Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
1,086
questions
2
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0answers
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Diagonalization of a Hamiltonian in a different frame of reference
Under a time-dependent transformation $V(t)$ of the state vectors $|{\psi}\rangle$
\begin{equation}
|\psi'(t)\rangle = V(t) |\psi(t)\rangle
\end{equation}
The Hamiltonian $H(t)$ has to transform as $H'...
4
votes
0answers
46 views
Derivation of operator version of the classical wave equation
I have the following summarised derivation for the operator version of the classical wave equation for massless and material particle.
My question is about the statement:
However, a problem is that ...
-1
votes
2answers
67 views
Does the Schrodinger equation obey the rule for differentiating a function if the function is in terms of the wavefunction?
Does the Hamiltonian operator act like a derivative when acting on a functional in terms of wavefunctions?
For example, does $$H\psi^2=2\psi H\psi$$ hold true? More generally, if the functional, $F(\...
0
votes
1answer
64 views
Diagonalization of Hamiltonian in second quantization
I am solving a problem from second quantization involving a Hamiltonian
$H=\hbar\omega a^\dagger a+\epsilon ab^\dagger +\epsilon ba^\dagger$ ,
which needs to be diagonalized by using the ...
0
votes
0answers
25 views
Does time-dependent perturbation theory work for time-independent perturbations?
Are the results from time-dependent perturbation theory for time dependent Hamiltonians of the form $H = H_0 + \Delta H(t)$ (such as the result below) equally valid for time independent Hamiltonians ...
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vote
0answers
63 views
How do I compute the Hamiltonian generating the CNOT gate?
I have the matrix for the C-Not gate acting in a system of two qbits, $|q_{2}q_{1}\rangle$. In the basis $ \left[ |00\rangle, |10\rangle, |01\rangle, |11\rangle \right]$:
$$
C_{N} =
\left[
\begin{...
1
vote
0answers
30 views
How to compute the Hamiltonian matrix elements in momentum basis? [closed]
I'm struggling to get it right, I'm having a lot of ideas but no one of them are making sense to me. So, the problem is as follows.
"Evaluate:
\begin{equation}
\langle q'_t\mid\hat{H}\mid q_t\...
3
votes
2answers
203 views
Fundamental Understanding of Hamiltonians
First of all, my major is CS for several months I have been exploring the area Quantum Computing, therefore my background in Quantum Mechanics is a bit lacking.
I know that a Hamiltonian is a self-...
2
votes
1answer
32 views
Peskin and Schroeder, where is the mass in the denominator of the simple harmonic oscillator Hamiltonian?
This relates to page 20 of Peskin and Schroeder.
They state that the Fourier transform of the Klein-Gordon field satisfies the following:
$$\left[\frac{\partial^2}{\partial t^2}+(|\vec p|^2+m^2)\right]...
1
vote
1answer
31 views
Hamilonian as a sum of quantum oscillators with symmetric matrices
I'm watching the Introduction to Quantum Field Theory course by Tobias Osborne (the lecture notes for which can be found here https://raw.githubusercontent.com/avstjohn/qft/master/QFT.pdf). In the ...
0
votes
0answers
33 views
Help with a step in a derivation in Peskin & Schroeder
In page 53 of P&S, they postulate the commutation relations for the annihilation and creation operators of spin 1/2 particles. From there, they start computing the commutation relations of $\psi$ ...
0
votes
0answers
25 views
Show the effect of a disturbance is not linear in $p$
Considering the Hamiltonian of the harmonic oscillator, written as:
$$\hat H_0 = (\hat a^\dagger_x \hat a_x + \hat a^\dagger_y \hat a_y + \hat1)$$
and a perturbation $\hat W = p \hat L_z$, show that ...
0
votes
2answers
63 views
The Hamiltoninian function always verifies $H(-p,q)=H(p,q)$?
Is it true that the Hamiltoninian function of a system always verifies $H(-p,q)=H(p,q)$? If $H$ is the sum of the potential energy plus the kinetic energy then it seems true. But is there a possible ...
