Questions tagged [ground-state]

The ground state of a quantum/classical mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. When a quantum system has infinite possible ground states, it is gapless with massless modes; if a quantum system has finite ground states, it is known as gapped and potentially topologically ordered. The ground state of a quantum field theory is usually called the vacuum state or the vacuum.

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Absolute zero and zero-point energy

What will be the theoretical temperature of ideal solid state matter where all the atoms have only zero-point quantum fluctuation? Is it exactly 0.00000 K or still some 0.000000x K?
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How to find the ground state of a system via its Hamiltonian density?

I am trying to find the ground state of the following Lagrangian (with $\lambda> 0 , g > 0$): $$\tag{1} \mathcal{L}= -\frac{1}{2}(\partial_\mu \partial^\mu \sigma + \partial_\mu \pi \partial^\mu ...
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Ground state in radial quantization -- Why isn't $\phi(0) |0\rangle = |0 \rangle$?

I am trying to reconcile two perspectives on the ground state defined through the path integral. In Tom Hartman's gravity lectures (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf) he says ...
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Why is the ground state of an atom never degenerate?

In this paper https://link.springer.com/article/10.1007/BF01391720 , Kellner argues that the ground state of the helium atom must be spherically symmetric because "it is known that the ground ...
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Allowed k values for Spinless Fermi-Hubbard in 2D

Suppose we have the following Hamiltonian of spinless Fermi-Hubbard model: $\hat H = -t \displaystyle\sum_{<i,j>} (c^+_ic_{j} + h.c.)$ The Ground state of the free fermion Hamiltonian can be ...
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How do I find the ground state of a hamiltonian?

On some notes on Hamiltonian field theory it shown that the hamiltonian density can be obtained from the Hamiltonian via $$\tag{1} H = \int d^3x \cal{H} $$ and the example shown gives $\cal{H}$ to be $...
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Ground state of sums of commuting, translated projectors

I have in mind a spin chain of length $L$ with local Hilbert space dimension $d$ and projectors $\{ P_i \}$ that act on $r$ sites $i, i+1, ..., i+r-1$. The projectors are identical besides which sites ...
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Can we measure the ground state energy?

If we consider a quantum harmonic oscillator, the ground state energy $\hbar\omega/2$ is typically stated to be not measurable, as energies are always measured as relative values (energy differences). ...
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How to construct a set of local Hamiltonians sharing the same ground state?

In the quantum approximate optimization algorithms, the superposition state $|+\rangle^{\otimes N}$ is usually prepared as the initial state, which is the ground state of driven Hamiltonian $H_M=\...
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Iterative projection into ground state

In this paper, it says that one can "determine the ground state wave function by applying the projection operator $\exp(-\tau H)$ to an arbitrary initial state $|\Psi\rangle$," and that in ...
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Quantum Harmonic Oscillator eigenfunction

I'm trying to understand why in quantum harmonic oscillator when finding ground state eigenfunction we don't use $a^\dagger$. For a simple harmonic oscillator the Hamiltonian is given by $$H=\hbar\...
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Scalar field theory with two scalars

Consider the following scalar field theory with a kinetic term as follows $$ \mathcal{L} = \frac{1}{2}\partial_{\mu}\phi_1\partial^{\mu}\phi_1-\frac{1}{2}\partial_{\mu}\phi_2\partial^{\mu}\phi_2 + V(\...
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Perturbation theory for triangular barrier

My teacher asked us to use perturbation theory to get the energy of the ground state in such potential: $$ V = \left \{ \begin{aligned} \infty \quad x < 0 \\ Ax \quad x \geq 0 \end{aligned} \right ...
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Time free evolution of the harmonic ground state (Quantum mechanics)

I have an atom in the ground state for a harmonic potential. At time $t = 0$ the parabolic potential is switched off. How can I derive the time evolution of the wave function $\psi(t; x)$ during the ...
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Ground state in $k$-space convert back to real space - spinless non-interacting fermionic system

Say, you have the following spinless non-interacting fermionic (1D) Hamiltonian: $$\hat H = -t \displaystyle \sum_{\langle i,j\rangle} (\hat c_i^+ \hat c_j + \hat c_j^+ \hat c_i)$$ Diagonalizing it (...
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Existence of Ground State of Dirac equation

