New Quantum Spin Perspective, Matter Waves and Uncertainty Principle

Introduction

Physical Significance of New Quantum Spin Perspective

Scale transition in Quantum Domain

Reduced Planck constant is not only constant but also a validation mechanism and mathematical tool of quantum domain [27,30]. Beyond reduced Planck constant (and Planck scale) all known physical laws are violated. Here it should be noted that Domain of quantum physics is quite huge. since the discovery of Planck’s distribution law for black body [27,30], this quantum hypothesis has been applied to almost every natural phenomenon and at every scale whether it is atomic scale, nuclear scale or quantum gravity scale (in short quantum scale)[27,30].

In general novel equation of Reduced Planck constant ℏ for whole quantum domain is written as [1]

$$\begin{array}{c}\hfill \hslash =\sqrt{k_{\beta }{T}_{scale}{l}_{scale}^2{m}_{scale}}\end{array}$$

(3)

since the magnitude of ℏ is constant at every scale of cosmos (atomic, nuclear and Planck), subscript scale is named accordingly. It is these three fundamental physical quantities such as temperature, length and mass which are well synchronized with each other in the above formula just to have quantum domain in the end, while Boltzmann constant $k_{\beta }$ is constant. It is auto-correct orauto-balance mechanism that is responsible for smooth scale transition of quantum domain in nature that holds constancy of ℏ.

The smooth transition of quantum geometry from Planck scale to nuclear scale and atomic scale can be explained using this new quantum spin perspective. Hence, the constancy of the reduced Planck constant ℏ can provide a way through which smooth transition of Planck scale to nuclear or atomic scale can be understood.

For instance, the value of ℏ for nuclear scale is,

$$\begin{array}{c}\hfill |\hslash |=\sqrt{k_{\beta }{T}_{nuc.}{r}_{nuc.}^2{m}_{nuc.}}=1.054\times {10}^{-34}J·s\end{array}$$

(4)

where, $m_{nuc.}=1.66\times {10}^{-27}kg$(atomic mass unit or $1amu$ or $1u$) , $T_{nuc.}=0.48\times {10}^{12}K$ - (from Equation (2)) and $r_{nuc.}=1\times {10}^{-15}m$ (one nuclear radius) for nuclear scale [28,38].

The value of ℏ for atomic scale is,

$$\begin{array}{c}\hfill |\hslash |=\sqrt{k_{\beta }{T}_{atm.}{r}_{atm.}^2{m}_{atm.}}=1.054\times {10}^{-34}J·s\end{array}$$

(5)

where, $m_{atm.}=9.109\times {10}^{-31}kg$ (mass of electron in atomic unit), $T_{atm.}=3.1\times {10}^{5}K$ - (from Equation (2)) and $r_{atm.}=5.291\times {10}^{-11}m$ (Bohr radius in atomic units) for atomic scale [28,38].

Here, one should note that just to verify constancy of reduced Planck constant ℏ for nuclear and atomic scale, numerical value of temperature can be obtained from eqn (2) which tells approximate value of temperature for nuclear scale and atomic scale.

Here, the values of temperature $T_{scale}$, mass $m_{scale}$, and length $l_{scale}$ within root change in the Equation (3) such a way that the reduced Planck constant ℏ remains constant. Due to this mechanism, the quantum geometry (or LQG) makes smooth transition from the Planck scale to the nuclear scale and atomic scale.

In other words, since $k_{\beta }$ is Boltzmann constant, eqn. (3) can also be written as

$$\begin{array}{c}\hfill \therefore \hslash =\sqrt{k_{\beta }}·\sqrt{T_{scale}{l}_{scale}^2{m}_{scale}}\end{array}$$

(6)

The term $\sqrt{T_{scale}{l}_{scale}^2{m}_{scale}}$ results into same magnitude for every scale in scale transition, so that ℏ remains invariant. The approximate value of this term is,

$$\begin{array}{c}\hfill \therefore |\sqrt{T_{scale}{l}_{scale}^2{m}_{scale}}|=2.84\times {10}^{-23}K·kg·m\end{array}$$

(7)

$$|\hslash |=\sqrt{k_{\beta }}·\sqrt{T_{scale}{l}_{scale}^2{m}_{scale}}$$

$$|\hslash |=3.7148\times {10}^{-12}·2.8366\times {10}^{-23}$$

$$\begin{array}{c}\hfill \therefore |\hslash |=1.053\times {10}^{-34}J·s\end{array}$$

(8)

Thus, collective physical quantity $\sqrt{T_{scale}{l}_{scale}^2{m}_{scale}}$ is important in scale transition and for constancy of ℏ.

