A Unified Explanation of Some Quantum Phenomena

There has been wide interest in beam splitter, quantum tunneling, and double-slit experiments for a long time. The states of single material particles are typically explained as being independent of other objects (and/or electromagnetic fields) and interactions. We argue that this independence causes counterintuitive explanations. By analyzing the research conducted separately by Wineland’s group and Haroche’s group, we naturally deduced that the states are dependent on other objects. Based on this dependence, our unified explanation is more intuitive. We design a double-slit thought experiment that can solve the center problem, i.e., which-path information. The dependence can also work for qubits in quantum information. To emphasize and use the dependence in the future, we propose a fundamental postulate that no material particle is free as the zeroth postulate of quantum mechanics.


Introduction
Currently, explanations of some quantum phenomena (e.g., the motions of single particles) still violate common sense or are counterintuitive, for example, the interference pattern of single material particles passing through beam splitters [1][2][3][4][5] and the quantum tunneling effect [6][7][8][9][10][11]. The typical explanations of these phenomena are as follows. After passing through a beam splitter, a particle has been explained to be in a coherent superposition of two path states, that is, it travels along two paths simultaneously. By changing the relative phase between the two path states and making the two paths meet in a detector, the probabilities of single particles being detected change with the phase. This is the interference of the two path states. Common sense dictates that a macroscopic object cannot travel along two paths simultaneously. The quantum tunneling effect has been used to explain that a particle has a nonzero probability of penetrating a potential barrier with a maximum potential energy that is higher than the energy of the incident particle. A macroscopic object cannot penetrate such a potential barrier.
The interference or coherence phenomenon of single particles passing through a double slit is the most typical example of a counterintuitive explanation [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Usually, particles approach the double slit one by one in different time intervals, and each particle has a probability of passing through the two slits. The well-known experimental results show that if one particle is detected just behind one slit, then this particle cannot be detected just behind the other slit at the same time, and no detector detects only part of a particle. These results naturally let one think that all particles reaching the screen have passed through only one slit, and in particular, the later explanation of the doubleslit experiment should rely on these results. However, many of the experiments conducted have shown that the larger the obtained probability information about a particle passing through a slit is, the more blurred the contrast of the interference pattern or the fringe visibility is. This has been expressed as a duality formula [26], where P is the probability information about a particle passing through a slit, and W is the fringe visibility. If one has obtained the information P = 1 , then W = 0 , and the fringe disappears completely. This is the famous which-path or which-way information dilemma. This has inversely made physicists revise the idea of a particle passing through only one slit and think that each particle in the wave packet passes through both of the slits simultaneously. This means that two parts or the two path states of the wave meet and interfere at different positions. A particle passing through the two slits simultaneously must violate common sense or be counterintuitive.
Actually, the motion states of a material particle always accompany or are dependent on interactions with other objects (and/or an electromagnetic field). Any external potential energy of a particle is an approximation of the interaction potential energy. Unfortunately, most quantum mechanics books have not emphasized this approximation. Furthermore, this approximation makes the theoretical calculation of the motion state of a particle easier, which usually lets calculators forget the approximation. These reasons have thus induced the explanations involving a particle state or behavior being independent of other objects. Section 2 introduces a common-sense speculation. Then, Sect. 3 explains why the independence is the main reason inducing the counterintuitive explanations. In Sect. 4, we present the method how to make the past counterintuitive explanations of some quantum phenomena, or, the coherent superposed pure states of a particle come closer to common sense. This method has been obtained by analyzing the research of single atoms passing through a beam splitter conducted by Haroche's group [3]. In Sect. 5 and Sect. 6, we use the method presented to introduce a unified, more intuitive explanation of quantum tunneling effect and other quantum phenomena. Our explanation of double-slit experiment is the subject of Sect. 7. In Sect. 8, we present a double-slit thought experiment. We designed it so that the neutrons having passed through the up (down) slit have the spin-up (down) state. The usual screen is replaced by a movable slit 3. After neutrons passed