In chapter four of Ryder, the author showed that there exists a ground state $|0\rangle$ for the Kelin-Gordon equation, just like the case of the linear harmonic oscillator. However, I was not able to ...
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Exciton state, ground state and completeness relation

Considering the exciton eigensystem $\mathcal{H} | \lambda \rangle = E_\lambda |\lambda\rangle$ with the Hermitian Hamiltonian $\mathcal{H}$ and wave function $|\lambda\rangle$. I'm thinking about the ...
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Energy values in case of 1D infinite potential well

As it is depicted in the picture, there are 4 types of potential wells and if $E_1, E_2, E_3, E_4$ represent the ground state energies of the particle confined in those wells, what should be the ...
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Exact ground state energy of the noninteracting one-dimensional Bose Hubbard model? [closed]

The Hamiltonian of Bose-Hubbard model reads as $$ H = -J\sum_{<i,j>}a_i^\dagger a_j + h.c. + \frac{U}{2}\sum_i n_i(n_i - 1) $$ In the non-interacting case where $U=0$, what is the ground state ...
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Exact eigenfunctions of two interacting identical particles [closed]

While I was reading about quantum states of $N$ interacting identical particles, I realized that I don't understand some fundamental things. So In order to clear my confusion, I decided to consider a ...
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What is the ground state of the Hubbard model for $t = 0$?

I am new to the Hubbard model and it is not clear to me how we go to the so called 'atomic limit' where $t = 0$. So we only have a $U > 0$ term in the Hamiltonian. This should be a trivial problem ...
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Tensor Networks with Julia and implementing given Hamiltonian

I have this Hamiltonian: (ref: https://arxiv.org/abs/1302.5843) I want to solve this Hamiltonian by using tensor networks. I wanted to make the implementation with ITensors, Julia. However, I am ...
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Ground State calculation for defined 2D Ising Model with tensor networks

I have a Hamiltonian and 2D spin-lattice system. I am trying to find a ground state configuration. Spin interactions are long-ranged so I am trying to use PEPS to approximate. My question is this: ...
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WKB approximation problem [closed]

I have some issues on solving this problem: use the WKB method to estimate the ground state energy of a particle of mass $m$ that moves in a three dimensional potential $V(r)=kr$, where $k$ is a ...
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Vacuum manifold and fermion condensation

Vacuum manifold is just another name for the manifold spanned by the ground states of quantum field theory. It is also called moduli space. According to https://en.wikipedia.org/wiki/Vacuum_manifold, ...
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Is the zero point energy of this system zero?

Consider the following Hamiltonian: $$\hat H=\frac{\hbar\omega}{2}(\hat x^2+\hat p^2)-\frac{\hbar\omega}{2}\hat 1 =\frac{\hbar\omega}{2}(\hat x^2+\hat p^2-\hat 1 )$$ After defining annihilation and ...
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Minimum energy eigenvalue [duplicate]

Why is the energy eigenvalue is always greater than minimum potential for a particle moving in a certain potential?
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How to calculate the ground state of bosonic hamiltonian?

I am trying to find the ground state of the Hamiltonian: $H=\sum^N_{K=i}\omega_kb_k^\dagger b_k$ where $b_k=\frac{1}{\sqrt{N}}\sum^N_{N=i}\exp(\frac{2\pi ikn}{N})b_n$. Any hints on how to proceed?
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Boson Einstein condensation of free particles

Under Schrödinger's representation, for free particles without spin, each eigenstate vector is $\delta(x-x_0)$, corresponding to the eigenvalue $x_0$, each position in the 3-dimensional configuration ...
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Elements Below Ground Energy State

I have been hearing things about people getting elements like hydrogen below the ground energy state. I don't understand how that is even possible. The more interesting thing is how this actually ...
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Do the Gell-Mann and Low Theorem and Haag's Theorem contradict one another?

I was wondering if these two theorems do not conflict with one another in some sense. It sees as if one allow us to move from the free to the interacting theory, while the other forbids it.
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What type of regulation is being employed?

