Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
749 questions
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How to get standard symplectic form so that one can get a vector field from a Hamiltonian function? [on hold]
In https://en.m.wikipedia.org/wiki/Hamiltonian_vector_field under 'Examples' a standard symplectic form w=(summation)dq(i)^dp(i) where p and q are the momentum and position of a particle. My question ...
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0answers
23 views
Symplectic form on a Hamiltonian
What are the Hamiltonian equations with respect to standard symplectic form? What is the standard symplectic form?
0
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25 views
How to derive the analytical expression for the retarded Green's function with quadratic Hamiltonian?
For two operators, $A(t)$ and $B(t)$ the retarded Green’s function is defined as
\begin{equation}
G^R(t,t') \equiv \langle \langle A(t)|B(t) \rangle \rangle^R
= -i\theta(t-t')\langle \{A(t),B(t')\} \...
2
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0answers
69 views
+50
Symmetry transformations that are self-inverse and global symmetries of the Hamiltonian
I have the simplified Ising model. The Hamiltonian is given by
$$
\mathcal{H} = -\mathrm{J}\sum_{<ij,i' j'>} \sigma_{ij} \sigma_{i'j'}.
$$
Where the sum over $<ij,i'j'>$ means just the ...
1
vote
1answer
46 views
Domain of the infinite square well hamiltonian
I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'.
In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well.
For ...
0
votes
2answers
39 views
Wave function of a particle under $V(x)$ (QM)
Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well)
Also given is the wave function at time $t=0$, $\psi(x,0)$.
What is ...
10
votes
2answers
991 views
Symmetry in quantum mechanics
My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$ UH(\psi) = HU(\psi) $$
Can you give me an easy explanation for this definition?
2
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0answers
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Virtual terms in the Dyson series (time dependent perturbation theory)
Let the interaction evolution operator in the interaction picture be
$$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$
where $T$ is the time order operator and $H_I=H-H_0$ is the ...
4
votes
2answers
113 views
Hamiltonian with one constant of motion (besides the Hamiltonian itself)
It is well known that a Hamiltonian system with $n$ degrees of freedom with $n$ constants of motion is integrable. My question is about the case in which there are only two constants of motion, one ...
0
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1answer
34 views
Two-level system Hamiltonian from electric-dipole approx
After making the electric-dipole approx., I can express the interaction of a monochromatic field with angular frequency $\omega$ and a dipole moment ${\bf \mu(x)}$ as
$V({\bf x},t) = - {\bf \mu(x)} \...
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0answers
35 views
A particle on a ring: orthogonality of eigenstates
Let us consider the quantum-mechanical problem---a particle on a ring of a circumference as $2\pi$ with a magnetic flux $A$ inserted through it:
\begin{eqnarray}
H=(-i\partial_\phi-A)^2,
\end{...
2
votes
0answers
57 views
Hamiltonian directly expressed in $(q,\dot{q})$ : how to find what is $p$?
I am reading a book about non relativistic quantization of E.M field. But first we do classical field theory.
We directly wrote the Hamiltonian of our study, and a part of our Hamiltonian is the ...
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1answer
61 views
Quantum mechanics on operator [closed]
If any operator is commute with Hamilton then they are labelled such a way that the energy eigenstate are equal and we also know it is a constant of motion. I don't related constant of motion with ...
8
votes
2answers
192 views
Lagrange multiplier in spin liquid mean-field theory (Paper by X.G. Wen)
My question is about a step in this paper: PhysRevB.65.165113 (X.G. Wen) page 6
Or alternatively: PhysRevB.90.174417 page 3. All the papers concerning spin liquids and the projective symmetry group ...
1
vote
0answers
49 views
Spontanous emission Hamiltonian model
I am looking for a clear (and not too long) model of spontaneous emission, for an atom modeled by a two level system in a cavity where the field is multimode
I am looking for model bases on ...
1
vote
2answers
101 views
Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator
In following Prof. Toyer's Computational Quantum Physics lecture notes, I came across the following:
In computing the Schrödinger equation in real space, one can make a "split operator" Ansatz, for ...
0
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0answers
32 views
What does “substantial changes” in material properties mean in geometric optics?
We know that (I read it from Kip Throne's Modern Classic Physics) if a wave's wavelength is smaller than the length scale over which "substantial changes" of material properties occur, then the wave ...