through the slit 3, the spin-up and spin-down ones can be separately counted. To emphasize and use the dependence of the particle behaviors with other objects and/or field in the future, we propose a fundamental postulate that no material particle is free as the zeroth postulate of quantum mechanics in Sect. 9. The conclusion is in Sect. 10. In the scenario of bouncing a table tennis ball against a table using a racket, the ball moves  up and down. The collisions between the ball and the racket and the seemingly unmovable  table and gravity cause these motions of the ball. If the up and down motions (or momentum states) do not require collisions or any interactions with the other objects, then the phenomenon must violate common sense or be counterintuitive. If the ball is a particle, then we can speculate that the coherent superposition of the particle's up and down momentum states is caused by or dependent on interactions with other objects. Wineland's group [27] prepared the Schrödinger cat state of an atom through interactions with Raman beams. The Schrödinger cat state of the atom is the entangled state of its two position states and its two internal energy states, which seems counterintuitive. Actually, the atom exchanges momentum and energy with the Raman beams. The atom corresponds to the ball and the Raman beams correspond to the racket and the table. Therefore, the Schrödinger cat state of the atom is dependent on the Raman beams. If the Raman beams stop interacting with the atom, then the cat state cannot exist since there is no object or electromagnetic field exchanging momentum and energy with the atom.

Independence Induces Counterintuitive Explanations
If two particles have charges (masses), then there must be a nonzero Coulomb (gravitational) interaction energy, which is inversely proportional to the distance between the particles regardless of how far away from each other they are. Therefore, the particle with a charge (mass) always interacts with all objects with charges (masses), except for the particle itself, or it is always acted upon by the electromagnetic (gravitational) field at the particle's position. The total interaction with the particle cannot be zero. Here, the considered particles have low energies. Since the momentum and position (or coordinates) of a particle are continuous variables, the total interaction always changes the particle's momentum, position, and state, regardless of how tiny the interaction is. Therefore, no particle is free. Any particle is always a subsystem of a larger composite system. Investigating all of the interactions between a considered particle and all objects is impossible. Thus, we only consider an approximately isolated composite quantum system (AICQS), in which the interactions between the subsystems are stronger than that between the AICQS and the environment, so the latter is omitted, though many authors explore approaches in which the system-environment (system-heat-bath, system-meter) interaction plays a fundamental role in quantum mechanics [28][29][30][31][32].
For simplicity, we mainly discuss an AICQS with two parts. The two parts are two particles or a particle and an object or an electromagnetic field. In quantum mechanics, the Hamiltonian of an AICQS can be considered to be strict, and its dynamic equations are fundamental postulates. If an AICQS consists of a particle and a macroscopic object or a strong electromagnetic field, then investigations of the object's motion can omit the particle's action, whereas investigations of the particle's motion can consider the macroscopic object's state as being approximately unchanged. The interaction energy belongs to the AICQS and is approximately equal to the external potential energy of the particle. This is similar to the gravitational potential energy of an object on Earth, which is an approximation of the gravitational interaction energy between the object and the Earth. Thus, the Hamiltonian and Schrödinger equations for a particle are always approximate. The particle wave function is also approximate and is dependent on the macroscopic object. In addition, in other dynamic equations, such as the Heisenberg equation and Feynman path integral equation, the Hamiltonians or Lagrangians of single particles are all approximate. Moreover, the processes for solving many dynamic equations for single particles are very long, which cause one to forget these approximations.
In a common example, if the interaction between the electron and the nucleus of a hydrogen atom is much stronger than that between the atom and the environment, then the hydrogen atom is an AICQS. The wave function of the ground state is 2 a 3∕2 e (−r∕a) , where r = | | r 1 − r 2 | | , r 1 and r 2 are the position vectors of the electron and the nucleus, and a is the Bohr radius. The state cannot be expressed as the product of the wave functions of the electron and the nucleus and is thus an entangled state. If the nucleus state and physical quantities are considered to be approximately unchanged, then the entangled state approximates the product state. The electronic state is an approximately coherent superposed state in the momentum representation and is still dependent on the nucleus and the interaction. Thus, the principle of superposition of states for an AICQS can be considered to be strict, and this principle can be approximately used for single particles in limit cases.
