2answers
709 views
Representation of Hamiltonian in terms of “creation” and “destruction” operators
Let's have Schrodinger equation or Dirac equation in Schrodinger form:
$$
i \partial_{0}\Psi = \hat {H}\Psi .
$$
Sometimes we can introduce some operators $\hat {A}, \hat {B}$ (the second is not ...
1
vote
1answer
2k views
Anharmonic oscillator solution function
I am solving a CLASSICAL an-harmonic oscillator problem with Hamiltonian given by
$$H= (1/2)\dot{x}^2+(1/2)x^2-(1/2)x^4$$ with all the constants ($k$'s) and mass being taken as 1 (one).
I find that $...
1
vote
0answers
252 views
What is the difference between broken and unbroken sypersymmetry?
In a physics article I read recently, the author introduces the notion of supersymmetry by saying basically that the system described by the Hamiltonian $H$ is supersymmetric if $H$ can be decomposed ...
1
vote
2answers
6k views
Getting Energies and Probabilities from the Hamiltonian
So I need to find the possible energies and the probabilities of these using the eigenvalues of a Hamiltonian.
Once I obtain the eigenvalues, are those the energies E_n in and of themselves?
Or do ...
8
votes
2answers
3k views
How to get Hamiltonian of QED from lagrangian?
I have the QED lagrangian:
$$
L = \bar {\Psi}(i \gamma^{\mu }\partial_{\mu} + q\gamma^{\mu}A_{\mu} - m)\Psi + \frac{1}{16 \pi}F_{\alpha \beta}F^{\alpha \beta} .
$$
I tried to get hamiltonian by ...
10
votes
2answers
803 views
Does Hamilton Mechanics give a general phase-space conserving flux?
Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
1
vote
2answers
299 views
Sign in the time-independent Schrödinger's equation
In the time-independent Schrödinger's equation:
$$ -\frac{\hbar^2}{2m} \frac{d^2} {dx^2} u + Vu ~= Eu, $$
why there is a minus sign before the first term?
0
votes
1answer
1k views
Hamiltonian operator apply to a wavefunction
When a Hamiltonian operator apply to a wavefunction, how could we write the hamiltonian as,
$$H \psi = (E_n-\hbar \omega_0) \psi \ \ ? $$
Is this because $E_n= H+ \hbar \omega_0$?
where $\...
6
votes
1answer
3k views
Hamiltonian matrix off diagonal elements?
I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I don'...
5
votes
1answer
2k views
What does it mean for a Hamiltonian to be SU(2) invariant?
Can somebody explain what it means when one says a Hamiltonian is SU(2) invariant? I know Heisenberg Hamiltonian is SU(2) invariant but why?
0
votes
0answers
445 views
Quantum Hamiltonian commuting with the Pauli-Runge vector
I have to prove that $[A_j, H] = 0$, with;
$$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$
$$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$
And, $Z, e, m$...