0
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1answer
89 views
Hamiltonian in a Master Equation
I am going through this paper on the complete positive map with memory. The bath operators $\Gamma_k (t)$ are told to satisfy the correlation $\langle \Gamma_j(t) \Gamma_k(t^\prime) \rangle = a_k^2 e^{...
0
votes
1answer
30 views
Can I express the Hamiltonian in terms of $L_z$ operator only, not $L_x$ and $L_y$? Is it generally true, that $-\omega L_z = H$?
I encountered the relation in the Solution of Problem 5.1 in the book by Kyriakos Tamvakis titled "Problems and solutions in quantum mechanics":
$$\frac{i}{h}[H,\textbf{L}]=-\frac{i \omega}{h}[L_z, \...
4
votes
2answers
83 views
Can the Hamiltonian operator act on a bra, if it was once acting on a ket?
I was watching a MIT Quantum Physics III class when I got a doubt about a specific bra-ket manipulation. My doubt is about the step from the expression $(3.7)$ to the expression $(3.8)$ of the lecture ...
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0answers
42 views
Solving time evolution equations in Hamiltonian formalism
I have 4 time evolution equations and the Hamiltonian $H(X_{1},X_{2},P_{1},P_{2})$ that generates the time evolution depends on 4 canonical coordinates but I would like to solve the differential ...
1
vote
1answer
37 views
Hamiltonian-commutation, hermiticity and non-hermiticity (QM)
When we have a QM system in an energy eigenstate (say after a measurement of energy) then we can measure any time another quantity that is described by an hermitian operator that commutes with the ...
2
votes
2answers
58 views
What is the simplest possible Hamiltonian that yields an Antisymmetric Wavefunction?
I am using a Split-Operator Fourier Transform (SOFT) technique to solve the time-dependent electronic Schrödinger Equation (TDSE) for a Hydrogen molecule under the Born-Oppenheimer approximation. So I ...
12
votes
2answers
2k views
In the quantum hamiltonian, why does kinetic energy turn into an operator while potential doesn't?
When we go from the classical many-body hamiltonian
$$H = \sum_i \frac{\vec{p}_i^2}{2m_e} - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{...
2
votes
1answer
141 views
What does $|$ mean in the Schrödinger Equation?
I saw the $|$ symbol in the Schrödinger Equation $$i\hbar\frac{\partial}{\partial{t}}|\Psi(r,t)\rangle=\hat{H}|\Psi(r,t)\rangle$$ But I don't know what the $|$ means.
What does $|$ mean in the ...
0
votes
1answer
99 views
Matrix of Hamiltonian $H=\frac{ħω}{2}(|1〉〈1|−|2〉〈2|)+\frac{iħχ}{2}(|1〉〈2|−|2〉〈1|)$ [closed]
I have a second order system, with a Hamiltonian $$H=\frac{ħω}{2}(|1〉〈1|−|2〉〈2|)+\frac{iħχ}{2}(|1〉〈2|−|2〉〈1|)$$ where $|1〉,|2〉$ form a complete basis for the system. I'm trying to get the matrix that ...
0
votes
1answer
45 views
Why do we consider only one mass when solving linear harmonic oscillators in quantum physics?
While solving the Hamiltonian, books concentrate on the horizontal flow with only one mass attached to the string. Isn't there any consequences if we add more masses and why is friction always ignored?...
2
votes
2answers
81 views
Why does the $\phi$-cubed theory have no ground state?
In the book of Sredinicki's, he claimed that the $\phi^3$ theory has no ground state, hence this is not a physical theory.
My question is that I can't see why this system has no ground state. And I ...
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1answer
23 views
Does adiabatic quantum computation require the initial and final ground states to be non-orthogonal?
Background
At a recent talk, I was told by the speaker that it is not possible to adiabatically transfer from one ground state $|\psi_0 \rangle$ to another $|\psi_1 \rangle$ if these states are ...
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votes
2answers
104 views
Is a quantum harmonic oscillator always infinite dimensional?
Let us assume we have a quantum particle in a harmonic potential with the Hamiltonian
$$H = \sum_n n \omega |n\rangle\langle n|$$
If I am not mistaken.
Now when talking about harmonic oscillators ...
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0answers
39 views
Time units in simulation
I am hoping that someone could give me some insight about my problem. Currently I want to simulate the evolution of a gaussian wavepacket in a time-dependent potential using the Crank-Nicolson scheme. ...
