Pandamaran Interpretation of Quantum Mechanics: the Second Interpretation

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Pandamaran Interpretation of Quantum Mechanics: The Second Interpretation

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Pandamaran Interpretation of Quantum Mechanics: The Second Interpretation

Andrew Das Arulsamy Condensed Matter Group, Institute of Interdisciplinary Science, No. 24, level-4, Block C, Lorong Bahagia, Pandamaran, 42000 Port Klang, Selangor DE, Malaysia

e-mail: [email protected]

Abstract Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to obser- vations or reality. Consequently, interpretation of quantum mechanics should involve proper physical mathematics and physical logic, which is often not the case from the Copenhagen interpretation. Here, we shall revisit all the postulates of quantum mechanics with proper physics and physical logic and reconstruct them to establish the physically less-complex Pan- damaran interpretation of quantum mechanics.

Keywords: Foundations of Quantum Mechanics; Copenhagen interpretation; Pandamaran interpretation.

§1. Introduction

Due to unsettled issues within the foundations of quantum mechanics, we have three main options for scientists to choose from when asked about the position of a ‘quantum’ particle before measurement.[1] A quantum particle here usually means an electron or a photon, which can be detected as free particles. The ﬁrst is the realist position—After the measurement, it is found that the particle was at some point A. But quantum mechanics is incomplete because it clearly lacks the ability to predict that the particle was at point A before the measurement. This realist position was advocated by Einstein himself, but he was unable to defend it in an unambiguous manner where one such attempt is the well-known Einstein-Podolsky-Rosen (EPR) paradox.[2] The second position is known as the orthodox position—Before the measurement, the particle was not really anywhere such that quantum mechanics can only predict the probability distribution of the particle’s possible position. This probability distribution is unique in a sense that it implies the act of measurement forced the particle to take a stand and exposed its position at point A. Here, ‘taking a stand’ refers to the notion of wavefunction collapse (whatever the mechanism for this collapse is). This is the core Copenhagen-interpretation that was advocated, and enforced to be the truth by Bohr since 1925. The third one is called the agnostic position. This is also a valid position because where is the point in taking a side when quantum mechanics is unable to predict the position of that particle. Raman was an advocate of this position. This is the current situation with the foundations of quantum mechanics, and in the absence

typeset using PTPTEX.cls hVer.0.9i 2 Andrew Das Arulsamy of additional technical arguments, we are not entirely sure which position is viable, let alone the truth. The primary aim of this work is to expose the new rules con- tained in the Pandamaran interpretation systematically, including the proper and consistent physics needed to take a firm stand on only one of the positions listed above, which should be automatic. We begin by revisiting the primary notion that is related to the wave-particle duality, which is the core quantum mechanical idea embedded in the Copenhagen interpretation that is responsible for the Heisenberg uncertainty principle. However, we shall not revisit the well-known Copenhagen postulates one-by-one, which can be obtained from Refs.,[1],[3],[4] instead, we shall re-evaluate and completely reconstruct the Copenhagen postulates with additional physics. The revised and the newly constructed postulates shall make up the Pan- damaran interpretation with consistent and unambiguous physics. 1.1. Rationale for the Second Interpretation Physically, the Heisenberg uncertainty principle is a consequence of wave-particle duality. Here, we should note that the Heisenberg uncertainty only applies for r ≤ ∆r, or when the observation scale, r is within the probability distribution of a quantum particle, ∆r. Therefore, if r>∆r, then the Heisenberg uncertainty is inapplicable because we can identify the position of the particle or the position of the wave packet or the position of the particle’s or wave packet’s distribution function. Here, we elaborate and apply this Heisenberg uncertainty principle and describe how it is actually related to one’s ability to know both the momentum and the position of a quantum particle. However, unlike the momentum–position Heisenberg uncertainty discussed above and elsewhere, we cannot determine or predict all the spin components of a quantum particle for all or any scales or range of observation, either for r ≤ ∆r or for r>∆r. For example, r is the range of observation or the region of observation that is defined within l and r. In particular, l ⊙ r implies ⊙ is an atomic hydrogen with its outer circle (with radius r) representing an electron wave packet surrounding the proton, · in the center. In this case, ∆r = πr2, while r denotes the region between r left and right. Therefore, for r>∆ , we can know the position of the particle, which is defined by the Bohr radius, but the position is not known at a given Cartesian coordinate (x,y,z) because such a point-like position does not exist for any electron. The above new interpretation of Heisenberg uncertainty principle, which limits its validity, belongs to the Pandamaran interpretation that shall be exposed in the following section. But first, we shall observe the generalized Heisenberg uncertainty relation, which is fully valid for Copenhagen interpretation, but only valid in a limited sense in Pandamaran interpretation. The generalized Heisenberg uncertainty relation is given by,[1] ~ σ σ ≥ , (1.1) A B 2 2 2 where σA and σB are the statistical standard deviations for the observables, A and ~ B, respectively. Here, σAσB is positive definite by definition and therefore, − 2 is automatically excluded. Pandamaran Interpretation of Quantum Mechanics 3

