Stationary states revisited
In recent posts, we have been discussing how external forces are represented in our proposal of local-realistic model for quantum mechanics. We have discussed some results with both constant and variable force fields, in particular the case of quantum harmonic oscillator. Such a scenario is instructive also because it introduces stationary states.
In standard QM, stationary states are defined as quantum states with observable that are independent of time. For a single particle, this means that the probability distribution of position, momentum, etc., is constant. In the standard picture, these states are found as the eigenvectors of the system's Hamiltonian (eigenstates).
The Hamiltonian of the quantum harmonic oscillator presents several eigenvectors, forming a family
=\frac{1}{\sqrt{2^n&space;n!}}&space;\cdot&space;\Omega^{1/4}&space;\cdot&space;e^{-\frac{\pi&space;\Omega&space;x^2}{2}}&space;\cdot&space;H_n(\sqrt{\pi&space;\Omega}x) "\psi_n(x)=\frac{1}{\sqrt{2^n n!}} \cdot \Omega^{1/4} \cdot e^{-\frac{\pi \Omega x^2}{2}} \cdot H_n(\sqrt{\pi \Omega}x)")
The Hamiltonian of the quantum harmonic oscillator presents several eigenvectors, forming a family
where n = 0,1,2,... and the functions Hn are the Hermite polynomials.
In order to check if the proposed model retrieves these stationary states, we prepare the particle ensemble to represent an eigenstate. We attribute a source probability
=\frac{1}{2^n&space;n!}&space;\cdot&space;\Omega^{1/2}&space;\cdot&space;e^{-\pi&space;\Omega&space;x^2}&space;\cdot&space;H_n^2(\sqrt{\pi&space;\Omega}x) "P_0(x)=\frac{1}{2^n n!} \cdot \Omega^{1/2} \cdot e^{-\pi \Omega x^2} \cdot H_n^2(\sqrt{\pi \Omega}x)")
over a sufficient number of possible source nodes x and a source phase, ε(x) = angle(ψn(x)). Then we use the accelerated numerical code presented in an earlier post to model an ensemble of similarly-prepared particles.
The resulting distribution of arrivals of Np = 5000 particles after Nt = 200 iterations, for values Ω = 0.001, n = 1, is shown in the figure below. The red curve reflects the initial (source) probability. As the figure clearly shows, the probability distribution has not changed with time.
If we now look at the probability distribution of the momentum, we can easily calculate the theoretical probability density by taking the Fourier transform of the initial position density using the properties of Hermite polynomials. However, as discussed in this post, the proposed model does not need this mathematical apparatus and evaluates the quantum momentum from the rules of motion.
We observe (see the figure below for Ω = 0.0001 and n = 1) a substantial coincidence between the calculated distribution (histogram with momentum bins of 0.001 and a total of Np = 5000 particles) with the theoretical distribution (in red). Other results for higher values of n are displayed in the 2017 ArXiv paper.
These momentum "modes" do not correspond to the energy levels that form the energy spectrum of the QHO and that standard QM evaluates as the eigenvalues of the Hamiltonian operator. For example, for n = 1, there is a single eigenvalue with energy (n+1/2)Ω/π = 4.775 10-5 (in lattice units). Howver, this value is precisely retrieved if one considers the probability distribution of the total energy
This distribution is shown in the figure below, together with the expected value of this quantity (4.768 10-5 for the simulation above), with coincides almost exactly with the theoretical value. Similar coincidences are found for other energy levels n, as it was shown in the Arxiv paper.
In summary, stationary states (standard QM: eigenvectors of the Hamiltonian) are found as those source preparations whose probability distribution for large times is the same as the initial one. In the 2017 ArXiV paper we have even presented a formula for them, which we reproduce here below.
With a stationary state preparation, while the momentum distribution (standard QM: Fourier transform) is retrieved as the usual large-time distribution of the expected quantum momentum for an ensemble of similarly-prepared particles, the energy levels (standard QM: eigenvalues of the Hamiltonian) are retrieved as the average values of the corresponding total energy distribution.
In order to check if the proposed model retrieves these stationary states, we prepare the particle ensemble to represent an eigenstate. We attribute a source probability
over a sufficient number of possible source nodes x and a source phase, ε(x) = angle(ψn(x)). Then we use the accelerated numerical code presented in an earlier post to model an ensemble of similarly-prepared particles.
The resulting distribution of arrivals of Np = 5000 particles after Nt = 200 iterations, for values Ω = 0.001, n = 1, is shown in the figure below. The red curve reflects the initial (source) probability. As the figure clearly shows, the probability distribution has not changed with time.
If we now look at the probability distribution of the momentum, we can easily calculate the theoretical probability density by taking the Fourier transform of the initial position density using the properties of Hermite polynomials. However, as discussed in this post, the proposed model does not need this mathematical apparatus and evaluates the quantum momentum from the rules of motion.
We observe (see the figure below for Ω = 0.0001 and n = 1) a substantial coincidence between the calculated distribution (histogram with momentum bins of 0.001 and a total of Np = 5000 particles) with the theoretical distribution (in red). Other results for higher values of n are displayed in the 2017 ArXiv paper.
These momentum "modes" do not correspond to the energy levels that form the energy spectrum of the QHO and that standard QM evaluates as the eigenvalues of the Hamiltonian operator. For example, for n = 1, there is a single eigenvalue with energy (n+1/2)Ω/π = 4.775 10-5 (in lattice units). Howver, this value is precisely retrieved if one considers the probability distribution of the total energy
In summary, stationary states (standard QM: eigenvectors of the Hamiltonian) are found as those source preparations whose probability distribution for large times is the same as the initial one. In the 2017 ArXiV paper we have even presented a formula for them, which we reproduce here below.
With a stationary state preparation, while the momentum distribution (standard QM: Fourier transform) is retrieved as the usual large-time distribution of the expected quantum momentum for an ensemble of similarly-prepared particles, the energy levels (standard QM: eigenvalues of the Hamiltonian) are retrieved as the average values of the corresponding total energy distribution.
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