When big bang [27,39,40,41] occurred, the temperature of universe was of the order of ${10}^{32}K$ at Planck scale. Afterwards, universe cooled off and various era of timeline of universe came into existence such as Planck, GUT, Inflation, quark, hadron, lepton, photons (radiation dominated era) and molecule, stars, galaxies era (matter dominated era). From big bang to galaxy era, temperature, length, and mass scale changed and various eras came to into existence gradually. It is auto-correct or auto-balance mechanism which maintains quantum domain in nature by maintaining constancy of reduced Planck constant and simultaneously changing temperature, mass and length scale of universe. Table 1 shows New Quantum Spin Perspective as Scale transition in Quantum Domain.

Table 1. New Quantum Spin Perspective as Scale transition in Quantum Domain

Table 1. New Quantum Spin Perspective as Scale transition in Quantum Domain

Sr. No.	scale	Temperature	Mass	Length
1	Planck	$T_P=1.416\times {10}^{32}K$	$m_P=2.176\times {10}^{-8}kg$	$l_P=1.616\times {10}^{-35}m$
2	Nuclear	$T_{nuc.}=0.48\times {10}^{12}K$	$m_{nuc.}=1.66\times {10}^{-27}kg$	$r_{nuc.}=1\times {10}^{-15}m$
3	Atomic	$T_{atm.}=3.1\times {10}^{5}K$	$m_{atm.}=9.109\times {10}^{-31}kg$	$r_{atm.}=5.291\times {10}^{-11}m$

Max Planck [30] defined ℏ as elementary quantum of action in his seminal paper. How new quantum spin perspective is directly connected with quantum of action is given below.

Canonical Conjugates and New Quantum Spin Perspective

De Broglie Hypothesis and New Quantum Spin Perspective

Introduction De Broglie Hypothesis

In 1905, Albert Einstein [31] firstly described the wave–particle duality of light (photon). In 1924, de Broglie [34] proposed that matter particle like electrons can also behave as wave similar to wave behaves as particle (photon - light quanta). In other words, matter particle can also have wave-particle duality and this duality has been observed experimentally [43]. such matter waves also exhibits wavelength $\lambda$ same as photon., i.e.,

$$E=hf=m{c}^2,E=pc,\lambda ={\displaystyle \frac{c}{f}}$$

Therefore,

$$\begin{array}{c}\hfill \lambda ={\displaystyle \frac{h}{p}}\end{array}$$

(11)

which is de Broglie wavelength. Both matter wave, Heisenberg’s uncertainty principle [35] play important role in the interpretation of quantum mechanics [42,43,45,46] that is one of the main topics of research in theoretical physics which arise because of wave-particle duality.

Non-localized harmonic wave is a type of wave that is extended over a volume of space and it is associated with the particles being confined to a small volume. One should note that de Broglie waves cannot be harmonic waves rather a combination of number of harmonic waves. With the aid of Fourier analysis, one can have combination of harmonic waves in the range $\Delta k$, (where k is wave number) will create a localized wave. The superposition of harmonic waves which are having small difference in frequency, cancels each other everywhere except a small region that will create a single localized wave which is known as wave packet and it spreads over a space $\Delta x$ [42,43].

The wave function of this wave packet is given by [42,43]

$$\begin{array}{c}\hfill \mathsf{\Psi }(x,t)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{+\infty }A\left(k\right)e^{i(kx-\omega t)}dk\end{array}$$

(12)

where k is wave number, $\omega$ is angular frequency, $A\left(k\right)$ is amplitude and ${\displaystyle \frac{1}{\sqrt{2\pi }}}$ is a factor. Fig. 1 shows superposition of harmonic waves and wave packet.

Figure 1. Superposition of harmonic waves and wave packet.

Figure 1. Superposition of harmonic waves and wave packet.

Heisenberg’s Uncertainty Principle and New Quantum Spin Perspective

Conclusions

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