For the coherent superposed pure state � 1 ⟩ + e i � 2 ⟩ of a particle (probability amplitudes are omitted in some cases in this paper), the interference or coherence of the two path states � 1 ⟩ and � 2 ⟩ is reflected in the different interference terms , then the particles reach different spatial locations with different probabilities. By detecting many such particles, one can obtain an interference pattern. We call this a spatial interference pattern. After a particle passes through the beam splitter BS1 (Fig. 1), it stays in the superposed pure state � 1 ⟩ + e i � 2 ⟩ with two different path states. Adjustment of and use of a second beam splitter BS2 are needed to obtain an interference pattern. A particle in states � 1 ⟩ + e i � 2 ⟩ , omitting a global phase, enters the two inputs of BS2 . At output 3 of BS2 , a particle appears with different probabilities as changes. One can also obtain an interference pattern by detecting many such particles as changes. If changes with time, then we call the obtained pattern a temporal interference pattern. The interference pattern obtained from a double-slit experiment is a spatial interference pattern, while that obtained from a beam splitter experiment can be a temporal interference Particles enter an input of the first beam splitter BS1 one by one, and the state of a particle exiting from BS1 includes two path states � 1 ⟩ and � 2 ⟩ . By adjusting , the particle with two path states � 1 ⟩ and e i � 2 ⟩ enters the two inputs of the second beam splitter BS2 , and the interference pattern can be obtained using detector D at output 3 pattern. Each momentum eigenstate of a particle can be considered a path state. The momentum eigenstates with the same direction and different magnitudes are also considered to be different path states. Except for a momentum eigenstate or plane wave of a particle, any wave function of a particle can be expanded into the coherent superposition of different momentum eigenstates. Thus, the probabilities of the particle appearing at different spatial locations are not exactly the same, and thus, many such particles produce a spatial interference pattern. According to the above approximations, the coherent superposed pure states of a particle with different path states are actually dependent on beam splitter BS1 , the double slit, and the environment, due to momentum exchanging between the particle and one of them. If the states are all independent, the explanations of the phenomena must violate common sense or be counterintuitive. However, the chapters about the Schrödinger equation, the Heisenberg equation, and the Feynman path integral equation in many books [33][34][35][36][37][38][39][40] do not emphasize that the Hamiltonians or Lagrangians of single particles are all approximated in these equations, although Zhang's book [39] mentions the external field approximation. All of these books assert that the principle of superposition of the states of a particle is independent of any object or field or is the intrinsic wave property of a particle. Therefore, the superposition principle of a single particle has led to the belief that the ideal coherent superposed pure states of a particle can be exactly obtained. Thus, the hypothetical intrinsic or independent property of a particle is the main reason for the counterintuitive explanations of some quantum phenomena.
The counterintuitive explanations have been considered to indicate that there is no good interpretation of quantum theory or mechanics. Subsequently, some interpretations were proposed, such as the hidden variables interpretation by Bohm [41] and the many-worlds interpretation of Everett [42]. Bohm presented an interpretation of quantum theory in terms of hidden variables and considered the wave function to be a Schrödinger -field. This interpretation provides the classical potential and quantum mechanical potential of a considered particle, which act as a force on the particle. The force and the particle's initial state determine the ensemble of particle trajectories, which are the solutions of the equations of motion. Bohm used these concepts to explain the experimental results for single particles passing through a double-slit apparatus and penetration of or tunneling through a potential barrier. We believe that the potential barrier for a particle must be an approximation of the interaction energy between some neglected objects and the particle. The particle and the neglected objects or the double slit constitute an AICQS. However, Bohm did not consider any neglected objects or the double slit as a subsystem of the AICQS. Several interpretations [43][44][45][46][47][48] of quantum mechanics have discussed the double-slit experiment and/or tunneling through a barrier. Similar to the Bohm's interpretation, none of these studies considered such an AICQS. Therefore, they all did not pointed out that the particles behaviors are dependent on other objects of field, and then their explanations still violate common sense or are still counterintuitive.