1
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0answers
32 views
How do I find the state and the energy exchange of a system after time $t$?
Assuming that the Hamiltonian of my system is of the form:
$$H=k\left( \left| 0\right> \left<1 \right| + \left| 1\right> \left< 0\right| \right)$$
and that my initial state can be ...
0
votes
2answers
39 views
Understanding the density matrix for systems in Thermal Equilibrium
If the eigenstate $| i\rangle$ of the hamiltonian ($\hat H$) has energy $E_i$ the relative probability of the system being in that state is $e^{-\beta E_i}$ where $\beta = 1/k_BT$.
The density matrix ...
2
votes
1answer
104 views
A problem with the BCS energy expectation value of an excited state
I want to calculate the energy expectation value of the following state.
\begin{align}
|\Psi_{ex}\rangle = \hat{c}_{-k'\downarrow}^\dagger \hat{c}_{k''\uparrow}^\dagger \prod_{k \neq k', k''}(...
2
votes
3answers
67 views
How degenerate time independent perturbation theory works? [duplicate]
Let's consider the usual setup for time independent perturbation theory:
$$H=H_0+\varepsilon H'$$
and we can then set up the usual expansion:
$$(H_0+\varepsilon H')[|n_0\rangle+\varepsilon |n_1\rangle+...
1
vote
1answer
39 views
Last step derivation of hamiltonian of particle in electromagnetic field
My textbook quantum mechanics does an analogues derivation of the hamiltonian as given here, but I'm struggling to understand the last step: The final obtained hamiltonian is (in my textbook's case)
$$...
0
votes
0answers
34 views
Does localized energy operator make sense in quantum field theory?
In quantum field theory in weak gravity regime, it is possible to trace out some region to obtain reduced density matrix of states restricted to hypersurface $V$.
I believe the same idea carries to ...
0
votes
1answer
47 views
Why does the hamiltonian in the operator act on the wave function?
Suppose we have the following relation:
$H_0 | \phi_1\rangle = E_1 |\phi_1 \rangle $
Why is it that if we take the unitary function $$U_{0} = \exp\left(\frac{-iH_{0}t}{\hbar}\right)$$
and apply it to ...
1
vote
0answers
27 views
How is Pauli's exclusion principle conserved in transformations from real space to $k$ space for a Fock space hopping Hamiltonian?
$$\hat{H}= -t\sum_{\langle i,j\rangle} c_{i\sigma}^{+}c_{j\sigma}+h.c$$
For this Hamiltonian a lattice with all sites doubly occupied would give 0 in real space and for single occupancy it gives $-4t^{...
1
vote
1answer
35 views
Time evolution in an oscillating magnetic field for spin-1/2 particles
This might be a rookie mistake.
For a magnetic field oriented in the z-direction of form, $B = B_0 \cos(\omega t) \hat{k}$.
The Hamiltonian in this case will be $H = \omega_0 \cos(\omega t) \hat{S_z}...
-1
votes
3answers
83 views
What physical quantity does the Hamiltonian operator represent? [closed]
In a equation in quantum mechanics for example
$$\mathrm i\hbar \frac{d}{dt}|\phi\rangle=H|\phi\rangle \tag{1}.$$
Would it also be a hermitian operator?
1
vote
1answer
51 views
Formalism of $H=E$ (Hamiltonian mechanics)
An answer to the question When is the Hamiltonian of a system not equal to its total energy? is:
In an ideal, holonomic and monogenic system (the usual one in classical mechanics), Hamiltonian equals ...
1
vote
1answer
60 views
Confusion Regarding the Derivation of Graphene Dispersion Using Annihilation and Creation Operators
I am going through a text which derives the energy bands in graphene (https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/3/67057/files/2018/09/graphene_tight-binding_model-1ny95f1.pdf) and am stuck on a ...
