# Questions tagged [hamiltonian]

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

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### Expectation value of coordinate mixed operator with ground state

I have a Hamiltonian of the Hydrogen atom: $H=H_0+H_1+H_2$ , when: $H_0$ is the hamiltonian from central force and from electron momentum , $H_1$ is the relativistic kinetic fixing, and $H_2$ is the ...
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### Energy in spherically symmetric space times

In deriving the equations of motion for geodesics in spherically symmetric spacetimes through Hamiltonian formalism, we can find some constants of motion, namely, $E$ and $L$, the energy per unit of ...
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### What is the physical meaning of Hamiltonian eigenstates for a single particle?

Let us assume we have one 2-dimensional quantum system with a Hamiltonian $$H = \sum_{n=1}^2 n \omega \mid n\rangle\langle n\mid$$ Do I understand it correctly when I assume that the eigenstates of ...
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### Commutation relations

Given that the Hamiltonian for Muonium spin in zero magnetic field is $$\hat{H} = a \vec I \cdot \vec J$$ where $\vec I$ is the spin of a muon, and $\vec J$ is the spin of the electron, what is the ...
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### Alternate definitions of Thermal states

The definition of thermal states I'm used to is: $$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$ where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
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### Harmonic oscillator energy difference between $(n+\frac{1}{2})h \omega$ and $(n+\frac{1}{2})\hbar \omega$

When I was studying the Harmonic Oscillator using the Schrödinger equation, I was told in lectures to pay attention to the units. There were 2 different equations given for the Energy of a Harmonic ...
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### How to construct a 2-partite matrix

Let's assume we have an internal hamiltonian $H_0 = \mid 1\rangle \langle 1\mid$. Now let's assume we have two systems with identical Hamiltonians $H^1_0$,$H^2_0$ and I want to compute the joint ...
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### How to calculate Hamiltonian when Lagrangian has higher order derivatives? [duplicate]

If we have a Lagrangian density $\mathcal{L}$ for a scalar field $\phi$ depending on $\phi$, $\partial _{\mu} \phi$, and $\partial _{\mu} \partial _{\nu} \phi$, what is the Hamiltonian? Additionally, ...
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### Constructing a Hamiltonian for $N$-qubits

Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$. Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like? I ...
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### How to calculate the ground state of Ising model at non-zero temperature

I'm studying the quantum Ising model, i.e. with Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$. I know conceptually how to compute the ground state of the Ising model at zero ...
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### How does the ground state of the quantum Ising model relate to Schrodinger equation?