The Heisenberg uncertainty principle means that there is an Heisenberg uncertainty relation for every pair of non-commuting operators, such that their respective pair of observables are known as incompatible observables. Such a pair of incompatible observables or incompatible eigenvalues do not have a common or shared eigenfunctions. For example, the complete set of eigenfunctions for the position operator,x ˆ is not the same as the complete set of eigenfunctions for the momentum operator,p ˆ. Note this, we will not always label all the operators with a hat. In contrast, the observables for commuting operators permit the complete set of eigen- 2 functions for L to be the same as for Lz where their eigenfunctions are denoted by ml 2 Yl . In other words, L and Lz have the complete set of common and simultaneous eigenfunctions, while such a situation cannot occur for x and p because x and p are incompatible observables. We have to wonder why incompatible observables have to mean that we cannot observe or measure them simultaneously? In particular, if we could measure the position (x) of an electron accurately, then it is not possible to measure its momentum (p) as accurately as x, and vice versa. The proper answer to this question is only given in the so-called Pandamaran interpretation. Copenhagen interpretation assumes the answer to be due to the notion known as the wavefunction collapse. For example, according to Copenhagen interpretation, once the measurement on the position of an electron is made, then the act of measurement collapses the said wavefunction to a narrow spike and the information about its momentum is lost forever. If we happen to measure the momentum first, then the wavefunction collapses into a proper wave with a well-defined wavelength, and the information about the electron’s position is lost forever. The reason why the above incompatibility is not observed for compatible observables is because the collapsed wavefunction for compatible observables carry the information for both observables. For example, if we measure L2 of an electron in an eigenstate, we can still measure Lz for another electron in the same eigenstate because both electrons shared the same eigenstate, and both operators shared the same ml eigenfunction, Yl . For compatible observables, the wavefunction collapse mechanism remains the same. This is a physically false statement because we can do the same for x (measure x for one electron in a eigenstate) and p (measure p for another electron in the same eigenstate) even if the wavefunction-collapse mechanism for x and p differs. For example, can we measure both ~2l(l+1), the eigenvalue for L2 of an electron in an eigenstate, and measure again the eigenvalue for Lz, ~ml for another electron in the same eigenstate? such that we do not measure one eigenvalue and calculate the other? If this is possible, then we can definitely do the same for x and p. In 2 2 particular, we just need to make these switches L −→ xˆ, Lz −→ pˆ, ~ l(l + 1) −→ x and ~ml −→ p in the above sentence. There is another possibility for us to consider in our attempt to save the Copen- hagen’s wavefunction-collapse scenario. Can we measure both of these compatible 2 eigenvalues, ~ l(l + 1) and ~ml simultaneously from the same electron in an eigen- 2 state? The answer is definitely yes because both ~ l(l+1) and ~ml are automatically determined if we could measure l alone. This does not happen for incompatible ob- 4 Andrew Das Arulsamy servables, x and p. However, ml = 0, ±l and the orbital angular momentum state (0, ±l) is not precisely known and is still subject to Heisenberg uncertainty principle because Lx,y,z do not commute with Lz,x,y. Historically, the position and momentum of a quantum particle are the parameters exploited to expose the Heisenberg uncertainty principle. We shall do the same here because these parameters shall also expose the limitation of this uncertainty principle. Let us calculate the statistical variance of x and p for the ground state electron of an atomic hydrogen. In this case, instead of x, we need r and we first evaluate, 3a 9a2 hψ |r|ψ i = B , hri2 = B , (1.2) 100 100 2 4 2 2 12aB where ψ100 is the ground state wavefunction for atomic hydrogen. Next, hr i = , 2 2 4 2 ~ 2 ~ hpi = − 2 , hp i = − 2 . We can finally obtain (from the standard variance formula), aB aB 2 2 3aB 2 √3 σr = 4 and σp = 0, which lead us to, σrσp = 2 aB · 0 = 0, and this does not mean the Heisenberg uncertainty principle has been violated. Here, σrσp = 0 means that the momentum is known precisely with zero uncertainty, and therefore, the uncertainty is maximally activated for the ground state electron’s position, which is indeed the case here. In particular, we only know the 3 radius, r = ix + jy + kz = 2 aB. Hence, there is an infinite amount of uncertainty to locate the electron at a given point, (x,y,z) on an atomic hydrogen Bohr sphere where aB is the Bohr radius, which is not a Cartesian coordinate. Rightly so, the precisely known momentum is not px,y,z, but rather pr. This hints to the fact the Heisenberg uncertainty principle is valid, is uniquely quantum mechanical effect and it has got nothing to do with uncertainties arising in classical mechanics. We shall make use of these concepts properly to construct the second interpretation of quantum mechanics. In classical mechanics, the uncertainty of the particles motion or its position arises entirely from the unknown microscopic physics, and it is just a matter of finding and working out the hidden or missing details such that there is no such thing as non-observability. What is uncertain in classical physics is the unknown physics that are in play, that can always be found and fine-tuned. As a consequence, the uncertainty in classical physics is entirely due to experimental errors, provided that all the physical forces have been properly accounted for and there is no uncertainty in the constructed classical theory itself. 1.2. Summary Our main findings derived from the Pandamaran interpretation that are exposed in the following section can be briefly elaborated as follow. The mathematics of quantum mechanics is argued to have direct correspondence with physical reality even though the wavefunction by itself is not a real physical entity because it cannot be made to realistically represent any one particular quantum particle. However, the wavefunction can still be regarded as ontological with respect to the physical variables attached to, or carried by the wavefunction. The variables that are attached to the Pandamaran Interpretation of Quantum Mechanics 5

Hamilton operator (as in the Schrödinger equation) and the wavefunction correspond to real physical observables. These variables can be made to be complete without requiring any (local or non-local) hidden parameters as proposed in Refs.[5]–[7] In this second interpretation, the EPR paradox[2] is invalid because it has been proven to be a false alarm[8] where the issue of entanglement is only true for as long as there is an interaction between the entangled quantities, namely, the spins. If the separation between the entangled spins is made to be large with negligible interaction, then their correlation can be invalidated with a properly set-up experiment.[8] The newly constructed postulates within the Pandamaran interpretation support the realist’s interpretation, and these postulates revisit and reconstruct the Copen- hagen interpretation such that quantum mechanics is shown to be not, and cannot be fundamentally stochastic,[9], [10] nor do we need these alternatives—consistent histories[11] and wavefunction collapse.[12] Apart from that, the Pandamaran interpretation also does not rely on the information-based interpretation,[13] quantum Bayesianism,[14], [15] Bohm’s pilot-wave theory,[16] Everett’s many-world interpretation[17] and time-symmetric theories.[18], [19] There are several unique features found within the Pandamaran interpretation, namely, quantum mechanics is shown to be deterministic macroscopically, but not microscopically. Quantum mechanics is microscopically not deterministic due to limited validity of the Heisenberg uncertainty principle as discussed above. For example, the position of a quantum particle is deterministic macroscopically (i.e., for r>∆r), but not so microscopically when r<∆r due to Heisenberg uncertainty where r denotes the range or size of observation. Here, r should be larger than the size of a quantum particle (defined by the wavefunction and the Hamilton operator) or the wavepacket (for massless particles). This means that all the relevant variables can be found within the properly constructed (or derived) wavefunction from the Hamilton operator without the need to invoke any hidden variables (locally or non- locally or without requiring additional variables from the future nor from the past). Instead of the position, if one were to theoretically determine the spin, then this is not possible for all range of observation. Hence, quantum mechanics is not deterministic for the variable spin. Consequently, the quantum mechanical variables such as the position, wave-property, orbital and the spin angular momenta and the energy at the microscopic level are actually critical variables. This means that, the said variables for quantum particles are simply the critical variables that are defined at the critical point that can be transformed with certain and sufficient perturbation or disturbance (see Pandamaran Postulate 9). Given this background, it is easy to deduce that quantum mechanics can be deterministic conditionally, which means, determinism is subject to whether an observable is observable in both classical and quantum physics. Next, the wavefunction is not ontic but the variables defined by the wavefunction are, primarily because the variables found in the wavefunction and in the Hamilton operator are ontic even though the wavefunction itself may not be ontic. Note this, the electronic wavefunction for the ground-state atomic hydrogen is ontic. Different sets of observables of a physical system may need different sets of wavefunctions such that a single physical system may need different sets of wavefunctions to fully capture that system. 6 Andrew Das Arulsamy

