Derivation of the Schrödinger equation II: Boltzmann’s entropy

Silva Filho, Olavo L. da; Ferreira, Marcello

doi:10.1590/1806-9126-RBEF-2024-0219

Olavo L. da Silva Filho

Universidade de Brasília, Instituto de Física, Centro Internacional de Física, Brasília, DF, Brasil.

https://orcid.org/0000-0001-8078-3065

Marcello Ferreira ^**Correspondence email address: marcellof@unb.br

Universidade de Brasília, Instituto de Física, Centro Internacional de Física, Brasília, DF, Brasil.

https://orcid.org/0000-0003-4945-3169

*Correspondence email address: marcellof@unb.br

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Figure 1
Formal connections already made (in color) and some connections scratched upon (in gray).

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Figure 2
Left column: The phase space probability density functions for n=0 and n=1. Right column: the average fluctuation profiles defined upon configuration space.

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Figure 3
Left column: The phase space probability density functions for n=2 and n=3. Right column: the average fluctuation profiles defined upon configuration space.

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Figure 4
Contours of the Hamiltonian and of the probability density function for the levels n=1, 2 and 3 of the harmonic oscillator problem. For the Hamiltonian, each contour represents a constant energy shell, while, for the probability density function, each contour represents a constant probability density.

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Figure 5
Behavior of the functions F_n(q, p) for n = 1 and n = 2. The images show the strong fluctuation behavior of the phase space density near the divergence points in the momentum fluctuations, while they satisfy Liouville’s equation far from the divergence points.

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Figure 6
New connections between formal results regarding quantization and the formalism of Quantum Mechanics. Bohmian Mechanics, in a new perspective, and a derivation based on Boltzmann’s entropy are now part of the set of results.

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Table 1
Comparison between the energy partition function and the momentum partition function as defined in the present work.

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Table 2
Phase space probability density functions for various values of the quantum number n for the harmonic oscillator problem.

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Table 3
Momentum fluctuations and energies for some states of the hydrogen atom problem given by the quantum numbers in the first column (ξ = cos θ).

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Table 4
Phase space probability density functions for some levels of the hydrogen atom (ξ = cos θ) and the functions are not normalized.

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Energy Partition Function	Momentum Partition Function
$Z_{e} = \sum_{r} e^{- β E_{r}}$	$Z = \int F (q, p; t) e^{\frac{i}{ℏ} p δ q} d p$
$〈E〉 = - \frac{\partial \ln Z_{e}}{\partial β}$	$〈p〉 = \lim_{δ q \to 0} - i ℏ \frac{\partial \ln Z}{\partial (δ q)}$
$〈E^{2}〉 = \frac{1}{Z_{e}} \frac{\partial^{2} Z_{e}}{\partial β^{2}}$	$〈p^{2}〉 = \lim_{δ q \to 0} - \frac{ℏ^{2}}{Z} \frac{\partial^{2} Z}{\partial {(δ q)}^{2}}$
$〈Δ E^{2}〉 = \frac{\partial^{2} \ln Z_{e}}{\partial β^{2}}$	$〈δ p^{2}〉 = \lim_{δ q \to 0} - ℏ^{2} \frac{\partial^{2} \ln Z}{\partial {(δ q)}^{2}}$

n	F_n(q, p)	E_n
0	$\frac{1}{π} e^{- β q^{2} - \frac{p^{2}}{ℏ^{2} β}}$	$\frac{1}{2}$
1	$\frac{2 {\|\sqrt{β} q\|}^{3}}{π ℏ \sqrt{β q^{2} + 1}} e^{- β q^{2} - \frac{p^{2} q^{2}}{ℏ^{2} (β q^{2} + 1)}}$	$\frac{3}{2}$
2	$\frac{{\|2 β q^{2} - 1\|}^{3}}{2 π ℏ \sqrt{2 β^{2} q^{4} + 4 β q^{2} + 5}} e^{- β q^{2} - \frac{p^{2} {(2 β q^{2} - 1)}^{2}}{ℏ^{2} β (2 β^{2} q^{4} + 4 β q^{2} + 5)}}$	$\frac{5}{2}$
3	$\frac{{\|\sqrt{β} q (2 β q^{2} - 3)\|}^{3}}{3 π ℏ \sqrt{4 β^{3} q^{6} + 9 β q^{2} + 9}} e^{- β q^{2} - \frac{p^{2} q^{2} {(2 β q^{2} - 3)}^{2}}{ℏ^{2} (4 β^{3} q^{6} + 9 β q^{2} + 9)}}$	$\frac{7}{2}$