Everett presented the concept of a relative state, which was used in the AICQS of a considered system and a measuring apparatus. His relative state means that the state for a subsystem has a relative state for the remainder of the AICQS. The product of the subsystem's state and the relative state is similar to one term of an entangled state of the AICQS. In this sense, his interpretation of particles behaviors is close to common sense. But his many-worlds interpretation inversely is more counterintuitive. He did not use his concept to explain double-slit experiments and quantum tunneling phenomena. Similar to Everett's interpretation, many other interpretations [49][50][51][52][53][54][55][56] of quantum mechanics have not used their concepts to explain double-slit experiments and quantum tunneling phenomena.
To explain many quantum phenomena more intuitively, we can postpone conducting the approximations that the external fields of a particle are obtained from the interaction energies. The interaction drives the composite system of the particle and at least one object (or an electromagnetic field) to evolve into an entangled state. Since the interactions always involve exchanges of momentum and energy between the particle and the object, the composite system always stays in the entangled state in terms of momentum and energy representations. Thus, the particle can only be in a mixed state, not a coherent, superposed, pure state. If the subsystem interacting with the particle is a large object or a strong field, the subsystem's state remains approximately unchanged. For example, the nucleus of a hydrogen atom is approximately motionless in the center-of-mass reference frame. Thus, the entangled state approximates a product state, and the particle stays in an approximately coherent superposed pure state, which is not independent. Additional examples will be presented in the subsequent sections. Because thousands or more studies have investigated the above quantum phenomena, we searched the literature using a group of six key words: beam splitter (or tunnel, double slit, or two slit), composite system, entangled state, approximate (or approximation), product state, and mixed state. We only found one related publication [57], and it does not contain a discussion similar to ours.

Entangled States Approximate Product States
In the experiment conducted by Haroche's group [3], they let rubidium atoms in the circular Rydberg energy level state �e⟩ (or a lower energy level state �g⟩ ) enter and interact with a beam splitter. The beam splitter is a coherent light field. They obtained the combined atom + beam splitter state as the entangled state States � g ⟩ and � e ⟩ represent the beam splitter states with mean photon numbers of N + 1 and N , respectively. The entangled state means that the atom and the beam splitter exchange the energy of a photon. The combined atom + beam splitter can be assumed to constitute an AICQS. Regardless of the state the atom or the beam splitter is initially in, their interaction always drives the AICQS state to evolve into an entangled state. Strictly speaking, if an AICQS stays in an entangled state, then each subsystem can only be described by a mixed state, not a pure one. Thus, the state of such a subsystem is always dependent on the other subsystem.
For the classical limit, the mean photon number is N ≈ N + 1 ≫ 1(the case N = 0 is the quantum limit), and Haroche's group proposed that � g ⟩ ≈ � e ⟩ . The coherent field states, which are different from the energy eigenstates of a particle, are not orthogonal, ⟨ e � g ⟩ ≈ 1 , then, such slightly different states cannot be distinguished in physics. Naturally, the entangled state approximates the product state: However, they did not write out this approximation equation. In 2013, Zeng [58] proposed the approximation equation. In the same year, Fayngold et al. [59] also proposed a similar approximation equation. They discussed the momentum conservation of a particle and its passage through a beam splitter, but they did not provide a further discussion similar to ours. The approximate product state indicates that the atom approximately stays in a superposed pure state, 1 √ 2 (�g ⟩ + �e ⟩) , and the state is actually dependent on the beam splitter and the interaction.
