note-math

Approximation of Relativistic Scalar Field Action to Non-Relativistic Scalar Field Action

Using a massive field, extracting the rest energy phase $\text{exp}\left(-𝑚{𝑐}^2\frac{1}{\mathrm{\hslash }}𝑡\text{ i}\right)$, using time ${𝑥}_0=𝑐𝑡$ and the speed of light limit $\underset{𝑐\to \infty }{\text{lim}}$

for #link(<Klein--Gordon-Lagrangian>)[], restore Planck constant $ℎ$, speed of light $𝑐$, time ${𝑥}_0=𝑐𝑡$

$${\int }_{\mathrm{ℝ}}𝑑𝑐𝑡{\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\frac{1}{2}\left(\frac{1}{{𝑐}^2}\left({\partial }_{𝑡}{\mathrm{\varphi }}^{\ast }{\partial }_{𝑡}\mathrm{\varphi }\right)-{\partial }_{𝑥}{\mathrm{\varphi }}^{\ast }\cdot {\partial }_{𝑥}\mathrm{\varphi }+\frac{{𝑚}^2{𝑐}^2}{{ℎ}^2}{\mathrm{\varphi }}^{\ast }\mathrm{\varphi }\right)$$

use $\mathrm{\varphi }=\text{exp}\left(-𝑚{𝑐}^2\frac{1}{\mathrm{\hslash }}𝑡\text{ i}\right)𝜓$

let $𝜃=𝑚{𝑐}^2\frac{1}{\mathrm{\hslash }}\text{ i}$

use ${\partial }_{𝑡}\mathrm{\varphi }=𝜃{𝑒}^{𝜃𝑡}𝜓+{𝑒}^{𝜃𝑡}{\partial }_{𝑡}𝜓$

$$\begin{array}{cccc}{\partial }_{𝑡}{\mathrm{\varphi }}^{\ast }{\partial }_{𝑡}\mathrm{\varphi }& =& -\frac{{𝑚}^2{𝑐}^4}{{\mathrm{\hslash }}^2}{𝜓}^{\ast }𝜓& \text{ use }{\text{ i }}^2=-1\\ & & +\frac{𝑚{𝑐}^2}{\mathrm{\hslash }}\text{ i }\left({𝜓}^{\ast }{\partial }_{𝑡}𝜓-𝜓{\partial }_{𝑡}{𝜓}^{\ast }\right)\\ & & +{\partial }_{𝑡}{𝜓}^{\ast }{\partial }_{𝑡}𝜓\end{array}$$

the term $-\frac{{𝑚}^2{𝑐}^4}{{\mathrm{\hslash }}^2}{𝜓}^{\ast }𝜓$ multiplied by $\frac{1}{{𝑐}^2}$ will cancel out the mass term $\frac{{𝑚}^2{𝑐}^2}{{\mathrm{\hslash }}^2}{\mathrm{\varphi }}^{\ast }\mathrm{\varphi }$

the term $\frac{𝑚{𝑐}^2}{\mathrm{\hslash }}\text{ i }\left({𝜓}^{\ast }{\partial }_{𝑡}𝜓-𝜓{\partial }_{𝑡}{𝜓}^{\ast }\right)$ multiplied by $\frac{1}{{𝑐}^2}$ will cancel out the speed of light $𝑐$ factor

the term ${\partial }_{𝑡}{𝜓}^{\ast }{\partial }_{𝑡}𝜓$ multiplied by $\frac{1}{{𝑐}^2}$ will become $𝑂\left(\frac{1}{{𝑐}^2}\left({\partial }_{𝑡}{𝜓}^{\ast }{\partial }_{𝑡}𝜓\right)\right)$, vanishing in the limit $\underset{𝑐\to \infty }{\text{lim}}$

$𝑑{𝑥}^0=𝑑𝑐𝑡$ might not affect it?