(n, ℓ, m)	Momentum Fluctuations	E
(1,0,0)	$\frac{1}{r}$	$- \frac{1}{2}$
(2,0,0)	$\frac{1}{2} \frac{(8 - 5 r + r^{2})}{r {(r - 2)}^{2}}$	$- \frac{1}{8}$
(2,1,0)	$\frac{1}{2} \frac{r ξ^{2} + 1}{r^{2} ξ^{2}}$	$- \frac{1}{8}$
(2,1,±1)	$\frac{1}{2 r}$	$- \frac{1}{8}$
(3,0,0)	$\frac{1}{3} \frac{2187 - 1944 r + 648 r^{2} - 84 r^{3} + 4 r^{4}}{r {(27 - 18 r + 2 r^{2})}^{2}}$	$- \frac{1}{18}$
(3,1,0)	$\frac{1}{6} \frac{108 r ξ^{2} - 27 r^{2} ξ^{2} + 2 r^{3} ξ^{2} + 108 - 36 r + 3 r^{2}}{r^{2} {(r - 6)}^{2} ξ^{2}}$	$- \frac{1}{18}$
(3,1,±1)	$\frac{1}{6} \frac{(- 27 r + 108 + 2 r^{2})}{r {(r - 6)}^{2}}$	$- \frac{1}{18}$

(n, ℓ, m)	F (r, θ, ϕ, p_r, p_θ, p_ϕ; t)
(1,0,0)	$r^{3 / 2} e^{- 2 r} e^{- \frac{3}{2} [p_{r}^{2} + \frac{p_{θ}^{2}}{r^{2}} + \frac{p_{ϕ}^{2}}{r^{2} \sin^{2} θ}] r}$
(2,0,0)	$\frac{r^{3 / 2} {\|r - 2\|}^{5}}{{(8 - 5 r + r^{2})}^{3 / 2}} e^{- r} e^{- 3 [p_{r}^{2} + \frac{p_{θ}^{2}}{r^{2}} + \frac{p_{ϕ}^{2}}{r^{2} \sin^{2} θ}] \frac{r {(r - 2)}^{2}}{8−5 r + r^{2}}}$
(2,1,0)	$\frac{\| r ξ \|^{5} e^{- r} e^{- 3 [p_{r}^{2} + \frac{p_{θ}^{2}}{r^{2}} + \frac{p_{ϕ}^{2}}{r^{2} \sin^{2} θ}] \frac{ξ^{2} r^{2}}{ξ^{2} r + 1}}}{{(ξ^{2} r + 1)}^{3 / 2}}$
(2,1,1)	$r^{7 / 2} \sin^{2} θ e^{- r} e^{- 3 [p_{r}^{2} + \frac{p_{θ}^{2}}{r^{2}} + \frac{{(p_{ϕ} - 1)}^{2}}{r^{2} \sin^{2} θ}] r}$

\begin{array}{l} M_{2} (q; t) - p^{2} (q; t) ρ (q; t) \\ \begin{matrix} = & \int [p^{2} - p^{2} (q; t)] F (q, p; t) d p \\ = & \int {[p - p (q; t)]}^{2} F (q, p; t) d p . \end{matrix} \end{array}

ρ (q, δ q; t) = ρ (q + δ q; t) = ρ (q; t) \times e^{[\frac{1}{k_{B}} {(\frac{\partial S (q + δ q; t)}{\partial q})}_{δ q = 0} δ q + \frac{1}{2 k_{B}} {(\frac{\partial^{2} S (q + δ q; t)}{\partial q^{2}})}_{δ q = 0} δ q^{2}]},

