Questions tagged [hamiltonian]

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

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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'...
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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 ...
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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(\...
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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 ...
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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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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{...
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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\...
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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-...
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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]...
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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 ...
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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 $\psi$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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''}(...
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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+...
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1answer
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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) $$...
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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 ...
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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 ...
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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^{...
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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}...
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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?
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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 ...
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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 ...
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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" $$...
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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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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 ...
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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,...
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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,\...
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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 $UU^\dagger H = H$,...
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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}.\...
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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 ...
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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 ...
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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_{...
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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 ...
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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 ...
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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)\...
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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 ...
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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}\...
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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^...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...

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