1
vote
2answers
105 views
Vacuum fluctuations of quantum scalar field
Consider a free scalar quantum field
$$ H = \int d^3 x \left( \, \Pi(x)^2+(\nabla\phi(x))^2 \right). $$
Introducing the creation and annihilation operators we find the "vacuum catastrophe"
$$...
3
votes
3answers
387 views
Why do Hopping Hamiltonians have physical significance?
When studying quantum systems, we often use "hopping" Hamiltonians to represent the system energy. For example, we might represent a 1D ring of N sites with a single particle as:
$H = -t\...
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vote
1answer
39 views
Does Hamiltonian including spin involves tensor products?
I'm trying to learn introductory QM from a book and I'm confused about how spin is incorporated into the formalism.
From what I have gathered so far, the state of a particle can be fully described by ...
1
vote
1answer
27 views
Rewrite hamiltonian from integral over momenta to integral over energies
How do I rewrite a Hamiltonian written as some integral over momenta, e.g.,
$$ H = \int\frac{d^d \vec k}{(2\pi)^d}\omega(\vec k) a_{\vec k}^\dagger a_{\vec k} $$
in terms of energy eigenmodes?
That is,...
0
votes
0answers
70 views
How to define the partial of the logarithm of a density operator?
I've been thinking about self-information ($I:=-\log (\rho)$) lately and I have a question on how to evaluate the derivative of such an operator... I realize that $i\hbar(\partial/\partial t)\rho=[H,\...
1
vote
3answers
70 views
Quantum unitary transformation
In quantum mechanics, we know $\dot{\psi}=-\frac{i}{\hbar}H\psi$,
but why is $U\dot{\psi}=-\frac{i}{\hbar} \left(UHU^\dagger \right) U\psi$?
Does it mean $UHU^\dagger = H$ ? I think $UU^\dagger H = H$,...
0
votes
1answer
39 views
Electric dipole moment and magnetic moment
What is the intuitive explanation for electric and magnetic moments? How would we justify their presence in the Hamiltonians, for example, $$H=-\overline\mu.\overline{B}$$and,
$$H=-\overline{d}.\...
3
votes
0answers
28 views
Meaning of the concept of external parameters in Statistical Mechanics
I'm confused about the meaning of the concept of external parameter in Statistical Mechanics. According to my textbook, the Hamiltonian of a system is a function that depends on the generalized ...
1
vote
1answer
25 views
Are these Transformations of the Green's function equivalent?
The Green's function $G(E)$ can be constructed from the Hamiltonian $H$
$G(E) = [(E+i\epsilon)I - H]^{-1}$
where $I$ is the identity matrix. Say we want to perform a transformation into another basis ...
0
votes
0answers
21 views
Good basis for coupled modes
Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system looks something like this:
$$H=-\frac{J}{2} \sum_{m}b_{...
0
votes
2answers
42 views
Deriving the Hamiltonian for a simple pendulum using mechanical momentum as a free parameter
So when we covered the derivation of a simple pendulum we , and from what ive found on the web, defined our free parameter as $q=L\theta$ and arrive at the Hamiltonian for a Harmonic oscillator.
But ...
0
votes
0answers
19 views
Effective Hamiltonian for Transmission Coefficient
Suppose we have a ring resonator. For the m-th resonant mode, its wave vector along the waveguide $\beta_{m}$ satisfies $\left(\beta_{m}-\beta_{0}\right) L=2 \pi m$. where L is the circumference for ...
0
votes
1answer
36 views
Can one assign a Hamiltonian under a general time-dependent transformation in quantum mechanics?
The time evolution of states under a time-dependent Hamiltonian $H_S(t)$ in the Schrödinger picture is determined by
$$
\label{TDS}
i\hbar \frac{d |{\psi(t)}\rangle}{dt} = H_{\mathrm{S}}(t) |\psi(t)\...