〈δ q {(q; t)}^{2}〉 = \frac{\int_{- \infty}^{+ \infty} {(δ q)}^{2} \exp (β δ q - γ δ q^{2}) d (δ q)}{\int_{- \infty}^{+ \infty} \exp (β δ q - γ δ q^{2}) d (δ q)} - {(\frac{\int_{- \infty}^{+ \infty} δ q \exp (β δ q - γ δ q^{2}) d (δ q)}{\int_{- \infty}^{+ \infty} \exp (β δ q - γ δ q^{2}) d (δ q)})}^{2},

〈δ p {(q; t)}^{2}〉 ρ (q; t) = \lim_{δ q \to 0} [- ℏ^{2} \frac{\partial^{2} Z (q, δ q; t)}{\partial {(δ q)}^{2}} + ℏ^{2} {(\frac{\partial Z (q, δ q; t)}{\partial (δ q)})}^{2}],

Z (q, δ q; t) = \{R {(q, t)}^{2} + {(\frac{δ q}{2})}^{2} [R (q; t) \frac{\partial^{2} R}{\partial q^{2}} - {(\frac{\partial R}{\partial q})}^{2}]\} \exp (\frac{i δ q}{ℏ} \frac{\partial S}{\partial q}),

Z (q, δ q; t) = R {(q; t)}^{2} + \frac{i δ q}{ℏ} R {(q; t)}^{2} \frac{\partial S (q; t)}{\partial q} + \frac{{(δ q)}^{2}}{2} [\frac{1}{4} R {(q; t)}^{2} \frac{\partial^{2} \ln R {(q; t)}^{2}}{\partial q^{2}} - \frac{R {(q; t)}^{2}}{ℏ^{2}} {(\frac{\partial S (q, t)}{\partial q})}^{2}] .

\bar{Δ p^{2}} \cdot \bar{Δ x^{2}} = - \frac{ℏ^{2}}{4} \int {(\sqrt{ρ (x, t)} \frac{\partial \ln ρ (x, t)}{\partial x})}^{2} d x \cdot \int {[\sqrt{ρ (x, t)} (x - \bar{x})]}^{2} d x,

E_{n} = \int \{ρ_{n} (q; t) \{\frac{p_{n} {(q; t)}^{2}}{2 m} + V (q) + \frac{ℏ^{2}}{2 m} {(\frac{\partial R_{n} (q; t)}{\partial q})}^{2}\} d q

F (\vec{r}, \vec{p}) = \frac{ρ (\vec{r})}{{[2 π m k_{B} T (q; t)]}^{3 / 2}} \exp \{- \frac{{[\vec{p} - \bar{p (q; t)}]}^{2}}{2 m k_{B} T (q; t)}\},

F (\vec{r}, \vec{p}) = \frac{ρ (\vec{r})}{{[2 π 〈δ \vec{p} {(\vec{r})}^{2}〉 / 3]}^{3 / 2}} \exp \{- \frac{3 {[\vec{p} - m]}^{2}}{2 〈δ \vec{p} {(\vec{r})}^{2}〉}\} .

E [ρ, \nabla ρ] = \int \int H (\vec{q}, \vec{p}) F (\vec{q}, \vec{p}) d \vec{r} d \vec{p} = \int \{​ [\frac{p {(q, t)}^{2}}{2 m} + V (\vec{q})] ρ (\vec{q}) + \frac{ℏ^{2}}{8 m} \frac{{|\nabla ρ|}^{2}}{ρ} ​\} d \vec{q},

\begin{array}{l} \frac{δ E [ρ, \nabla ρ]}{δ ρ} \\ \begin{matrix} = & - \frac{ℏ^{2}}{4 m} \nabla \cdot (\frac{\nabla ρ}{ρ}) - \frac{ℏ^{2}}{8 m} \frac{{|\nabla ρ|}^{2}}{ρ^{2}} + \frac{p {(\vec{q}, t)}^{2}}{2 m} + V (\vec{q}) - λ \\ = & - \frac{ℏ^{2}}{4 m} [\frac{\nabla^{2} ρ}{ρ} - \frac{1}{2} \frac{{|\nabla ρ|}^{2}}{ρ^{2}}] + \frac{p {(\vec{q}, t)}^{2}}{2 m} + V (\vec{q}) = λ, \end{matrix} \end{array}

[1] *Correspondence email address: marcellof@unb.br

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