5
votes
1answer
384 views
General derivative of the exponential operator w.r.t. a parameter
I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure:
\begin{equation*}
\frac{\...
0
votes
0answers
49 views
Interaction picture: why the Hamiltonian describing the dynamic doesn't change with the same law as other observables?
First: what happens in a general change of picture?
If I have the following equation:
$$ A | x \rangle = | y \rangle .$$
To do a change of picture is to apply a unitary $U$ on all vectors of the ...
1
vote
1answer
58 views
Discretization of Hamiltonian with first derivative
In a particular 1D system, the Hamiltonian can be writen as
$$H=\mathrm{i}\left(f(r)\frac{\partial}{\partial r}+\frac{1}{2}f'(r)\right)\; ,$$
wher $\mathrm{i}$ is the imaginary unit, and $f(r)$ is a ...
1
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0answers
37 views
How to interpret overlap in Hamiltonian if it is not a degeneracy?
In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $
\begin{bmatrix}
h_0+h_z & h_x-i h_y \\
...
0
votes
1answer
79 views
Eigenvalues of the Hamiltonian
Is every eigenvalue of the Hamiltonian a form of energy?
If not are there values of the Hamiltonian that do not correspond to the energy of the system?
0
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1answer
43 views
Is the tight binding model an effective free fermion model?
The tight-binding Hamiltionian has the form
$$H=-t\sum_i\left(c_i^\dagger c_{i+1} + c_{i}c_{i+1}^\dagger\right)$$
But does this mean that it can be represented in the form of free fermion modes?
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2answers
114 views
How does a Hamiltonian 'generate' a unitary?
I know that the unitary (propagator) is given by
$$U=e^{iHt}\tag{1}.$$
But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
1
vote
1answer
53 views
Hamiltonian of a quantum heat bath
I have seen the Hamiltonian for a heat bath written as:
$$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$
I was hoping to understand this equation better. This suggests that ...
0
votes
0answers
25 views
What is the 4x4 matrix for the charge inversion operator and how do you construct it?
I have a 4x4 Hamiltonian describing a part of my system. To get a holistic view of the situation I need to do a charge inversion on the matrix. What is the 4x4 charge inversion operator? And what is ...
3
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2answers
63 views
Wilson Sommerfeld Method to solve for Energy
I have an example in my notes to find the quantum energy levels when the Hamiltonian is $H(p,q)={p^2}/{2m}+(mw^2q^2)/2$. However when given the Hamiltonian $H(p,q)={p^2}/{2m}$, I'm having ...
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0answers
57 views
Mathematical representation of Symmetry Transformation
Consider a general Hamiltonian that is made up of three terms
$\mathcal{H}$ = term I + term II + term III .
Suppose the combination of charge conjugation and parity (CP) is a symmetry of this ...
0
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1answer
61 views
What people mean by “state evolving with the interacting/free theory”?
This is a quite basic question but I confess it is something I didn't get up to this point.
When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
0
votes
1answer
62 views
Potential must be real for Hamiltonian to be Hermitian?
I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that?
If we're solving the Schrodinger equation, we ...
0
votes
2answers
58 views
Good /not good quantum numbers in spin-orbit coupling
Given that the Hamiltonian associated with the spin-orbit interaction can be expressed in terms of the total orbital angular momentum and total spin operators as:
$$ H_{SO} = -\frac{e}{2m_e ^2 c^2} \...
0
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1answer
44 views
Time reversal of a QM Hamiltonian
I'm interested in the time reversal properties of a term in the non-relativistic QM Hamiltonian proportional (up to a true scalar) to
$$
H \propto (\vec S_1 \times \vec S_2) \cdot \vec L
$$
The ...
0
votes
1answer
82 views
Quantum walk and the Hamiltonian operator
The Hamiltonian operator is defined on the graph $G$ as $H_{A}(t) = \exp(itA)$ where $A$ is the adjacency matrix of the graph $G$. It is said that this operator is a transition matrix and represents ...
2
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3answers
113 views
Time translation invariance of Hamiltonian
I am learning about the time translation invariance of the Hamiltonian. I read that
the time translation invariance is already manifest in the fact that our
Hamiltonian is chosen an ...
4
votes
1answer
98 views
‘Supersymmetrizing’ an arbitrary quantum-mechanical potential
To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form
$$
H = A^\...
1
vote
1answer
41 views
Hamiltonian time-independent, partial derivative always zero?
For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$.
Hamiltonian is a function that maps a pair ...