In such cases, the variables in all of these wavefunctions are also ontic though the wavefunctions themselves are not, which is usually the case for realistic physical systems. Apart from these deductions, Pandamaran interpretation also demands unique history and the validity of the principle of locality, but denies the notion of wavefunction collapse, the existence of hidden variables of any kind and the influence of observers. Note this, the only observer’s role is to conduct the experiment and whoever (including robots that are activated remotely) can repeat the experiment to get the identical result (subject to experimental errors). In addition, the measured observables are definite if such observables are not subject to Heisenberg uncertainty principle. For example, for r>∆r the position of a ground state electron in an atomic hydrogen is definite, defined by the Bohr radius, but not for r<∆r. Whereas, the electron’s spin is never definite, either before or after the measurement because the spin is subject to the Heisenberg uncertainty principle for all range of observation. Finally, we can argue that a proper wavefunction can be constructed for our universe, which is indeed possible because the wavefunction need not be ontic nor unique. Thus, one could attempt to write down a physically reasonable wavefunction or as many wavefunctions as one possi- bly need containing all the relevant observables for the universe at a particular point of time, for as long as the introduced variables are quantum mechanically acceptable. Here, quantum mechanically acceptable implies even though constructing one or many wavefunctions for the Moon [a classical (non-quantum) object] is permitted in physics, but it is physically meaningless.

§2. Pandamaran Interpretation

We stress that the Pandamaran interpretation is complete because it incorpo- rates both classical and quantum mechanics consistently, which is another unique fea- ture that is absent in the Copenhagen interpretation. The Pandamaran interpretation for classical physics, with or without relativity have been exposed in Refs.[20]–[22] The core problem resides in the interpretation of wave-particle duality, massive and massless particles, and observability and non-observability in classical physics. Let us now provide further details on this wave-particle duality so as to properly expose the limitation to the Heisenberg uncertainty principle. The atomic hydrogen radial distribution functions for n = 1 and l = 0 is given by,[4]

3 1 2 r R1s = R10 = 2 exp − , (2.1) aB aB where these equations give the radial electron distributions for R1s. We can calculate the probability of ﬁnding the ground state electron (n = 1, l = 0) at r/a = 1 and again at r/a = 4 using the suitable spherical harmonics. For example,

2 2 2 2 2 r r a[R1s] r = a[R10] r = 4 exp − 2 = 0.54 , (2.2) a a r a =1

Pandamaran Interpretation of Quantum Mechanics 7

= 0.02 . (2.3) r a =4

Apparently, 0.54 and 0.02 are the probabilities of finding the electron at radii, r/a = 1 and r/a = 4, respectively, for the ground state electron (n = 1, l = 0). Now comes the interesting part, the interpretation. What do we make of these probabilities? Does this probability for each r/a means that we could also make the measurement for each r/a as in a Cartesian coordinate, (x,y,z)? The answer is definitely NO because each r/a refers to a radius, not a point in (x,y,z). As a matter of fact, even though we can calculate the probability of an electron for a given point-like position, r/a from the radial distribution function, but we cannot measure the position confined to a point, r/a. What we can measure is the whole radial distribution function curve (for a given radius r) in one go, not as individual r/a points as in Cartesian coordinate points, (x,y,z). This whole-curve measurement (for a given radius) ‘in one go’ that represents the electron’s position as a radial function would reproduce the probability distribution of that electron, depending on the electron’s orbital. Hence, our new interpretation here is that the electron’s charge, e and mass, mel are distributed in the shape of the probability distribution curve such that the mass and charge of this ground state or excited-state electron is never ‘fractionalized’ nor can it be broken into smaller pieces or parts. This means that the electron has a certain size, and its shape can change due to interaction with its surrounding such that the electron’s mass mass is distributed according to the electron’s size and shape. The size and shape of the electron is captured by the probability distribution function. Therefore, the electron remains as an indivisible fundamental particle and the group velocity of an electron, that is bound to a given principal quantum number, n cannot be defined if r ≤ ∆r due to its relation with mass distribution according to the probability distribution curve defined by the wavefunction (see Pandamaran Postulate 5). However, the charge (not the mass) is the one that gives rise to wave-like property for an electron with Pancharatnam phase velocity that is adjusted upon interaction as we have learned earlier.[23] The group velocity of an electron bound to a given principal quantum number, n cannot be defined if r/a ≤ ∆r due to its relation with mass distribution according to the probability distribution curve or function. The phrase, ‘cannot be defined’ here and elsewhere does not mean ‘do not exist’ or ‘does not exist’. Here, r/a ≤ ∆r refers to the range of observation (r/a) with respect to the size of the electron’s distribution function (∆r). Thus, the electron cannot be observed if r/a ≤ ∆r because the electron is never a point-like particle, but is defined by the distribution function. Note this, we do not know (and we can only guess) the physics that is in play within or inside an electron, and this particular ‘hidden’ or ‘unknown’ physics also stays unknown for all other elementary quantum particles like quarks and photons. I have been careful with these two words, ‘hidden’ and ‘unknown’ because I have to admit that I do not even know what I do not know about such physics. Anyway, even though the mass and charge of this ground state or the excited- state electron is distributed according to the probability distribution function, but 8 Andrew Das Arulsamy this does not imply, in any way, that the electron’s charge or mass can be broken into smaller or fractional parts such that e′ < e or mel′ < mel. Hence, in view of this new Pandamaran interpretation, we have a situation that the Born’s statistical interpretation cannot be entirely correct and needs to be replaced with the one that is compatible with Pandamaran interpretation. Before revising the Born’s statistical interpretation, let us understand what it says first. The Born’s original interpretation states that the position of an electron can only be measured for each r/a with a probability, and therefore, we cannot predict the electron’s position, r/a with certainty. This uncertainty is due to Heisenberg’s uncertainty principle. There are two physical problems with Born’s interpretation. The first problem is related to the assumption that the electron can be confined within a point-like position, r/a, and the second problem is that even though the position of the electron cannot be determined with certainty due to uncertainty principle, but this uncertainty is assumed to be true for all range of observation, namely, for 0 < r/a < ∞. Both of these assumptions are problematic that are derived from Born’s own interpretation, which are false On the other hand, the Pandamaran interpretation states that the mass and charge of an electron is distributed following the probability distribution function such that the point-like position of an electron, r/a cannot be defined properly for all r/a ≤ ∆r and within this range, Heisenberg’s uncertainty principle is valid where 2 2 2 2 ∆r is defined as the range where a[Rnl] r 6= 0, while a[Rnl] r = 0 for r/a > ∆r. Pandamaran interpretation implies that the said uncertainty principle is inapplicable for r/a > ∆r. For an electron in the free space or vacuum, ∆r refers to the size of the electron’s distribution function. The electron’s path that was detected in a cathode-ray tube (CRT) or in the bubble-chamber experiment,[24], [25] respectively, are possible simply because r/a > ∆r. In other words, the range of measurement is much larger than that of the electron’s size or the space occupied by the ‘shape- shifting’ electron. This shape-shifting refers to different orbitals for an electron in an atomic hydrogen, which is true for all electrons, regardless of whether the electron is bounded or free. Therefore, we should be able to identify the fact that the electron is correctly represented by a probability-distribution-shifting (or shape-shifting) function known as the wavefunction that changes due to wavefunction transformation. For example, the wavefunction transformation is needed when the electron in an atomic hydrogen is excited from the ground state (ψn=1,l=0) to an excited state (ψn=2,l=1). We should be able to deduce that the Pandamaran interpretation also automatically rules out the existence of Feynman’s nonclassical paths. We shall elaborate on this matter further later. Anyway, if ∆r is the size of the crystal, then r ≤ ∆r means our range of observation is smaller than the size of the crystal and therefore, the electron’s position is subject to Heisenberg uncertainty principle. Of course the situation is infinitely complex to determine the position of any single electron in a crystal because there are at least of the order of 1023 interacting valence electrons in a crystal. Whereas, for larger observation range, r>∆r, we are certain that the electrons can always be found within the crystal, while the probability to find any electron beyond the Pandamaran Interpretation of Quantum Mechanics 9 surface of the crystal is zero. Let us now begin constructing the Pandamaran postulates as listed below. We first recall the first Copenhagen Postulate and reconstruct it as given below. Pandamaran Postulate 1: The state function, Ψ(r,t), which is a function of time and space coordinates, is also known as the wavefunction. How- ever, Ψ(r,t) for massive and charged quantum particles does contain all the observable or measurable information about a system except for the particle’s mass, charge and the interactions in that system. We also de- mand that Ψ(r,t) to be single-valued (uniquely determines the eigenvalue), continuous and quadratically integrable. For continuum states, quadratic integrability should be excluded. The Hamilton operator contain the remaining information about charge, mass and interactions. For massless and chargeless photons however, we cannot invoke the state function to represent photons where photons themselves are treated as waving wave packets,[28] which are subject to wave equation.