The Hamiltonian $$H = -\sum_{i\in V} h_i \sigma_i^z -\sum_{(i,j)\in E} J_{ij} \sigma_i^z\sigma_j^z - \Gamma\sum_{i\in V} \sigma_i^x$$ is kind of the cost function of the quantum annealing optimization ...
The corrections to the wavefunctions and energies depend on $<\psi_m^0\,| \,H'|\psi_n^0>$ to some order. I would've thought that $<\psi_m^0\,| \,H'|\psi_n^0> \, =\, <H' \psi_m^0\,|\... 2answers 205 views ### Calculating expectation value of the Hamiltonian squared So the main idea of the problem was to find the error in the argument, which I think I have a good grasp of. Basically, the Hamiltonian of the wavefunction is a constant non-zero value inside the box ... 1answer 137 views ### Solving Schrödinger equation by neural networks - trial function explanation I'm reading this paper about solving Schrödinger equation using the combination of genetic algorithm and neural networks. But one part confuses me - the author defines his trial function, i.e. the ... 1answer 102 views ### Hamiltonian for a magnetic field An atom has an electromagnetic moment,$\mu = -g\mu_B S$where S is the electronic spin operator ($S=S_x,S_y.S_z$) and$S_i$are the Pauli matrices, given below. The atom has a spin$\frac{1}{2}$... 3answers 369 views ### What is the Hamiltonian in the “energy basis” for a simple harmonic oscillator? My textbook says that for a simple harmonic oscillator the Hamiltonian can be expressed in the "energy basis" in this way: $$\hat H=\hbar\omega\bigg(\hat a^{\dagger}\hat a + {1\over 2}\bigg).$$ I ... 1answer 54 views ### Boltzmann equation derivation for$H=v\sigma \cdot p$hamiltonian I am trying to write the Boltzmann equation for $$H=v_{F}\vec{\sigma}\cdot(\vec{p}-e\vec{A}).$$ This is a free charged particles gas. The velocity for this hamiltonian is $$\vec{v}=v_{F} \vec{\sigma}.... 2answers 236 views ### In quantum mechanics, why is \langle T\rangle=\frac{\langle p^2 \rangle}{2m} rather than \langle T\rangle=\frac{\langle p \rangle^2}{2m}? I'm a newbie reading quantum mechanics from "Inroduction to Quantum Meachanics" by Griffiths and in the early pages of the book the author defines:$$\langle x\rangle =\int_{-\infty}^{\infty} x|\Psi(... 1answer 100 views ### How to generate ladder operators for an arbitrary Hamiltonian? How to generate ladder operators for an arbitrary Hamiltonian? i.e. for a power-law potential. 1answer 35 views ### Intro QM representation of Abraham-Lorentz Force What does the Schrodinger equation look like if you add some term for the Abraham-Lorentz force? I get a self reference term I'm not sure how to handle. I realize this is probably addressed by QED, ... 1answer 146 views ### Hamiltonian for a 1D spin chain [closed] I am trying to implement the Lanczos algorithm to tridiagonalize the Hamiltonian for a 1D spin chain of length$L$, but I am unable to decipher from my professor's notes (here's a link), what the ... 1answer 45 views ### Is the ground state energy always larger for the system with higher potential energy? Say we have two Hamiltonians$\hat{H}_1$and$\hat{H}_2$that differ only in their potential energies and $$V_2(x) > V_1(x)$$ for all$x$. Is the energy of the ground state of system 2 necessarily ... 1answer 235 views ### Eigenstates of a Hamiltonian [closed] For a particle with a spin of 1/2, which was exposed to both magnetic fields$B_{0}=B_{z}e_z$and$B_1=B_xe_x$I already found the eigenvalues of its Hamiltonian which is given by \begin{... 2answers 62 views ### How to distinguish two different systems which have the same Hamiltonian in the Schrodinger equation? Suppose we have 4 hydrogen atoms and 2 oxygen atoms. If we write the Hamiltonian containing all the possible interactions for the Schrodinger equation, how can we distinguish the system is two ... 3answers 2k views ### What is meant by unitary time evolution? According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution? 1answer 44 views ### Total angular momentum in QM Dos the total angular momentum,$J=S+L$, commute with the hamiltonian of a general sistem, with no particularities? 1answer 162 views ### Perturbation theory: justifying expansion in terms of eigenstates of the basis Hamiltonian I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this ... 1answer 249 views ### How to diagonalise a hamiltonian which posesses symmetry? I have a large hamiltonian but I know that it posseses some symmetries. How do you reduce the hamiltonian in order to find the eigenenergies? 2answers 244 views ### Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric? The time-dependent Schroödinger equation is given as (with$\hbar=1$): $$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$ where$\psi$is some normalized column vector and$H(t)$is a Hermitian matrix with time-... 2answers 76 views ### QM: Time evolution with$H = H(t)In order to calculate time evolution in QM we use Schrödinger equation \begin{align*} i \partial_t |\psi\rangle_t = H(t) | \psi\rangle_t. \end{align*} IfH\neq H(t)then \begin{align*} i \partial_t ... 1answer 94 views ### Why does the Hamiltonian represent something different after plugging in the solution? so I am beginning to learn Hamiltonian mechanics. We have learned that the Hamiltonian is a function of q, p, and t. Once we have a Hamiltonian, we can use the Hamiltonian equations to derive the ... 1answer 85 views ### How to expand this equation?H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$[closed] $$H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$$ in the approximation$ \left |x_{1}\right |,\left |x_{2}\right |\ll R $we expand to obtain in lowest ... 0answers 107 views ### Definition of Hamiltonian in Quantum Mechanics [duplicate] Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$... 3answers 217 views ### Deriving or building a Hamiltonian from a Density Matrix Is it possible to create a Hamiltonian if given a Density Matrix. If you already the the Density Matrix, then is the Partition Function (Z) even needed? This Q is not about physics. Its about an ... 0answers 196 views ### How to prove time reversal symmetry in a system (given a hamiltonian) The generic hamiltonian for a particle that interacts with an electromagnetic field can be written as: $$H=\frac{1}{2M}\sum_{i}\left(P_i-\frac{q}{c}A_{i}(X_j)\right)^2+V(X_j)+q\phi (X_j)$$ Where$(\...
There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion To be explicit, suppose you're solving for a density matrix $\rho(t)$...