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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Confusion on Hamiltonian unbounded from below and Ostrogradsky Instability
This might be a silly question but I failed to get it. In Ostrogradsky Instability, we deduced that Lagrangian of higher-order derivatives leads to Hamiltonian linear to canonical momenta, and thus, ...
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Questions of lower boundness of Hamiltonians in quantum theories
In general spectral analysis, we have examples of unbounded from below hamiltonians with discrete spectrum. Is it okay to say that they have no sense in physical context, because for me it looks like ...
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Ground state energy of infinite Heisenberg XXX model with open or periodic boundary conditions equal?
I was wondering if there is anywhere a formal proof that shows that the ground state energy of a Heisenberg XXX model with periodic boundary conditions becomes equal to the ground state energy with ...
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Diagonalize a many-body Hamiltonian
Assume we start with a generic many-body Hamiltonian:
$$
H=\sum_{ij} t_{ij} a_i^\dagger a_j+\sum_{mnlk}U_{mnkl}a_{m}^{\dagger}a_{n}^{\dagger}a_la_k.
$$
Now if there is only the one-body part, which ...
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Expectation value of non-interacting groundstate
Assume that I have a tight binding model given in second quantized form as follows;
\begin{equation}
H = \sum_i f_i^{\dagger}f_i + t \sum_{i,j} f_i^{\dagger}f_j
\end{equation}
In real space, ...
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On the existence of ground state for a free particle
Given a free particle, the ground state of the system is the eigenstate with zero momentum. However, we also know that momentum operator does not have proper eigenstates, but rather it's spectrum is ...
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What is the ground state energy of $H = H_{0}-\mu N$?
Suppose we take $\mathscr{H} = L^{2}(\Lambda)$ our one-particle space, with box $\Lambda = [-L/2,L/2]^{d}\subset \mathbb{R}^{d}$ for some $L > 1$. Let $H_{0}$ denote the kinetic energy:
$$H_{0,1} = ...
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Problem about the ground state energy calculated with quantum Monte-Carlo
There are many quantum Monte-Carlo methods. Many of them can be used to calculate or estimate the ground state energy.
The problem is, is the estimated energy an upper bound of the true ground state ...
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Non-degeneracy of the ground states of quantum spin models
It is known that the ground state of some quantum spin models is non-degenerate. For example, the ground states of the quantum Ising model and the ferromagnetic Heisenberg model on the subspace of a ...
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Symmetry of the non-degenerate ground state
From Quantum field theory and condenced matter by Shankar, pp67, he mentioned that
In normal
problems, the symmetric state, or more generally the state with eigenvalue unity for the
symmetry ...
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Why does the variational method simplify in this way when $H$ Hermitian?
Ballentine (Quantum Mechanics: A Modern Development 2nd edition, page 290) writes the attached in his introduction of the variational method. My question is about his very last line: why does $H$ ...
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Lower bound on the ground state of 2D s-wave with a radial potential
Consider a Hamiltonian $H = -\Delta + V(r)$ in two dimensions, i.e. $L^2(\mathbb{R^2})$, where $V(r)$ is a smooth and bounded radial and non-positive potential. Let me define $\phi(r)$ as my wave ...
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Relation between ground state energy and the potential
I am given the following Hamiltonian:
$$H = \frac{p^2}{2m} + \lambda|x|^3$$
where $\lambda$ is a positive constant.
Is there a relation between the ground state energy of $H$ and $\lambda$ i.e. is ...
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Ground state interparticle distance in a system of interacting particles under harmonic confinement
Let us consider a system of two particles interacting via Coulomb force and being acted upon by a global harmonic oscillator potential. So, the system Hamiltonian is,
$$H=\sum_{i=1}^{2}\frac{p_i^2}{2m}...
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Does the state change, when the Hamiltonian changes?
Consider the Hamiltonian
\begin{equation}
H = \frac{p^{2}}{2m} + \frac{1}{2} m\omega^{2}x^{2} - \theta(t) qEx
\end{equation}
where $\theta(t)$ is $0$ for $t = 0$ and $1$ for $t > 0$. If at $t = ...