The complete information about a system can be obtained from the Schr¨odinger equation, ∂Ψ(r,t) ih = HΨ(r,t), (2.4) ∂t where we also need to know the state function, Ψ(r,t) and H is the Hamil- tonian that is composed of the relevant kinetic (K), potential (V ) and other interaction (Hint) energy operators. Here, Ψ(r,t) alone does not contain all or the complete information about a system.

On the other hand, the electric and magnetic field components of an electromagnetic wave do contain all the information about photons and these components are subject to the wave equation, ∂2(E, B) ∂2(E, B) = c2 , (2.5) ∂t2 ∂x2 where c is the speed of light. Therefore, the rest of the postulates do not apply for photons primarily because V =0= Hint and its energy is defined by, hc E = hν = , (2.6) photon λ where h is Planck constant, ν denotes the frequency and λ is the wavelength of an electromagnetic wave. Here, Ephoton is 100% kinetic energy. Here, the state function cannot be a physical wave for two reasons—(i) Ψ(r,t) is the linear combination of the many eigenfunctions, ψi and (ii) the particle-property cannot be incorporated into Ψ(r,t) nor into the eigenfunctions, ψi. We need the constants (mel and −e) attached to physical operators to do that, to give particle- property such as mass and charge to an electron. Whereas, the electric field (E) nor 10 Andrew Das Arulsamy the magnetic field (B) component of an electromagnetic wave cannot be a physical wave for the second reason, (ii). In particular, the particle-property of a photon cannot be incorporated into the said wave-property of the magnetic- and electric- field components. The second and third Copenhagen Postulates remain the same in Pandamaran interpretation. Therefore, we just reproduce them below. Copenhagen Postulate 2: Each linear Hermitian operator corresponds to a physical observable. To find this operator, we can exploit the correspon- ~ ∂ ~ ∂ ~2 ∂2 p2 dence rules, i ∂t −→ E, i ∂x −→ p and − 2m ∂x2 −→ 2m = K. In Copenhagen Postulate 2, we did not specify whether the total energy, E, the momentum, p nor the kinetic energy K refers to a particle or a wave or both. The proper specification is given in Pandamaran Postulate 6 and in the discussion after Pandamaran Postulate 8. Copenhagen Postulate 3: The physically observable property, A pro- duces measurable eigenvalues, ai subject to the following eigenvalue equation, Aψˆ i = aiψi where Aˆ is the Hermitian operator corresponding to the observable property, A and eigenvalues, ai. The eigenfunctions, ψi are of course required to be well-behaved as stated in the first Pandamaran Postulate. The fourth Copenhagen Postulate[4] is ad hoc and is limited in its validity. There- fore, we reconstruct it so as to make it suitable for Pandamaran interpretation. Pandamaran Postulate 4: The Hermitian operator, Aˆ of an eigenvalue equation, Aψˆ i = aiψi is also a linear operator, and is responsible for a physically observable property, A. Its eigenvalues, ai are all real and their respective eigenfunctions, ψi transform to a new set of eigenfunctions upon measurement of the physical property A. Here, we do not require the eigenfunctions to form a complete set because during each measurement, there is this thing called the wavefunction transformation such that the complete set for the eigenfunctions before the measurement is no longer valid during the measurement. Therefore, it is more meaningful to evaluate the wavefunction transformation during a measurement than to assume completeness before and after a measurement. Moreover, in quantum mechanical calculations, the guessed wavefunctions or eigenfunctions are constructed and adjusted to obtain eigenvalues that are comparable to experimental values, without any regard as to whether the constructed and adjusted eigenfunctions form a complete set or not. Pandamaran Postulate 5: If Ψ(r,t) is orthonormalized and well-behaved, and if it is the state function of a system, then the physical observable, A at time t is,

∞ 3 hAi = Ψ ∗AΨˆ d r, (2.7) Z −∞ where Ψ(r,t) = Ψ. Here the physical property, A of an electron can only be measured for a range of r with a probability such that we cannot predict A with certainty if the range of observation, (−∞, +∞) is replaced with [−a, +a] and ∞ ≫ |2a|. This uncertainty is due to Heisenberg’s Pandamaran Interpretation of Quantum Mechanics 11

uncertainty principle.

The Born’s modiﬁed statistical interpretation reads,

+b + 3 ∞ 3 Ψ ∗Ψd r ≈ 1= Ψ ∗Ψd r, (2.8) Z b Z − −∞ where ∞ ≫ |2b| > |2a| such that the following integral,

+a 3 Ψ ∗Ψd r, (2.9) Z a − cannot be deﬁned if 2a<∆r and ∆r refers to the size of the electron’s distribution function or the size of the wave packet determined by the wavefunction.

