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A harmoinic oscillator in the Heisenberg picture. Considering the Hamiltonian of a harmonic oscillator. H = p2 2m + mω2x2 2, H = p 2 2 m + m ω 2 x 2 2, the time evolution of the Heisenberg picture position and momentum operators is given by. x˙ p˙ = i ℏ[H, x] = p m = i ℏ[H, p] = −mω2x, x ˙ = i ℏ [H, x] = p m p ˙ = i ℏ [H, p ...
What is the harmonic oscillator? There are at least two fundamental incarnations of "the" harmonic oscillator in physics: the classical harmonic oscillator and the quantum harmonic oscillator. Each of these is a mathematical thing that can be used to model part or all of certain physical systems in either an exact or approximate sense depending ...
A harmonic oscillator, whether it's bosonic or fermionic, is a single-particle state that can be occupied by noninteracting particles. A fermionic state can be occupied by one particle at most, while a bosonic state can be occupied by an unlimited number of particles.
4. In Sakurai the derivation of the propagator leads to the expression. Which it says leads to the expression for the propagator by using the formula 1 √1 − ζ2exp(− ξ2 − η2 + 2ξηζ (1 − ζ2)) = exp[− (ξ2 + η2)]∑ n = 0(ξn 2nn!)Hn(ξ)Hn(η). I’m wondering, what exactly is the reasoning for using this formula?
1. For one classical harmonic oscillator with Hamiltonian. H = p2 2m + mω2 2 x2. the density of states can be calculated as by calculating the number of states with Energy smaller than E: Γ(E) = area of ellipse h = E ℏω. and then by carrying out the derivative dΓ dE one obtains: g(E) = dΓ dE = 1 ℏω as the density of states.
Let us first calculate the Q factor for the damped oscillator. Here, the energy of the oscillator E(t) is time dependent (oscillating with decaying amplitude ∼ e − t / τ), so the natural definition of the Q factor would be Q = 2π E(t) E(t) − E(t + T) = ωd E(t) P (t). Here, T = 2π / ωd is the period and ωd = √ω20 − (1 / 2τ)2 is ...
I have a problem regarding a forced, damped harmonic oscillator, where I'm trying to find the resonance frequency. I have calculated the frequency for free oscillations as $$\\omega_{free}=\\sqrt{\\fr...
Inverted Harmonic oscillator. what are the energies of the inverted Harmonic oscillator? H =p2 −ω2x2 H = p 2 − ω 2 x 2. since the eigenfunctions of this operator do not belong to any L2(R) L 2 (R) space I believe that the spectrum will be continuous, anyway in spite of the inverted oscillator having a continuum spectrum are there discrete ...
For the Harmonic oscillator the Ehrenfest theorem is always "classical" if only in a trivial way (as in case of the eigenstates). However in general the Ehrenfest theorem reduces to the classical equation of motion only on such localized wavepackets that concentrate near the classical trajectory as $\hbar$ goes to zero.
where [a(k),a†(p)] = (2π)3δ3(p − k) and [a(k), a(p)] = 0. You see that this is the Hamiltonian for infinitely many harmonic oscillators, one in every point of momentum space. Since the energy levels for a harmonic oscillator are evenly separated for every k, you get the particle interpretation of the free field theory given that the state ...