Multiply the entire expression by $\frac{{\mathrm{\hslash }}^2}{𝑚}$ to get

Non-relativistic scalar field Lagrangian, alias Schrodinger-Lagrangian Schrodinger-Lagrangian_(tag)

$${\int }_{\mathrm{ℝ}}𝑑𝑡{\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\frac{1}{2}\left(\mathrm{\hslash }\text{ i }\left({𝜓}^{\ast }{\partial }_{𝑡}𝜓-𝜓{\partial }_{𝑡}{𝜓}^{\ast }\right)-\frac{{\mathrm{\hslash }}^2}{𝑚}{\partial }_{𝑥}{𝜓}^{\ast }\cdot {\partial }_{𝑥}𝜓\right)$$

Can add a gauge field electrostatic potential term to the action, i.e., change ${\partial }_{𝑡}$ to ${\partial }_{𝑡}+\frac{\text{ i }}{\mathrm{\hslash }}𝑉$

Non-relativistic scalar field equation, alias Schrodinger-equation Schrodinger-equation_(tag)

$$\text{i }\mathrm{\hslash }{\partial }_{𝑡}𝜓=-\frac{{\mathrm{\hslash }}^2}{2𝑚}∆𝜓+𝑉𝜓$$

For either relativistic or non-relativistic scalar fields, it's possible to use a time plane wave mixed static solution $\text{exp}\left(-\frac{1}{\mathrm{\hslash }}𝐸𝑡\text{ i}\right)𝜓\left(𝑥\right)$ to obtain the eigenvalue equation

$$-\frac{{\mathrm{\hslash }}^2}{2𝑚}∆𝜓+𝑉𝜓=𝐸𝜓$$

One can also add static magnetic potential, ${\partial }_{𝑥}$ is changed to $\left({\partial }_{𝑥}+\frac{\text{ i }}{\mathrm{\hslash }}𝐴\right)𝜓$, with

${\left({\partial }_{𝑥}+\frac{\text{ i }}{\mathrm{\hslash }}𝐴\right)}^{†}\left({\partial }_{𝑥}+\frac{\text{ i }}{ℎ}𝐴\right)𝜓=\left(∆+\frac{\text{ i }}{ℎ}\text{ div }𝐴+\frac{2\text{ i}}{\mathrm{\hslash }}𝐴\cdot {\partial }_{𝑥}-\frac{1}{{\mathrm{\hslash }}^2}|𝐴{|}^2\right)𝜓$

Schrodinger-eq-potential-example_(tag) Example

• $𝑉=𝑘{𝑟}^2$

  Resonator (one or more) represents the electric potential of constant charge density in ${\mathrm{ℝ}}^{𝑛}$ or spherical regions. $\text{div }\text{ grad }{𝑟}^2=\text{ const}$. Opposite charges correspond to elliptical type, same charges correspond to hyperbolic type.

• $𝑉=𝑘\frac{1}{𝑟}$

  ${\mathrm{ℝ}}^3$ Hydrogen atom model (single particle, static, zero spin) represents a spherically symmetric electric potential generated by a point charge or a spherically symmetric charge sphere.

  Using symmetry + Gauss's theorem

  Or using differential equations, spherical symmetry $𝑓\left(𝑥\right)=𝑓\left(\left|𝑥\right|\right)=𝑓\left(𝑟\right)$ + charge concentrated at a point or spherical region, then outside $∆𝑓\left(𝑟\right)=0$ ==> $2\frac{1}{𝑟}\frac{𝑑𝑓}{𝑑𝑟}+\frac{{𝑑}^2𝑓}{𝑑{𝑟}^2}=0$ ==> $𝑓\left(𝑟\right)=𝑘\frac{1}{𝑟}$

  Warning The Schrodinger equation has no analytical solution for atoms and molecules other than hydrogen atoms for the time being. I also don't know the existing multi-particle models and how effective their numerical calculations are.