Therefore, the physical property A (including position, momentum or any other observable physical properties) of an electron cannot be defined if the range of observation 2a<∆r. Here, the average value, hAi of an arbitrary physical property, A of an electron cannot be defined for 2a<∆r because the electron does not have a point-like position, (x,y,z). Therefore, for 2a<∆r, the position of the electron cannot be determined with certainty due to Heisenberg uncertainty principle, while for 2b ≥ ∆r > 2a, the position is not subject to Heisenberg uncertainty principle. However, the intrinsic spin (or spin angular momentum) of a quantum particle is not subject to Pandamaran Postulate 5. The original Copenhagen Postulate 6 is redundant for Pandamaran interpretation because it has been incorporated in Pandamaran Postulate 1. Therefore, our next postulate is related to the applicability of Heisenberg uncertainty principle for certain physical observables that are also captured or observable in classical physics. Pandamaran Postulate 6: The state function, Ψ(r,t) that is subject to time-dependent Schrödinger equation captures the time evolution of a quantum mechanical system that is free from any external disturbances. This equation is given in Pandamaran Postulate 1. The state function is separable into products, which means,

ml iEt Ψ = Ψ(r,t)= R(r)Y (θ, φ)χ(ms)exp − , (2.10) l ~

ml where R(r) is the radial wavefunction, Yl is the orbital angular momentum eigenfunction or angular wavefunction or sometimes known as the spherical harmonics, χ(ms) = |smsi is the spin eigenstate (not an eigenfunction), and finally, exp (−iEt/~) is the time-dependent phase factor that takes the effect of time evolution of the state function subject to the 1 Schrödinger equation. Note this, ms = ± 2 . The above separable prod- uct solution implies that the quantum mechanical physical observables are physically separable from the observables in classical physics. Therefore, 12 Andrew Das Arulsamy

the Heisenberg uncertainty principle has limited application such that the said uncertainty is not and cannot always be true for all observables and for all range of observation. Let us elaborate on this Postulate 6 further. The momentum eigenvalue in quantum mechanics is physically different from the momentum in classical mechanics to the extent that the quantum mechanical momentum refers to the wave-property of an electron, while in classical physics, the momentum refers to the particle-property of the same electron. In addition, there are quantum mechanical observables such as orbital angular momentum states, ml = 0, ±l and spin angular momentum states, ms = 0, ±s, which are not observable in classical physics entirely because orbital and spin angular momenta states in quantum mechanics refer to the wave-property of an electron and there are no particle-property for these angular momenta states in classical physics. The group velocity of an electron, that is bound to a given principal quantum number, n cannot be defined if r ≤ ∆r due to its relation with mass distribution according to the probability distribution curve defined by the wavefunction (see Pandamaran Postulate 5). The original Heisenberg uncertainty principle is activated for all quantum mechanical physical observables if and only if the pairs of the physical operators are non-commuting. However, the said uncertainty becomes inapplicable for certain quantum mechanical physical observables even if the pairs of the relevant operators are non-commuting. In particular, Heisenberg uncertainty principle is inapplicable for quantum mechanical observables that are also observable in classical physics. Since, ml = 0, ±l and ms = 0, ±s are not observable in classical physics, then the Heisenberg uncertainty principle is always valid in such cases. For example, the position operator,r ˆ, even though it does not commute with the momentum operator,p ˆ, but both the position and momentum eigenvalues can be observed simultaneously if the range of observation, r>∆r. Here, r and p refer to particle-like position and momentum of a massive electron in classical physics. However, the following eigenvalues, ml = 0, ±l and ms = 0, ±s are not observable in classical physics, and therefore, ml = 0, ±l and ms = 0, ±s are subject to Heisenberg uncertainty principle for all range of observation, 0 ≤ r ≤∞. Consequently, the Heisenberg uncertainty principle is always true only for physical observables that are not observable in classical physics, and if the pairs of the operators for those observables are non-commuting. Whereas, for quantum mechanical physical observables that can also be observed in classical physics are not subject to Heisenberg uncertainty principle if r>∆r, even if the pairs of quantum mechanical operators are non-commuting. Let us now apply Pandamaran Postulate 6. This postulate means that for r ≤ ∆r, the quantum mechanical momentum operator refers to the momentum of a wave-like electron, while for r>∆r the observable momentum refers to classical momentum of a particle-like electron with mass, mel and velocity, v. Subsequently, we can apply Pandamaran Postulate 6 to understand why the wave-like momentum of a photon remains wave-like for all range of observation (0 ≤ r ≤ ∞), which is because photons are not subject to Schrödinger equation, instead, they obey the Pandamaran Interpretation of Quantum Mechanics 13 massless wave equation. For example, classical momentum cannot exist for massless photons and is denied by the the wave equation. Finally, Pandamaran Postulate 6 provides the physical justification as to why the Heisenberg uncertainty principle is not always true for physical properties that are observable in both quantum mechanics and classical physics as exposed above. It is apparent that Pandamaran Postulate 6 exposes the source of Bohr’s confusion, and this confusion then led to the acceptance of Feynman’s non-classical paths as physically valid. Einstein attempted to find the source of the Bohr’s confusion, but Einstein could not do so because he (together with Podolsky and Rosen) invented the irrelevant EPR paradox that turned out to be false.[8] Let us continue with the remaining two postulates related to spin. The sev- enth postulate stated in the Copenhagen interpretation section stays the same in Pandamaran interpretation. Copenhagen Postulate 7: The wavefunction of a system of electrons or particles with half-integral spins, must be antisymmetric with respect to interchange of particles with half-integral spins, including electrons.