In the Schrödinger picture, the initial state of an assumed free or isolated atom can be set as a superposition (for example, 1 (�g ⟩ + �e ⟩) is seemingly coherent, but its interference or coherence pattern cannot be directly detected since there are no different path states. The same is true for the spin state of a particle (or the polarized state of a photon) without different path states. Before obtaining an interference pattern, a particle should interact with another object, for example, a beam splitter or double slit, to achieve the superposed state of two or more paths. In Haroche et al.'s Figure 2 [3], after an atom leaves the coherent light field beam splitter ( BS1 ), the atom passes along two paths simultaneously. The quantum phase difference of the two path states can be tuned. A second beam splitter ( BS2 ) with two inputs was added to obtain the interference pattern of the two path states. In the caption of their figure, they mentioned that the cavity mode waist and the distance between the two beam splitters are exaggerated. Thus, this part of the experimental apparatus is really small, and thus, the interactions between the atom inside and the two beam splitters are not zero. The mean photon number N of BS1 is variable. When N = 0.31, 2, 12.8 , each has an interference fringe pattern at output D of BS2 , which is in the classical limit state. The patterns show that the smaller N is, the more blurred the fringe is.
Our explanation is that after an atom leaves BS1 , the atom and BS1 are in the entangled state: where �1 ⟩ and �2 ⟩ are the path states of the atom, and their phase difference can be tuned. The atom stays in a mixed state. At output D of BS2 , the path �g ⟩�1 ⟩ remains unchanged and �e ⟩�2 ⟩ becomes �g ⟩�1 ⟩ . The interference pattern at output D means that the original states �g ⟩�1 ⟩ and �e ⟩�2 ⟩ in Eq. (5) are coherent. Based on Haroche et al.'s experimental results, the smaller N is, the more blurred the fringe is, which means that the degree of coherence of �g ⟩�1 ⟩ and �e ⟩�2 ⟩ increases as N increases. Thus, the coherence is still dependent on BS1 (see Appendix A). This differs from Dirac's viewpoint that each photon interferes only with itself [33]. When N = 0 , BS1 stays in the quantum limit state, and there is no interference fringe. If BS2 also stays in the quantum limit state, then an interference fringe pattern reappears. Our explanation is provided in Appendix A.

Quantum Tunneling Effect
A particle has a nonzero probability of tunneling through a potential barrier when the highest potential energy is higher than the particle's incident energy. The potential barrier is actually an approximation of the interaction energy between a particle and at least one neglected macroscopic object or strong field, which make up an AICQS. The potential energy distribution of a real potential barrier changes from a small to large value and then returns to a small value. The energy of an incident particle is always larger than the potential energy of the particle just entering the barrier region. In the barrier, the interaction drives the AICQS state to evolve into an entangled state. According to the literature search in main text, no one has considered such an AICQS.
The Hamiltonian of such an AICQS is unknown, so we can only discuss the issue qualitatively. For simplicity, we consider one-dimensional motion. The total energy of the AICQS is the sum of the kinetic energies of the particle and the macroscopic object and the interaction energy in the barrier region. The interaction always leads to exchanges of energy and momentum between the two subsystems in the AICQS, and it drives the AICQS to evolve into an entangled state. In the momentum representation, the momentum of the particle is a continuous variable, and the entangled state must be expressed in an integral form. To simply explain the tunneling effect, we express the entangled state as a discrete form, �p 0 ⟩�P 0 ⟩ → � − p 1 ⟩�P 1 ⟩ + �p 2 ⟩�P 2 ⟩ , where �p 0 ⟩ and � − p 1 ⟩ are the initial and return momentum states, and �p 2 ⟩ is the forward state. �P 0 ⟩ , �P 1 ⟩ , and �P 2 ⟩ are the momentum states of the macroscopic object, which remain approximately unchanged. Thus, The particle is approximately in a coherent superposed pure state, which is similar to the case of the beam-splitter experiment. They all have one input and two outputs. A tunneled particle has obtained some energy and forward momentum from the macroscopic object with a nonzero probability, and the energy plus the initial energy of the particle p 0 2 ∕2m may exceed the barrier's highest potential energy. In this case, the particle actually does not penetrate but passes over the barrier, and the state of the macroscopic object remains approximately unchanged. Such energy and momentum exchanges are important for understanding the tunneling of the particle. Similarly, a particle can be reflected with a nonzero probability by a potential well.

