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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 ...
Quantization is straight-forward: [x(0), p(0)] = iħ, from which follows [x(t0), x(t1)] = iħsinω(t1 − t0) mω, which is all you actually need, since the brackets for v, p and f follow from this by algebra and differentiation. That's the simple harmonic oscillator in the Heisenberg Picture.
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?
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.
I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\\rangle$$ or, if solved in the spherical coordinate system: ...
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 ...
Absorb their inverses into the respective variables. This change of units is informally summarized by physicists as "Setting m = 1, ω = 1, ℏ = 1 m = 1, ω = 1, ℏ = 1 ", that is the natural units for the problem are used, so they are out of the way, and trivial to reintroduce, if needed, by elementary dimensional analysis. The hamiltonian.
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 ...