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Shape of 2D infinitely deep potential well with the lowest ground-state energy and fixed perimeter
In quantum mechanics context, consider an 2D infinitely deep potential well with fixed perimeter. That is, the shape could be arbitrarily changed on condition that the perimeter is constant. Now ...
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What is the physical meaning of the vacuum state in the BCS ground state?
the BCS ground state is:
$$|\psi_G\rangle= \prod_k(u_k+v_kc_{k,\uparrow}^+ c_{-k,\downarrow}^+)|\phi_0\rangle $$
now, i always believed that the vacuum state $|\phi_0\rangle$ was a state with no ...
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Ground eigenstate of the quantum harmonic oscillator with the interacting vacuum $| \Omega \rangle$
According to this video (at the timestamp), the professor writes down the derivation of the ground state of the 1D quantum harmonic oscillator.
Here is the screenshot from the slides of where she does ...
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Ground state of Bogoliubov quasi-particles (simpler version)
This is a simplified version of one of my previous questions. Let $b_1, b_2$ be two boson operators; their vacuum is denoted as $|0\rangle$, i.e. $b_i |0\rangle = 0$. We can make a canonical ...
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What is the quantum state of a particle sitting at rest at the minimum of its classical potential?
Suppose a particle, moving in a potential $V(x)$, is known to be at rest, at one of the local minima of its classical potential. Therefore, classically its total energy is at its local minimum. Given ...
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Quantum fluctuations are not real, but yet they can create observable phenomena?
I have read these
What are quantum fluctuations, really?
Quantum fluctuation
https://www.physicsforums.com/insights/vacuum-fluctuations-experimental-practice/
and found out that quantum fluctuations ...
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Why energy-positivity?
In any relativistic quantum field theory, we require that the spectrum is bounded from below. The typical explanation is that this condition enforces the stability of the theory. However, to me this ...
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Definition of the $n$-point Green's function and vacuum state in QFT
Reading different books, I've come upon two apparently different definitions of the $n$-point Green's function.
For simplicity, let's consider a real scalar field $\hat{\phi}(x)$ (in the Heisenberg ...
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Free energy of the BCS ground state in the weakly attractive limit
I have a rather involved question regarding the weakly attractive limit of the BCS ground state. We know for exampel from The book of Pitajevski and Stringari (Bose–Einstein Condensation and ...
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Does the ground state of interacting of interacting spinless/scalar bosons in a centrally symmetric potential have angular momentum 0? [duplicate]
Motivated by this question: If electrons were spinless/scalar bosons, would atomic ground state configurations necessarily have total orbital angular momentum zero?.
Can we make a group-theoretical ...
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If electrons were spinless bosons, would atomic ground state configurations necessarily have total orbital angular momentum zero?
My motivation for this question comes from the periodic table. There, the many-body ground state electronic configurations of certain atoms like boron or carbon can have a nonzero total orbital ...
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Paradox between Heisenberg and ground state? [closed]
According to Heisenberg principle if the electron is near the nuclei it is more likely to have a greater momentum $p$. So the energy must be greater. But the electron loses energy and has in fact ...
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Finite quantum harmonic oscillator and existence of a ground state
I am having some problems with a finite, shifted quantum harmonic oscillator potential, and the theorem that states:
Any attractive potential in one dimension must have at least one bound state.
Let'...
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How can a cosmological constant exist in flat Minkowski Space?
The ground state energy of a standard scalar field in Minkowski space diverges so we need normal ordering to get it to zero. This divergence is normally interpreted as coming from the cosmological ...
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How can we measure cosmological constant if we can't measure ground state energy?
From what I understand, we can only measure energy differences (see for example Peskin & Schroeder page 21, last paragraph), and therefore the ground state of a system cannot really be measured.
...
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Do different Hamiltonians result in different ground states?
I'm learning density functional theory. In the proof of Hohenberg–Kohn theorem I, we assume that different Hamiltonians result in different ground states. Is it true?
In general, for example, we can ...
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Is Hartree-Fock (HF) a ground state theory?
