Ground state Definition and 275 Threads

Homework Statement While playing around with basic QM, I tried using the hamilton operator to find the kinetic energy of a hydrogen atom in its ground state. I assume the wave function ##\psi_1## is known. However, I of course ran into problems... 1) in my solution attempt below, I end up with...

Homework Statement An electron in hydrogen atom can be described with normalized wavefunction ##\psi (r,\vartheta ,\varphi )=(\frac{1}{64\pi r_B^3})^{\frac{1}{2}}exp(-\frac{r}{4r_B})##. Calculate the probability that the electron is in ground state of hydrogen atom. How much is the energy...

I've recently realized I have completely taken for granted that electrons simply tend to be found in their ground state. I want to understand fundamentally what is causing the electron to drop back to its ground state. It feels a force from the positively charged nucleus, but if it was given...

Shouldn't the integrating factor be ##exp(\frac{m\omega x}{\hbar})##? \frac{\partial <x|0>}{\partial x} + \frac{m\omega x}{\hbar} <x|0> = 0 This is in the form: \frac{\partial y}{\partial x} + P_{(x)} y = Q_{(x)} Where I.F. is ##exp (\int (P_{(x)} dx)##

I have confounded myself with the following observation. Take the standard expression for the lowest energy Dirac 4-spinor solution of the Dirac equation with a Coulomb potential (the H atom ground state). Plug this into the standard expression for the 4-vector current and use values for the...

If a Hamiltonian is unbounded from below, say the hydrogen atom where the Hamiltonian is -∞ at r=0, is there a way to tell if the ground state is bounded (e.g. hydrogen is -13.6 eV and not -∞ eV)? It seems if the potential is 1/r^2 or less, then the energy will be finite as: \int d^3 r (1/r^2)...

Homework Statement Calculate the magnetic moment of the ground state of \,_{20}^{39}Ca. Compare to the experimental value in table 1. Homework Equations Nuclear Shell Model knowledge The Attempt at a Solution Well firstly the magnetic moment of the nucleus similar to the spin is...

Homework Statement A particle is in the ground state of a simple harmonic oscillator, potential → V(x)=\frac{1}{2}mω^{2}x^{2} Imagine that you are in the ground state |0⟩ of the 1DSHO, and you operate on it with the momentum operator p, in terms of the a and a† operators. What is the...

I am reading the book "Lecture notes on Electron Correlation and Magnetism" by Patrik Fazekas. It says, "The ground state (of Heisenberg FM model) is not unique. We have just found that the system has the maximum value of the total spin Stot = LS. Sztot = LS state which is maximally...

Hi, can I know what are the ways to show that the ground state particles of a Bose Gas possesses 0 entropy?

Homework Statement Consider a molecule with an electric dipole moment d. The Hamiltonian of a molecule in the external electric field E is: \hat{H} = \frac{\hat{L^2}}{2I} - dE \cos{\theta}, where the polar angle \theta characterises the orientation of the molecule. (We have chosen the field...

The wave function of an electron in ground state hydrogen atom is non-zero at points far away from the nucleus. Interpreted as the probability of finding the electron, this is telling us that there is a finite probability of finding an electron far away from the nucleus even if it is at ground...

Why does Vanadium have a ground state electron configuration of 4s2 3d3 and not 3d5? In introductory chemistry and the latter arrangement seems to be lower in energy. Why is the former configuration the actual one?

Hi, I've been working through my lecture notes from last year and I encountered this example of Russel-Saunders term symbols. I'm confused by the value of L being given as 1 + 2 = 3. What I recalled was that L is defined for two electrons by the Clebsch-Gordan series L = l(1)+l(2), l(1) +...

Homework Statement The question is to find the total angular momentum of the following atoms in their ground state - Na (11 electrons), and Rb (37 electrons). That's all the info given. Homework Equations I have no idea - that's what I can't find! The Attempt at a Solution I've...

Is there an experimental verification of the radial probability density for the hydrogen ground state given in the introductory texts. See the following link as an example. Thank you in advance. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydrng.html#c1

Homework Statement . Hi, could someone look at the attachment & comment on whether I'm anywhere near getting the expectation value correct, please. In the grnd state; 1. terms such as AA†A†A, with lowering operator on RHS has zero expectation value, 2. terms such as AA†A†A† with uneven...

