Approximation of Relativistic Scalar Field Action to Non-Relativistic Scalar Field Action
Using a massive field, extracting the rest energy phase , using time and the speed of light limit
for Klein--Gordon-Lagrangian, restore Planck constant , speed of light , time
use
let
use
the term multiplied by will cancel out the mass term
the term multiplied by will cancel out the speed of light factor
the term multiplied by will become , vanishing in the limit
might not affect it?
Multiply the entire expression by to get
Non-relativistic scalar field Lagrangian, alias Schrodinger-Lagrangian [Schrodinger-Lagrangian]
Can add a gauge field electrostatic potential term to the action, i.e., change to
Non-relativistic scalar field equation, alias Schrodinger-equation [Schrodinger-equation]
For either relativistic or non-relativistic scalar fields, it's possible to use a time plane wave mixed static solution to obtain the eigenvalue equation
One can also add static magnetic potential, is changed to , with
Eaxmple [Schrodinger-eq-potential-example]
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Resonator (one or more) represents the electric potential of constant charge density in or spherical regions. . Opposite charges correspond to elliptical type, same charges correspond to hyperbolic type.
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Hydrogen atom model (single particle, static, electrostatic, 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 + charge concentrated at a point or spherical region, then outside ==> ==>
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.
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box/ball potential well/barrier
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constant electric field or constant magnetic field
Regarding the non-relativistic approximation limit
- Static energy phase extraction is not a gauge transformation.
- Can be generalized to ? Replace with any unit element. doesn't work? Or is used for hyperbolic mass KG rather than elliptical mass KG, in one-dimensional the equation 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
Similar to the case of relativistic scalar field, energy-momentum tensor [energy-momentum-tensor-Schrodinger]
Time
Space
Energy
Using the product rule + divergence term + zero at boundary, it becomes
If there is an electrostatic potential , it will still be , but it might become non-positive definite
Due to the zero divergence of the energy-momentum tensor, energy is conserved with respect to time ,
For non-relativistic scalar fields, the energy of the Schrodinger field is real, positive and time invariant
Rotational angular momentum is [angular-momentum-Schrodinger]
Gauge current of the Schrodinger field
Phase change and its ฮด change belong to variations with fixed boundaries near the solution, so
The spatial part of the current is similar to the relativistic case,
The time part of the current ( )
The result will be the quantity
The entire current, after dividing by , is [current-gauge-Schrodinger]
The time component of the Schrodinger field gauge current is real, and the spatial component is purely imaginary
Assume is an integrable quantity in
time-invariant by [conserved-spatial-integral-charge-Schrodinger]
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 "charge density"?
[motivation-of-quantization]
Most treatments of quantization assume
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Linear unitary evolution of field
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Possibly non-free fields, so plane waves cannot be used
If the non-relativistic approximation limit of the KG field is considered, these assumptions will be automatically obtained.
Then, the time component of the static gauge field acts as the electric potential. Both the harmonic oscillator potential (uniform Charge) (probably should not be interpreted this way) and the hydrogen atom potential (point charge) originate from simple charge densities.
Thus, the motivation problem becomes:
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Why use KG field?
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Why and how does it correspond to point particles?
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Why does the potential of a gauge field become the potential of a particle when the field is quantized into particles?
Classical correspondence refers to the expectation value version of the point particle Lagrange-equation (ref-15, p.116) (wiki:Ehrenfest_theorem), e.g.
there also new operator in the speed of expectation
Where the non-commutativity or is controlled by the very small Planck constant
Note the difference between classical energy and quantum energy, similar to the difference between mean and variance. For example, . We can consider the standard deviation . There is the uncertainty principle . The equality holds <==> harmonic-oscillator-ground-state
The Schrodinger eq is the non-relativistic limit of the KG eq, and Newton's equation is the non-relativistic limit of relativistic point particles. So, can it be proven that the KG eq also has a point particle limit? At this point, should the definition of "expectation" use the charge density of the KG, . However, the charge of the KG eq is not positive definite, making it even further from the meaning of classical particles. However, the energy of the KG eq is positive definite (even with electromagnetic potential). For the Dirac eq, the charge is positive definite but the energy is not positive definite.
How to make the expected value version of the point particle Lagrange-equation correspond to the Lagrange-equation of the field?
Other signs that the Schrodinger wave function corresponds to point particles:
- Feynman path integrals use the path statistics weighted by the point particle Lagrangian to calculate the propagator of the Schrodinger eq. Question proof that it satisfies Ehrenfest theorem and then satisfies Schrodinger eq
[quamtum-operator-motivation]
The derivation of the behavior of the Schrodinger wave function under Galileo transformation, in brief, assumes that in addition to the non-relativistic space-time coordinate transformation , it is also assumed that the wave function changes phase , the result will be (ref-16)
Question Simplified calculations and better motivated explanations for the phase transformation? Perhaps it can be related to the use of static energy phase when KG eq is approximated to Schrodinger eq. In short, neither Schrodinger eq nor electromagnetic field are purely Galileo transformation invariant.
Galileo boost gives the transformation of the wave function as , the ฮด action of this action is the operator
Time translation -> ฮด action , Hamiltonian
Spatial translation -> ฮด action , momentum operator
Rotation -> ฮด action , angular momentum operator
Phase -> ฮด action
For QM in space, the Lie bracket of boost and spatial translation , or equivalently
[motivation-of-eigenstate]
For the following discussion, why is it an infinite-dimensional generalization of a finite-dimensional quadratic form? Because the source of the Lagrangian is the finite-dimensional quadratic form
Mimicking the finite-dimensional case, (ref-3, p.143โ144. ref-23, p.218โ222) diagonalize one quadratic form with an orthonormal basis in a (positive definite) quadratic form space.
Using differentiation to find the extreme values or first-order stable values of the Hermitian operator
on , let (orthogonal in the sense of )
Another quadratic form on a finite-dimensional standard quadratic form corresponds to an infinite-dimensional case where the wave function is in some or Hilbert space, acting as a quadratic form space. The energy is another quadratic form.
Its first derivative forall ==> ==> exists
States with different eigenvalues are orthogonal
There exists an orthogonal basis for a finite-dimensional standard quadratic form, such that , and is in a form close to a standard quadratic form, i.e., diagonalizing . can be written as or . The corresponding infinite-dimensional case is that there exists an orthogonal basis of the Hilbert space such that the energy is diagonalized. However, in the case of Hilbert space, diagonalization of the energy expectation might be in the form of a countable sum.
However, sometimes a countable sum is not enough, and an integral form is required, and the orthogonal basis is not the space where the wave function resides, even if the coefficients of the wave function's orthogonal basis expansion might be in the space of the eigenvalues. (Question What is the specific construction method for the structure of the eigenvalue space?) An example is the existence of unbound states in the electrostatic hydrogen atom model.
Schrodinger eq evolution preserves the eigenvalue space
The energy is a quadratic form, and its second-order differential polynomial is
Divide the space into the eigenspace and its orthogonal complement . In the direction, the second-order differential is positive definite or negative definite, and the energy will increase or decrease. According to Lagrangian variational theory, the energy of the Schrodinger eq is a time-conserved quantity
are separated in the sense of limit, unless at . , so the evolution will be confined to the eigenspace
Specific solution for the Schrodinger eq evolution in the eigenvalue space. is a constant-coefficient linear ODE from for each spatial point , solution i.e., essentially static except for the phase factor which oscillates in time according to .