As for the final postulate on the intrinsic spin, we need to reconstruct it, and is given below. 1 3 Pandamaran Postulate 8: The intrinsic spin (s = 0, 2 , 1, 2 , · · · ) or the spin angular momentum is unique, specific and is immutable for each species or type of quantum particle. The said spin has no corresponding physical observable in classical mechanics and therefore, the Heisenberg uncertainty principle is always valid such that we cannot observe more than one (Sz or Sx or Sy) component of a quantum particle’s spin simultaneously for all range of observation, 0 ≤ r ≤∞. Pandamaran Postulate 8 establishes the fact that classically, the spin components (similar to orbital angular momentum components, Lx, Ly and Lz) cannot be defined because they are not observable physical properties in classical physics or the said observables are subject to Heisenberg uncertainty principle. In particular, Sx,y,z and Lx,y,z are associated to the wave-property of the quantum particle, which is not related to any particle-property in classical physics. Hence, Sx,y,z and Lx,y,z are subject to Heisenberg uncertainty principle for all range of observation, 0 ≤ r ≤∞. 1 3 Here, we have to be aware that s = 0, 2 , 1, 2 , · · ·= 6 l where l = 0, 1, 2, · · · as proved in Ref.[26] Having said that, we can now state that the new Pandamaran Postulates (1, 4, 5, 6 and 8) listed above have demolished Bohr’s philosophy (a particle takes a stand only upon measurement)[27] and Feynman’s non-classical paths (the possibility of many simultaneous paths (of different probabilities) for every particle). The above Pandamaran (1, 4, 5, 6 and 8) and Copenhagen (2, 3 and 7) postulates strictly imply that quantum mechanics is an ontic theory that deals with ‘real’ existing objects, but subject to the Heisenberg uncertainty principle or wave-particle duality. Here, wave- particle duality implies Heisenberg uncertainty principle. However, the wavefunction 14 Andrew Das Arulsamy does not represent any sort of physical real wave. Consequently, we can rule out the idea that quantum mechanics is an epistemic theory (based on Bohr’s philosophy), which defines that all quantum mechanical observables cannot exist (in whatever physical forms), until one measures them. This is simply because the wavefunction is not a real physical wave anyway. Consequently, we can now deduce unequivocally that the orthodox position has got to be false. We are left with the realist and agnostic positions as the only possible and viable candidates. As a matter fact, quantum mechanics is in favor of both realist and agnostic positions because even though the wavefunction is not a real physical wave (due to wave-particle duality, and from the definition of Ψ(r,t) given in Pandamaran Postulate 1) but the wavefunction, together with the Hamilton operators, correctly represent the observables such that they are real and they do exist before the measurement. However, even though these quantum mechanical observables do exist in a real physical sense, but they are subject to these two physical conditions, C1 and C2 (see below). (C1) The Heisenberg uncertainty principle. (C2) Whether the said quantum mechanical observables are also observable in classical physics. These two conditions exist entirely due to wave-particle duality of quantum particles. All quantum mechanical observables, regardless of whether they are also observable in classical physics or not, are guaranteed to satisfy the Heisenberg uncertainty principle if and only if r ≤ ∆r. For those quantum mechanical observables that are also observable in classical physics, are not subject to the Heisenberg uncertainty principle for r>∆r. In this range of observation, the quantum-classical observables are subject to classical uncertainties. On the other hand, for all quantum mechanical observables that are not observable in classical physics, the Heisenberg uncertainty principle is always valid for all range of observation, 0 ≤ r ≤ ∞. Therefore, C2 limits the applicability or validity of the Heisenberg uncertainty principle. Apart from that, quantum mechanics (based on Pandamaran postulates) has to be probabilistic because of the notion of wave-particle duality in the presence of massive and charged electrons, which are represented by wavefunctions, as well as for massless wave packets (photons). For example, the wave-property of quantum mechanical observables that are not observable in classical physics validate the Heisenberg uncertainty principle for all range of observation. While, the quantum mechanical observables that are also observable in classical mechanics are subject to the same Heisenberg uncertainty principle, but its validity is limited to r ≤ ∆r for as long as the size of the wave packet can be physically defined (subject to probability distribution function). For example, we can define, ∆r as the shape-shifting function or probability-distribution-shifting function. As such, wave-particle duality that gives rise to the Heisenberg uncertainty is responsible for the probabilistic nature of quantum mechanics. Hence, probabilistic nature here means that we still need the statistical description or statistical interpretation (exists entirely due to wave-particle incompatibility) to calculate or determine an eigenvalue of a physical property A. Pandamaran Interpretation of Quantum Mechanics 15

On the other hand, we also have massless and chargeless photons. Here, photons are also waving wave packets but then their electric and magnetic field components are subject to a pure wave equation because photons are massless and chargeless. In this case, all quantum mechanical observables for photons are wave-like with no corresponding particle-like observables in classical physics, and therefore, the probabilistic nature for photons comes entirely from the Heisenberg uncertainty principle for all range of observation, 0 ≤ r ≤∞. Probabilistic nature for photons exists because we cannot determine the particle-property of photons such that each photon’s energy, E = hc/λ with constant speed, c and momentum, p = h/λ where both E and p refer solely to wave-property of an electromagnetic wave. Now comes the proper connection to classical physics. Quantum mechanical observables such as position and momentum are also observables in classical mechanics, and therefore, the Heisenberg uncertainty principle does not apply for r>∆r. On the other hand, quantum mechanical observables such as spin components (Sx,y,z) are not observable in classical physics, and therefore, such observables are always (for all range of observation) subject to the Heisenberg uncertainty principle. Just because quantum mechanical observables satisfy the Heisenberg uncertainty principle, does not imply that these observables do not exist until one measures them. Tech- nically, when one measures a quantum mechanical observable, then one is actually operating on the wavefunction with the Hamilton or a suitable quantum mechanical operator. The momentum operator of an electron is actually observable for r>∆r. Only for r>∆r, the momentum of an electron refers to a classical momentum (due to attached mass to kinetic energy operator), while for r ≤ ∆r, this particular momentum, as well as the orbital and spin angular momenta refer to wave-property of the same electron. It so happens that the quantum mechanical operators for orbital angular and spin angular momenta are independent of mass. Therefore, the orbital angular and spin angular momenta always refer to the wave-property of an electron, which cannot be determined precisely theoretically (before the measurement) because these two operators are subject to Heisenberg uncertainty principle for all range of observation, 0 ≤ r ≤∞. The waving photons however, are subject to a wave equation and they always remain wave-like, and therefore, we can only predict and measure their wave-like properties, and not their particle-like properties such that, the classical position and the classical momentum of a photon’s wave-packet are not observable for all range of observation. Note this, not observable does not mean ‘does not exist’ and in this case for photons, we cannot define ∆r, which should represent the size of a massless photon’s wave packet. Recall that classical position and classical momentum are physically meaningful and observable only for massive particles or massive wave packets such as electrons and other massive elementary quantum particles. Conse- quently, particle-like properties of photons always obey the Heisenberg uncertainty principle for all range of observation, 0 ≤ r ≤∞. For any quantum mechanical observables that are subject to Heisenberg uncertainty principle, the concerned physical properties that correspond to their respective observables cannot be properly defined entirely due to wave-particle duality. This 16 Andrew Das Arulsamy duality is the reason why we need both the wavefunction and the Hamilton operators to have a complete picture of a physical system. In addition, the wavefunction that solves the Schrödinger equation can be separated into products, which implies that the wave-property and the particle-property observables can be separated too but for different range of observation, and the said observables are not independent. The discovery of these facts have led us to revise the Copenhagen postulates and construct the Pandamaran postulates, which then lead us to our second interpretation of quantum mechanics. On the other hand, for an electromagnetic wave with photons or waving wave packets, the electric field (E) and the magnetic field (B) components are separable, and they can also be treated independently, for example,