Homework Statement Calculate the expectation value for a harmonic oscillator in the ground state when operated on by the operator: $$AAAA\dagger A\dagger - AA\dagger A A\dagger + A\dagger A A A\dagger)$$ Homework Equations $$AA\dagger - A\dagger A = 1$$ I also know that an unequal number of...

I realize that at ground state of a harmonic oscillator the energy will be at zero. I'm assuming that the expectation value will also be at zero. Could someone confirm this & possibly explain just a little more. Thank you

Why the ground state HOMO level of a molecule with two opposite-spin electrons, is singlet while we know that a level with two opposite-spin electrons may be singlet(s=0,m=0) or triplet(s=1,m=0)?

Can somebody clear the fog in front of my eyes: how can a Hamiltonian describing free fermions have a ground state with non-trivial entanglement? My reasoning is: we can build the ground state in the independent electron approximation (which is now exact), hence it is a product state in the...

Hello PF members, I am trying to solve for helium atom ground state by using perturbation theory. It is very easy to calculate the first order correction of ground state energy but I have no idea for how to find its wave function. I have tried to use the first order correction formula for wave...

Is there some mathematical prove that for ground state in QM problems wave function doesn't have any zeros?

Hello Everybody, In page 86, in Peskin & Schroeders Introduction to QFT, the following expression is introduced to analyze \left | \Omega \right >; the ground state of the interacting theory: e^{-iHT} \left | 0 \right >. Where |0> is the ground state of the free theory and H is the...

Homework Statement Why does <0|\frac{1}{(2\pi)^3}∫ \hat{a}^{\dagger}(t,r) \hat{a}(t,r) d^{3} \textbf{k} |0> = \frac{1}{\pi^2}∫|β|^2 k^2 dk. Where \hat{a} and \hat{a}^{\dagger} and its conjugate are bogulobov transformations given by: \hat{a}(t,k) = \alpha(t)a(k) + β(t)b^{\dagger}(-k)...

This problem is 1.1b out of "Atomic Physics" by Budker, Kimball, and Demille. There are solutions in the book, but I am confused: I'm asked to find the ground state configuration of Phosphorus, which is has 3 P-state valence electrons. Following Hund's rule, we want to find a state with largest...

For a particle with a force acting on it whose potential is given by U(x) = g*lnx for x>1 and U(x) = ∞ for x = 1, how do I calculate the ground state energy of the particle? Supposedly, there is no need to use Schrodinger's equations for this question, which is why I have no idea how to start...

Homework Statement Homework Equations E_{1}=<ψ_{1}|V(r)|ψ_{1}> The Attempt at a Solution That is equal to the integral ∫ψVψd^3r So I'll just perform the integral, correct ? But r is not constant here right? So, I' ll keep it inside the integral? How should I continue? Please...

This is more of a math question I suppose, but its in the context of calculating the second order energy shift in the ground state energy for a non relativistic collection of electrons. We end up showing that the energy shift has a finite and divergent piece. The divergent bit is proportional...

Homework Statement What is the energy of H atom in the first excited state if the potential energy in the ground state is taken to be 0 ? Homework Equations Usually the energy of H atom in the ground state is -13.6eV and in the 1st excited state is -10.2eV E(n) = πme2/8ε2h2 Bohr's...

Homework Statement An electron is trapped in an infinite square-well potential of width 0.5 nm. If the electron is initially in the n=4 state, what are the various photon energies that can be emitted as the electron jumps to the ground state?Homework Equations ΔE=13.6(1/nf2-1/ni2)...

Are there any theories based upon Dark Matter (DM) being a precursor or sublayer or ground state to Observable Matter (OM)? Pre-Big Bang? Galaxy distribution linked to DM tendril intersections is consistent with DM providing a pre-existing structure. Any theories based upon DM interactions...

Homework Statement An atom of mass M decays from an excited state to the ground state with a change in mass of ΔM<<M. In the decay process, the atom releases a photon. Use the laws of energy and momentum to determine the energy of the photon, assuming the atom decays from rest. Homework...