To calculate the orbitals in Hartree-Fock (HF) theory we imply the variations principle, so we try to find the wavefunction which minimises the energy $\left<\psi|H|\psi\right>$.
This ...
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Beta-decay of a Tritium [closed]
Calculate the probability of a Tritium beta decay into ground state of a Helium ion with perturbation theory.
What should I start with? I lack any ideas.
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Ground state of hydrogen atom, $e^{-\kappa r }$ or $e^{-\kappa r }/ r$
The ground state must be an $s$-wave state, so it depends only on the radius $r$. I cannot remember the exact form, but I know it must be one of the two.
Is there any simple way to determine which one ...
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Is the average position for the ground state of a 1D simple harmonic oscillator zero? [closed]
My textbook claims the average position of the $n$-th state of 1D simple harmonic oscillator (SHO) is zero, which means
$$
\def\bra #1{\langle #1 |}
\def\ket #1{| #1 \rangle}
\def\braket #1{\langle #1 ...
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Excited state at room temperature and near 0 Kelvin
What does the excited state of the electron of a hydrogen atom look like at room temperature? What does the almost ground state look like slightly above 0 Kelvin?
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Ground state of electrons in an atom
When we talk about the ground state of the electrons in the atom, do we mean its state at room temperature?
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Relevance of the Jahn-Teller theorem
Jahn and Teller stated in their paper that
All non-linear nuclear configurations are therefore unstable for an orbitally degenerate electronic state. Thus if we know of a polyatomic molecule that the ...
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Ground state of Bogoliubov quasi-particles
Consider a set of boson/fermion creation and annihilation operators satisfying the canonical (anti-)commutation rules (CCR/CAR):
$$
[a_i, a_j]_\eta
= [a^\dagger_i, a^\dagger_j]_\eta = 0,
\quad
[a_i, ...
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General remarks on the Hubbard model in the strong coupling limit
Some results are known for the Fermi-Hubbard model under certain assumptions. For instance, it is known that at half-filling and in the strong coupling limit, the Hubbard model reduces to a Heisenberg ...
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Correlation functions of XY quantum chain
I'm trying to understand the calculation of the correlation functions in the XY quantum chain performed by Lieb, Mattis and Schultz in the paper "Two Solvable Models of an antiferromagnetic chain&...
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Is there a relationship between the degeneracy of the Hamiltonian and the set of non-commuting operators that commute with the Hamiltonian?
I'd like to know if there is any relationship between the number of the degeneracy of the Hamiltonian and the set of non-commuting operators that commute with the Hamiltonian.
If I know a set of $n$ ...
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Morse index for height function potential on $S^2$ from its Hessian [closed]
I was reading David Tong's notes on Supersymmetric Quantum Mechanics. In the third chapter, page no. 86, he gives an example of ground state counting using Morse Theory over $S^2$. The potential $h$ ...
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Integral of $\exp(-r^2)/r$
I came across the integral
$$
\int_0^\infty\frac1r\exp(-2\lambda r^2)\mathrm dr
$$
in variational method for hydrogen method when choosing the test wavefuntion as
$$
\psi=\exp(-\lambda r^2)
$$
since ...
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Un-equal time correlation via non-interacting tight-binding Hamiltonian
Let's assume we have a model, which is initially defined by the tight-binding Hamiltonian with a random on-site energy $f_n$, as follows:
$$H^i=-J\sum_n^{L-1}\left(a_n^\dagger a_{n+1}+h.c\right)+\...
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Expectation value interaction term BCS ground state
Let's define the pair-creation and pair-annihilation operators:
$$
b_{\bf{k}} = c_{-\bf{k}\downarrow}c_{\bf{k}\uparrow},\\
b_{\bf{k}}^{\dagger} = c_{\bf{k}\uparrow}^{\dagger}c_{-\bf{k}\downarrow}^{\...
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What eigenstates of the Fock operator gives the Hartree Fock ground state?
In the Hartree Fock method with $N$ electrons, the Fock operator is a one-electron operator defined by
$$f := h + \sum_{n=1}^N J_n - K_n,$$
where $h = -\frac{1}{2} \nabla^2 + V_{el-nu}$ is the one-...
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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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