As already mentioned in this post. In the context of QFT, the kernel of integration for the overlap of a field configuration ket, $| \Phi \rangle$ with the vacuum $|0\rangle$ in a free theory is given ...
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Estimating ground state of Yukawa potential using a variational method [closed]

I have to calculate an upper bound for the ground state energy $E_0$ given the Yukawa potential $$ V(r) = -\dfrac{g}{r} e^{-kr}\ ,\quad g,k > 0\ , $$ and a test function family $$ \phi_\lambda (r) =...
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Ground state energy of the closure of an essentially self-adjoint operator

In my lecture notes it says, that for an essentially self-adjoint operator $(H,D(H))$, that is bounded from below, and its self-adjoint closure $(\bar{H},D(\bar{H}))$ the ground state energy $E_0$ of $...
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3+ electron wave function expansion coefficients for atoms

I was curious if anyone knows of any published coefficients for 3+ electron ground state wave function expansions. For example, I have recently come across a paper "Ground State of the Helium ...
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Ground state energy

I am trying to get familiar with the ground state energy of an operator. In my lecture we defined the ground state energy of a self-adjoint operator $H$ that is bounded from below as $$ E_0= \inf_{\...
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Why aren't zero energy states paired in supersymmetry?

In a general supersymmetric theory, given a supercharge $Q$ and Hamiltonian $H$, one generally has $$[Q,H]=0$$ One can then say that given an energy eigenstate $|E\rangle$ with energy $E\ne 0$, then $$...
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Ground state energy for fermions with different spin orientation [closed]

Electrons are subject to a harmonic potential in one dimension, described by one-particle Hamiltonian $$H = \frac{P^2}{2m}+\frac{1}{2}m\omega X^2,$$ where $P$ are the momentum operator, $X$, is the ...
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Is this quantum mechanical proof of the virial theorem general?

I have seen the following proof for the virial theorem in QM using the variational method. It goes like this: Suppose an exact eigenstate of the system is $\psi(\vec{r})$ and consider a variational ...
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Can classical spin-glass problems be solved exactly in a more efficient way than bringing them to the quantum level?

It is known that classical spin-glass problems are NP-hard, just like their quantum counterparts. Does this truly mean that finding their Ground-State is equally hard in both cases? The most ...
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Is there a physical example where there are $N$ gapped degenerate ground states preserving $U(1)$ symmetry?

I am looking for systems (like QFTs, lattice models etc) where there are $N$ ground states which are all degenerate, and all are gapped, and preserving $U(1)$ global symmetry. For example, in certain ...
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Energy Renormalization and Vacuum Diagrams

I have been reading the lecture notes of Coleman's course on QFT. When developing scattering theory with the use of a cutoff function, he mentions that, in order to ensure that the free vacuum ...
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Ground state degeneracy of $\hat{H}= \sum_{i=1}^{L-1} \mathbf{\hat{S}}_i\cdot \mathbf{\hat{S}}_{i+1}$, given a possible ground state

Consider $\hat{H}= \sum_{i=1}^{L-1} \mathbf{\hat{S}}_i\cdot \mathbf{\hat{S}}_{i+1}$ where $$ \mathbf{\hat{S}}_i\cdot \mathbf{\hat{S}}_{i+1} = 1_1\otimes\cdots\otimes 1_{i-1}\otimes \mathbf{\hat{S}}_i\...
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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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Why is the ground state important in condensed matter physics?

This might be a very trivial question, but in condensed matter or many body physics, often one is dealing with some Hamiltonian and main goal is to find, or describe the physics of, the ground state ...
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Thermodynamic Limit of Entanglement-Entropy like quantities

Suppose i have, in one spatial dimension, a unique ground state $\Omega$ of a local, gapped, translational invariant Hamiltonian. Denote by $\sigma_s$ the density matrix of $\Omega$ on lattice sites $\...
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If nitrogen has 7 electrons, how come the ground state is $2s^2 2p^3$?

If nitrogen has 7 electrons, how come the ground state is $2s^2 2p^3$? This would mean that there are only 5 electrons in the nitrogen atom.
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XXZ chain exact ground state energy

I would like to know the analytical expression of the ground state energy of the XXZ model, if such formula exists (probably from a Bethe Ansatz solution) and if it is valid in all parameter regimes.
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What does a negative energy eigenvalue mean?

If I have a Hamiltonian $H$ that only has kinetic energy and no potential energy, do the energy eigenvalues have to be non-negative? Could the ground state of Hamiltonian have negative eigenenergy? If ...
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Why do we consider the ground state as the lowest energy state?

According to Bohr's theory, electrons move around the atom in orbits having specific energy. When it absorbs energy, it gets excited to higher energy states. In H2 atom, ground state is -13.6 eV while ...
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