Ground State Wave Equation: ψ0=(a/∏)(1/4)e(-ax2/2) Prove the Heisenberg Uncertainty principle ≥h(bar)/2 by way of expectation values. First I found <x>=0 because it was an odd function then I found <Px>=0 because it was an odd function Then <x2>=∫(a/∏)(1/2)x2e(-ax2)/2dx=1/2a by way of...

I hope this is a coherent question: Solving the Schrödinger equation for energy eigenvalues for a harmonic oscillator leads to the ground state of the energy for a particle being ω/2. What is the experimental evidence that this is, in fact, the lowest achievable energy and that zero energy is...

Homework Statement The deuteron ground state is made up of l = 0 and l = 2 states; a)Show this mixture cannot be an eigenstate of a central potential Hamiltonian b)Using first-order time independent perturbation theory, argue the potential must contain a term proportional to some combination of...

Hello Everyone! I have a question regarding a Quantum problem I am trying to solve in L. Brown's Quantum Field Theory book, Chapter 1, Problem 4.f. Homework Statement I have a question which asks me to compute [p][/n], i.e. the probability that the ground state (n=0) is brought to the...

In its ground state, an atom has no net electric dipole momentum ,but it can have a nonzero angular momentum. Is this a spontaneous symmetry breaking? why the ground state is not of zero angular momentum?

Say two girls spin a jump rope, of length ∏ meters, so that the only nodes are where the girls hold the rope, ψ= sin(x)exp(iω_1t) (actually there are no nodes at the girls hands as where they hold the rope goes in a smallish circle). Now instruct the girls how to shake the rope so as to most...

I have having difficulty understanding why the ground state configuration of helium has a wavefunction that is the product of two hydrogenic 1s wavefunctions and the spin eigenstate of a singlet state. Firstly, why is the space wavefcn hydrogenic, and secondly, why is the spin wavefcn a...

Homework Statement The problem is from Ashcroft&Mermin, Ch32, #2(a). (This is for self-study, not coursework.) The mean energy of a two-electron system with Hamiltonian \mathcal{H} = -\frac{\hbar^2}{2m}(∇_1^2 + ∇_2^2) + V(r_1, r_2) in the state ψ can be written (after an integration by...

Homework Statement A particle of mass (m) moves in the one-dimensional potential V(x) = V0 0 ≤ x ≤ a = ∞ otherwise Wave function of the particle is ψ(x,t) = C sin (\frac{x\pi}{a}exp[-iωt] Determine V0 Homework Equations Schrödinger's Equations...

Quantum field theory predicts a value for the cosmological constant that is 123 orders of magnitude larger than the observed value (if one assumes the Standard Model to be correct up to the Planck scale of 10^19 GeV)! To theoretically predict the value of the cosmological constant, one must, I...

Homework Statement In a hydrogenlike ion with atomic number Z, the energies of the allowed states are given by E(n) = (-13.6eV) (Z^2/n^2) What is the wavelength asociated with the transition between first excited state and ground state of hydrogen-like helium? (He+) Homework...

Homework Statement Consider a quantum mechanical particle moving in a potential V(x) = 1/2mω2x2. When this particle is in the state of lowest energy, A: it has zero energy B: is located at x = 0 C: has a vanishing wavefunction D: none of the above Homework Equations The...

Homework Statement For potassium in ground state configuration (Z=19) how would you expect the energy of the least tightly bound electron to compare with the energy of the electron in hydrogen excited to a state of the same principal quantum number n. Explain your answer. Homework...

Hello My question is about the ground state of vibrations for a solid. I'm working with graphite and have found out that for k=0 (The Gamma symmetry point), the vibrational modes can be decomposed into irreducible represenations in the following way Vibration = 2 * E1u + 2 * E2g + 2 * A2u...

Homework Statement Consider the Hydrogen atom in a magnetic field of 2T. If the atom is in its ground state (orbital angular momentum L=0): (a) Write down the magnetic moments of the proton and the spinning electron. [1]Homework Equations I am aware that the magnetic moment μ =...

Hi, In one of my advanced quantum mechanics classes, the instructor posed a problem, namely to show that the ground state of a one dimensional quantum harmonic oscillator is unique, without getting into differential equations. I know that the equation a\left|0\right\rangle = 0 when...

when proton and electron combine to form hydrogen atom(ground state) what is the source of radiation emitted