• $𝑉=$ box/ball potential well/barrier

• constant electric field or constant magnetic field

Regarding the non-relativistic approximation limit

• Static energy phase extraction $\text{exp}\left(-𝑚{𝑐}^2\frac{1}{\mathrm{\hslash }}𝑡\text{ i}\right)$ is not a gauge transformation.
• Can $\mathrm{ℂ}$ be generalized to $\mathrm{ℍ},\mathrm{𝕆}$? Replace $\text{i}$ with any $\text{Im}\left(\mathrm{𝕂}\right)$ unit element. $\mathrm{ℝ}$ doesn't work? Or $\mathrm{ℝ}$ is used for hyperbolic mass KG rather than elliptical mass KG, in one-dimensional $\mathrm{ℝ}$ the equation ${𝑥}^2=-1$ cannot be solved)
• This method of non-relativistic approximation limit is coordinate-dependent. On curved manifolds, since multiple coordinates may be needed to cover the entire manifold, the definition problem of the non-relativistic approximation limit will be more difficult.

Symmetry and Conserved Current

let

$$𝐿\left(𝜓,{\partial }_{𝑡}𝜓,{\partial }_{𝑥}𝜓\right)=\frac{1}{2}\left(\mathrm{\hslash }\text{ i }\left({𝜓}^{\ast }{\partial }_{𝑡}𝜓-𝜓{\partial }_{𝑡}{𝜓}^{\ast }\right)-\frac{{\mathrm{\hslash }}^2}{𝑚}{\partial }_{𝑥}{𝜓}^{\ast }\cdot {\partial }_{𝑥}𝜓\right)$$

Similar to the case of relativistic scalar field, energy-momentum tensor energy-momentum-tensor-Schrodinger_(tag)

$$𝑇={𝑇}_{𝜈}^{𝜇}=\{\begin{array}{cc}\frac{\partial 𝐿}{\partial \left({\partial }_{𝜈}𝜓\right)}\cdot {\partial }_{𝜈}𝜓-𝐿& \text{ if }𝜇=𝜈\\ \frac{\partial 𝐿}{\partial \left({\partial }_{𝜇}𝜓\right)}\cdot {\partial }_{𝜈}𝜓& \text{ if }𝜇\ne 𝜈\end{array}$$

Time

$$\begin{array}{ccc}\frac{\partial 𝐿}{\partial \left({\partial }_{𝑡}\mathrm{\varphi }\right)}\cdot {\partial }_{𝜈}𝜓& =& \frac{1}{2}\mathrm{\hslash }\text{ i }\left({𝜓}^{\ast }{\partial }_{𝜈}𝜓-𝜓{\partial }_{𝜈}{𝜓}^{\ast }\right)\\ & =& \mathrm{\hslash }\text{ i }\text{Im}\left({𝜓}^{\ast }{\partial }_{𝜈}𝜓\right)\\ \end{array}$$

Space

$$\begin{array}{ccc}\frac{\partial 𝐿}{\partial \left({\partial }_{𝑥}\mathrm{\varphi }\right)}\cdot {\partial }_{𝜈}𝜓& =& -\frac{{\mathrm{\hslash }}^2}{2𝑚}\left({\partial }_{𝜈}{𝜓}^{\ast }{\partial }_{𝑥}𝜓+{\partial }_{𝑥}{𝜓}^{\ast }{\partial }_{𝜈}𝜓\right)\\ & =& -\frac{{\mathrm{\hslash }}^2}{2𝑚}\text{Re}\left({\partial }_{𝑥}{𝜓}^{\ast }{\partial }_{𝜈}𝜓\right)\end{array}$$

Energy

$$𝐸={\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\left({𝑇}_0^0\right)={\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\left(\frac{{\mathrm{\hslash }}^2}{2𝑚}|{\partial }_{𝑥}𝜓{|}^2\right)$$

Due to the zero divergence of the energy-momentum tensor, energy is conserved with respect to time $𝑡$, ${\partial }_{𝑡}𝐸=0$