4
votes
1answer
103 views
Value of tensor product of projectors
If I have two projectors $\pi_1, \pi_2$ such that for some $|{\phi}\rangle$:
$\langle {\phi}| I \otimes \pi_1 |{\phi}\rangle \geq e$ and $\langle {\phi}| \pi_2 \otimes I | {\phi}\rangle \geq e$
What ...
0
votes
1answer
58 views
Direct Hamiltonian Operator computation in polar coordinate
Suppose a classical Hamiltonian of the form
$$\mathcal{H}=\frac{1}{2m}(p^2_x+p^2_y)+a(x^2+y^2)^{1/2}$$
We know that this change to following quantum operator
$$\hat{H}\rightarrow -\frac{\hbar^2}{2m}\...
0
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0answers
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Deduction of Hamiltonian using Fourier coeficients on a vector potential $\vec{A}$
Given the operator $\vec{A}=\frac{1}{\sqrt{V}} \sum_{\vec{k}, \alpha} \varepsilon^{\alpha}\left(c_{\vec{k}, \alpha}(t) e^{i\left(\vec{k} \cdot \vec{r}-\omega_{k} t\right)}+c_{\vec{k}, \alpha}^{*}(t) e^...
2
votes
2answers
46 views
Is $E=0$ included in the energy spectrum of the free particle in 1d?
In finding the eigenfunctions, $\psi_E$'s, of the free-particle Hamiltonian in 1d,
$$
H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2},
$$
with eigenvalues $E$'s, subject to the conditions that they are ...
1
vote
0answers
43 views
What mathematical property must the Hamiltonian have for the system to evolve to an entangled state? [closed]
For a closed system composed of two subsystems in a pure, non-entangled state, what mathematical property must the Hamiltonian have for the system to evolve to an entangled state?
What property must ...
6
votes
1answer
440 views
Is it right to replace Hamiltonian with Lagrangian in the Schrödinger equation?
The Time-dependent Schrödinger equation is given by
$$i\hbar \frac{d}{dt}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle $$
From Classical Mechanics, we know that
$$\mathcal{L}=\dot{q}p-H$$
which should change ...
1
vote
1answer
45 views
Time evolution of an electron in an homogeneous magnetic field: get rid of $S_y$ in the exponential
Context
We have a particle of spin $1/2$ and of magnetic moment $\vec{M} = \gamma\vec{S}$.
At time $t=0$, the state of the system is
$$
|\psi(t=0) \rangle = |+\rangle_z
$$
We let the system evolve ...
2
votes
1answer
67 views
How to determine the ground state of quantum harmonic oscillator like Hamiltonian?
For the time-dependent Hamiltonian
$$H = \frac{\hat{P}^2}{2m} + \frac{1}{2} m\omega^2\hat{X}^2 + m\omega^2vt\hat{X} +v\hat{P}$$
I would like to calculate the ground state, more precise, the stationary ...
1
vote
1answer
80 views
Is the Hamiltonian an Observable
From the book Quantum Mechanics by Cohen-Tannoudji it seems that the only requirement for an Operator to be an Observable is to form an orthonormal basis in the state space (finite or infinite ...
0
votes
1answer
42 views
Unique properties of Hamiltonian
Given a general (time-independent) system where I have some Hermitian operator $O$, is there a way of knowing if $O$ happens to be the Hamiltonian?
In other words, are there special mathematical ...
2
votes
2answers
123 views
How does the hamiltonian operator $contain$ the angular momentum operator? [closed]
The Schrödinger equation is written in terms of the hamiltonian
$$
\hat H \Psi = i\hbar \partial_t \Psi
$$
When solving this equation for the hydrogen atom (in position space) by saperation of ...
4
votes
4answers
303 views
Looking for a Basic ( or “for Dummies” ) Explanation of the Lagrangian - Hamiltonian Relationship. ( Mathematician ) [duplicate]
(Mathematician here - first time physics.stack poster).
I'm basically looking for as simple as possible explanation of the Hamiltonian - Lagrangian relationship.
$\textbf{My understanding :}$
$\textbf{...