E(x,t)= E0 sin (kx − ωt), B(x,t)= B0 sin (kx − ωt), (2.11) where the one-dimensional components for E(x,t) and B(x,t) are solutions to the same wave equation (see Pandamaran Postulate 1). Therefore, this both ‘separate’ and ‘independent’ solutions mean that we can also observe or produce the electric (or the magnetic) field independent of the magnetic (or the electric) field. The Heisenberg uncertainty principle for photons is always valid for all range of observation simply because photons are always wave-like with no observable particle-like property. In other words, for photons or waving wave packets, we cannot define the size of a waving wave packet, ∆r because there is no such thing as waving entity within a photon. Moreover, a waving wave packet or a photon is defined by E(x,t) and B(x,t) and therefore, ∆r is actually meaningless for photons. For example, photons are massless, which implies ∆r cannot even exist, and they are chargeless too, which means there is no waving entity within each photon. The wavefunction that solves the Schrödinger equation is composed of radial wavefunction, angular wavefunction, spin eigenstate and the time-dependent phase factor, which are written as separable products but they are not independent (see Pandamaran Postulate 6). Therefore, the quantum mechanical observables are also separable, but then, they are not independent because each of the wavefunction (radial, angular, spin and time) are not independently defined. In other words, the complete wavefunction is composed of terms that are written as separable but not- independent products. These two facts imply that the wave-property and particle- property of an electron are separable but not independent, which correctly captures the wave-particle duality. Similarly, there are quantum mechanical operators that contain particle-like (kinetic energy) and wave-like (potential or interaction energy) information such as mass and charge, respectively, but then, they operate only on the wave-like property of the wavefunction. As a consequence, the Pandamaran postulates of quantum mechanics are subject to Heisenberg uncertainty principle, and only for quantum mechanical observables that are also observable in classical physics, the said uncertainty can be disobeyed, provided that ∆r can be defined properly such that for r>∆r the electron has a classical (particle-like) momentum. Let us now answer the question—how does the ground-state wavefunction trans- Pandamaran Interpretation of Quantum Mechanics 17 forms to occupy the excited state? We can invoke Pandamaran Postulate 1 to state that the state function, Ψ(r,t) is only properly defined for the following quantum numbers, principal quantum number, n = 1, 2,... , orbital angular momentum, l = 0, 1, · · · and ml = 0, ±l, and therefore, the existence of forbidden energy gap is entirely due to our definition of Ψ(r,t) with quantized energy levels, which are all valid. Therefore, it is not that the electron should behave according to the conditions imposed on Ψ(r,t) by us, but rather, Ψ(r,t) is the one that should be defined properly, and should capture everything about the electron, which is indeed the case after defining the Hamilton operators too, except when the electron makes a transition between energy levels. This transition is not captured by the Schrödinger equation, and we have seen exactly why and how that is the case from the theory of wavefunction transformation exposed in Ref.[29] For example, the time-dependent state function (defined in Pandamaran Postu- late 1) evolution does not include the transition between energy levels because such transitions involve the existence of ‘stateless’ wavefunction, which is not captured by any other Copenhagen nor the Pandamaran postulates of quantum mechanics exposed thus far. In particular, we have no other choice but to accept the following facts as truths, which are straightforward to be understood physically and technically. F1: An electron does not disappear from one initial state and then reappear on another state by not occupying the forbidden energy gap. F2: The time-evolution of Ψ(r,t) does not capture the transition between energy levels because the moment we define Ψ(r,t) to be so, then we do not need to guess the eigenfunctions for each energy level, where the time-evolution of Ψ(r,t) can be exploited to derive all the eigenfunctions for all the energy levels, including the ‘stateless’ eigenfunctions from the Schrödinger equation. We know for certain that this is never the case. We know for certain that both F1 and F2 are absolutely true, and these facts lead us to the ninth Pandamaran Postulate. Pandamaran Postulate 9: When an electron makes a transition from one energy level (E1) to another (E2), then Ψ(r,t) evolves in such a way that this particular evolution is not captured by the Schrödinger equation. During this time-dependent evolution, the energy of the electron, E′ is defined to exist between E1 and E2 such that E′ 6= E1, E′ 6= E2 and in addition, n′, l′ and ml′ no longer represent the initial state wavefunction, Ψ(r,t)n,l,ml . Therefore, the wavefunction during the transition is known ′ ′ ′ as the stateless wavefunction, Ψ(r,t)n ,l ,ml such that n′, l′ and ml′ are r ′ ′ ′ undefined because Ψ( ,t)n ,l ,ml is not subject to the Schrödinger equation where the electron is either in the process of emitting one (spontaneous) or two (stimulated) photons (if E1 > E2) or absorbing a photon if E1 < E2. Obviously, E′ exists in the forbidden gap and this gap exists because the Schrödinger equation is blind-by-definition to the existence of E′. In Pandamaran Postulate 8 stated earlier, the technical reason why Ψ(r,t) can- 18 Andrew Das Arulsamy not represent a stateless wavefunction is due to the fact that Ψ(r,t) is not a physical wave in the first place. Secondly, the changes to n′, l′ and ml′ refer exclusively to wave-like properties of the electron that is making the transition by emitting or absorbing photons with appropriate energy. The changes to the position and momentum of an electron during the transition are observable if r>∆r, and these changes to the position and momentum implies that one can indeed stop an electron from making a transition between energy levels as recently observed by Z K Minev et al.[30] Recall here that ∆r is the size of the electron or the radial distribution of the electronic wavefunction subject to Pandamaran Postulate 5. The existence of ′ ′ ′ this stateless wavefunction, Ψ(r,t)n ,l ,ml also strictly implies the electron can never be a point-like particle because the excitation of an electron implies transformation of the wavefunction. Therefore, we can detect the disturbance to this wavefunction by switching on or off the disturbance during an electronic transition between two energy levels, which has been reported by Z K Minev et al.[30] recently. If an electron is a point-like particle, then stopping an electron from making the transition is like flipping a coin—now you can, now you cannot. We have derived the complete postulates that capture quantum mechanics in its full glory such that we have also identified what can be known from the wavefunction and from the Schrödinger equation or the Hamilton operator. We also have removed two major flaws in quantum mechanics, namely, the so-called wavefunction collapse and entanglement for quantum particles that are no longer interacting.[8] Apart from that, we have argued logically with proper physical mathematics on the limited validity of the Heisenberg uncertainty principle, which has its origin in the wave-particle duality such that the position and the momentum of an electron can be observed simultaneously if and only if r>∆r. We have to acknowledge that Bohr was correct in respecting classical physics such that quantum mechanics is not and cannot be independent of classical physics. For example, we have to find the link between quantum mechanics and classical physics, which we did when we discuss the quantum mechanical observables and under what circumstances that such observables are also observable in classical physics. For example, the classical position and momentum of an electron refers to the particle-property of an electron, while the momentum in quantum mechanics exclusively refers to the wave-property of an electron (see the correspondence rules in Copenhagen Postulate 2 and the discussion after that). Now, how does special relativity comes into quantum physics? How about gravitational effect in quantum mechanics? Note this, there is no gravitational potential operator that could exclusively operate on an electronic wavefunction. As for the relativistic effect, I can be certain of the answer. However, gravity can only be included by realizing the fact that both special relativity and gravity are connected to the time dilation effect. Any other approach to include gravity into quantum physics is doomed to fail because gravity treats both massless photons and massive electrons on an equal footing—consistent time dilation effect and spacetime curvature. This argument is valid because electrons obey Schrödinger equation, while photons obey wave equation, but gravity affects both electrons and photons with consistent spacetime curvature or time dilation effect. Pandamaran Interpretation of Quantum Mechanics 19