For non-relativistic scalar fields, the energy of the Schrodinger field is real, positive and time invariant

Rotational angular momentum is angular-momentum-Schrodinger_(tag)

$${𝑀}_{𝑖𝑗}^{𝑘}=\left[{𝑥}_{𝑖},{𝑇}_{𝑗}^{𝑘}\right]$$

Gauge current of the Schrodinger field

Phase change ${𝑒}^{𝜃\left(𝑥\right)}\mathrm{\varphi }\left(𝑥\right)$ and its δ change $𝜃\mathrm{\varphi }$ belong to variations with fixed boundaries near the solution, so

The spatial part of the current is similar to the relativistic case, $\frac{{\mathrm{\hslash }}^2}{2𝑚}\left({𝜓}^{\ast }{\partial }_{𝑥}𝜓-𝜓{\partial }_{𝑥}{𝜓}^{\ast }\right)=\frac{{\mathrm{\hslash }}^2}{𝑚}\text{Im}\left({𝜓}^{\ast }{\partial }_{𝑥}𝜓\right)$

The time part of the current ($𝜃\left(𝑥\right)\in \text{Im}\left(\mathrm{ℂ}\right)$)

$$\mathrm{\hslash }\text{ Im }\left(\left(-𝜃{𝜓}^{\ast }\right){\partial }_{𝑡}𝜓+{𝜓}^{\ast }{\partial }_{𝑡}\left(𝜃𝜓\right)\right)=\mathrm{\hslash }\text{ Im }\left({𝜓}^{\ast }𝜓{\partial }_{𝑡}𝜃\right)$$

The result will be the quantity $\text{i }\mathrm{\hslash }{𝜓}^{\ast }𝜓$

The entire current, after dividing by $\text{i }\mathrm{\hslash }$, is $({𝜓}^{\ast }𝜓,-\frac{\text{i }\mathrm{\hslash }}{2𝑚}\left({𝜓}^{\ast }{\partial }_{𝑥}𝜓-𝜓{\partial }_{𝑥}{𝜓}^{\ast }\right)$ current-gauge-Schrodinger_(tag)

The time component of the Schrodinger field gauge current is real, and the spatial component is purely imaginary

Assume $𝑗$ is an integrable quantity in ${\mathrm{ℝ}}^3$

time-invariant ${\partial }_0{\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\left({𝑗}^0\right)=0$ by ${\partial }^{†}𝑇=0$ conserved-spatial-integral-charge-Schrodinger_(tag)

$$\begin{array}{ccc}{\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\left({𝑗}^0\right)& =& {\int }_{{\mathrm{ℝ}}^3}𝑑𝑥\left({𝜓}^{\ast }𝜓\right)\\ & =& {\int }_{{\mathrm{ℝ}}^3}𝑑𝑥|𝜓{|}^2\end{array}$$

The time component of the Schrodinger field current is positive and its spatial integral is time invariant

Should this quantity be "particle number density" or "probability density" or "electric charge density"?

Warning No one can currently solve atoms other than hydrogen atoms using the Schrodinger equation. What form of many-particle interaction model should be used is also not an obvious matter. No repeatable evidence or downstream applications

motivation-of-quantization_(tag)

For harmonic oscillator $𝑘{𝑟}^2$ and hydrogen atom $𝑘\frac{1}{𝑟}$, if we assume the phase of the wave function oscillates as ${𝑒}^{-\text{ i }𝐸𝑡}$ and the amplitude is static $𝜓\left(𝑥\right)$, then ${𝑒}^{-\text{ i }𝐸𝑡}𝜓\left(𝑥\right)$ satisfies the Schrodinger or Dirac eq <==> $𝜓\left(𝑥\right)$ satisfies the eigenvalue equation $\text{H }𝜓=𝐸𝜓$ for a Hermitian operator, and $𝐸$ is discrete, for bound states of elliptic harmonic oscillators and hydrogen atoms.