Therefore, special relativity and the presence of gravity do not change the physics such that relativity and gravity do not introduce new physics. This means that the Galileo’s relativity principle is always valid for as long as the speed of light is a constant. Recall that only a constant speed of light can lead us to a consistent time dilation effect.[21] Only a consistent time dilation effect can connect special relativity to gravity. Therefore, the stated connection (between gravity and special relativity) is like saying, higher and higher speed of a particle is equivalent to larger and larger gravity, and they both activate consistent time dilation effect. Fortu- nately, this connection makes the physics so much easier (technically) and powerful (physically unambiguous) because the said connection is both technically and physically straightforward for gravity and relativity do not introduce new physics, instead, only the time gets dilated in a consistent manner. Indeed, the said connection is true because gravity affects all particles on an equal footing, be they massive or massless, classical or quantum mechanical, with consistent time dilation effect. We shall now focus only on special relativity or relativistic effect in quantum physics such that the postulates and results presented here also apply for atoms that are found in non-zero gravity. Non-zero gravity means that the gravitational field is not strong enough to tear apart the atoms. Tearing apart the atoms with extreme gravity involves a type of (quantum) phase transition that is caused by gravity or the gravity provides the energy or the required force. But then, how does this phase transition should occur is entirely determined by quantum physics, such that gravity only provides the consistent spacetime for the quantum mechanical phase transition to occur. From Refs.,[20], [21] we can construct the tenth Pandamaran postulate, which is associated to time dilation effect. Pandamaran Postulate 10: Time dilation is always the effect that does not affect any physical processes, regardless of whether the processes are classical or quantum mechanical. Here, the connection between time dilation and spacetime curvature can be captured by the fact that we cannot stretch the space without time dilation effect, nor permit time dilation without space stretching. In addition, time dilation effect implies non- universal ticking of time, in other words, consistent time dilation effect connects special relativity to gravity. In fact, we can physico-logically validate the correctness of Postulate 10. Physico-logical validation means we exploit established physics and logical analysis to validate a proper physical statement (see Ref.[21] for details). Apart from that, we also have demonstrated in Ref.[20] that any measured value with relativistic correction cannot be a true value, instead, it is just a value measured from one observer’s frame of reference. As such, relativistic effect does not introduce any new physics into non-relativistic physics, which is as demanded by the Galileo’s relativity postulate. This means that the relativistic effect only implies correction to the non-relativistic value by adjusting its magnitude. Therefore, we can obtain a truthful value of a measurement by letting the observer and the observed to reside in the same frame of reference, and there is no need for relativistic correction. This leads us to the eleventh Pandamaran Postulate. Pandamaran Postulate 11: Truthful value either measured or observed means, a value without any relativistic correction. 20 Andrew Das Arulsamy

A measured value with relativistic correction becomes the correct value from our frame of reference, which has been discussed in Ref.[20] such that any relativistic correction does not introduce new physics as demanded by Galileo’s relativity. Pan- damaran Postulates 10 and 11 together give rise to the possibility that relativity and gravity do not introduce new physics. There is one more postulate that validates Galileo’s relativity postulate, which is given below. Pandamaran Postulate 12: If the relativity postulate is violated, then the speed of light cannot be a universal constant. This means that all physical laws cease to be invariant if the speed of light is not a constant. We have proved the correctness of Pandamaran Postulate 12 in Ref.,[21] which en- forces the fact that Galileo’s relativity postulate demands the speed of light to be a constant. Let us now derive the ﬁnal postulate. It is worth noting that many researchers falsely and implicitly assumed the physical interaction to be reversible in the same way as time as recently reviewed in Ref.[31] In particular, the following two false statements are implicitly assumed. (a) Time reversibility automatically implies reversible physical interaction. (b) Each forward process is assumed to be reversible without any other changes. Therefore, we can construct the thirteenth Pandamaran Postulate (given below) to invalidate the above statements, (a) and (b), which are always assumed or activated a priori to prove the possibility of the second law of thermodynamics violation. Pandamaran Postulate 13: Time reversibility never implies reversible physical interaction and any forward physical process cannot be assumed to be reversible without any other changes. Pandamaran Postulate 13 has been formally proved in Ref.[22] for any timescale and for any system size.

§3. Conclusions

We have constructed a total of 13 postulates, 9 of which belong to quantum mechanics, while the last 4 are associated to classical physics. The primary conclusions based on these postulates are listed here. (1) The realist position is the only valid position in classical mechanics such that the agnostic position is meaningless, while the orthodox position is simply nonsensical. (2) In quantum mechanics, indeterminism has its origin in wave-particle duality and it has got nothing to do with indeterminism in classical physics, which is related to how accurate one measures the observables. (3) In quantum mechanics, we have to adopt the agnostic position for r ≤ ∆r because in this range of observation, both momentum and position of an electron are undeﬁned. For r>∆r however, we can switch to realists position because one can measure both the particle-like momentum and position of an electron. (4) Indeterminism in quantum mechanics is not about how accurate one can measure the observables, but is determined by an intrinsic property known as the wave-particle duality, which then gives rise to the Heisenberg uncertainty principle. (5) The validity of this principle is shown to be limited—whether an eigenvalue is observable in classical physics, and whether the eigenvalue is conﬁned or spread beyond Pandamaran Interpretation of Quantum Mechanics 21

the range of our observation. Finally, (6) for both classical and quantum physics, the orthodox position turns out to be physically unacceptable.

Acknowledgments

Madam Sebastiammal Savarimuthu, Mr Albert Das Arulsamy and Madam Roosiya- mary Lourdhusamy have supported this work.

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