Other Quantum Phenomena
An assumed free particle can be in the superposed state of different momentum states according to the principle of superposition of states. This is counterintuitive. For example, for a particle in a one-dimensional infinite potential well, its ground state can be extended to different momentum eigenstates in an integral form. These eigenstates include right and left moving eigenstates. The momentum of a particle is a continuous variable. For simplicity, we suppose that the particle is in the superposed state of the right and left momentum eigenstates, 1 √ 2 (�p⟩ + � − p⟩) . If we consider the entire potential well as a macroscopic box, then the particle in initial state �p⟩ interacts with the macroscopic box, and they evolve into an entangled state in momentum form: where �P M ⟩ and �P M + 2p⟩ represent the macroscopic box's states. The particle exchanges momentum 2p with the macroscopic box, and their total momentum is a constant vector, p + P M . Because the macroscopic box's motion state remains approximately unchanged (static or uniform linear motion), �P M ⟩ ≈ �P M + 2p⟩ , i.e., the magnitudes of the two vectors are approximately the same and the angle between the two vectors is zero, so the entangled state approximates a product state. If the macroscopic box is so large that we can hardly see the shell of the box, then the particle inside is easily mistakenly assumed to be free. Such a free particle, which approximately stays in the superposed state, 1 , is then not very strange or counterintuitive. This explanation may become closer to our intuition or common sense when it is linked with classical understandable phenomena, for example, the table tennis ball scenario in the main text. If there is a real free particle staying in the superposed state, then the phenomenon will still be counterintuitive. The above approximations have been used in explaining statistical and strict conservation of momentum [60]. The entangled state in the integral form shows that the conservation of the momentum of an AICQS is strict. This is similar to Eq. (7). If such an AICQS consists of a particle and a macroscopic object, the latter momentum can be considered approximately as a constant vector due to its motion state being approximately unchanged, and their entangled state approximates a product state. The approximately coherent superposed pure state in the product state shows that the conservation of the momentum of the particle is statistical.
If a particle is assumed to be free and has incompatible observables A and B , and there are no different path states, then one of the eigenstates of A is naturally the superposition �b 1 ⟩ + �b 2 ⟩⋯ + �b n ⟩ of the eigenstates �b j ⟩ of B according to the principle of superposition of states. However, no interference pattern can be directly obtained by detectors if there are no different path states. Therefore, we consider that such a superposition of the eigenstates of B has no coherence. If there is a macroscopic object that interacts with the considered particle before it reaches the detectors, for example, a beam splitter, a macroscopic object with slits, or a diffraction crystal, then the interaction drives the AICQS's state to evolve into an entangled state with different path states and B eigenstates for the particle. The state of the macroscopic object remains approximately unchanged, and the entangled state approximates a product state. The particle stays in an approximately entangled state �b 1 ⟩�1⟩ + �b 2 ⟩�2⟩⋯ + �b n ⟩�n⟩ , with different path states �j⟩ and B eigenstates, and then, an interference pattern or coherence can be observed by the detectors. Even if the object is a microscopic object and the particle is not in an approximately coherent superposed pure state or is in a mixed state, we can still detect the interference pattern using many identical particles in the same state. Thus, without a beam splitter or other object, the superposition of the eigenstates of B obtained using the principle of superposition of states is not coherent in the experiment and can therefore be called the incoherent superposition. For example, the spin eigenstate of a particle can be an incoherent superposed state in different spin eigenstate representations. This is discussed below.
The Stern-Gerlach experiment [61] provides a typical example of the superposed pure state of a particle. If the up state � ↑⟩ of the spin-1/2 electron is assumed to be free, it is theoretically considered to be the coherent pure superposition of two horizontal spin states . If a different base is theoretically selected, e.g., { �↑ ′ ⟩, �↓ ′ .⟩ }, then where a and b are the superposition coefficients. These two superposed states seem to have coherence, but one cannot directly detect the coherence of the superposed states. To prove the coherence, one can experimentally select different magnetic fields, and each field and one electron constitute an AICQS, which is in an entangled state. In these two cases, the states of the magnetic fields remain approximately unchanged, i.e., �M � 1 ⟩ ≈ �M � 2 ⟩ and �M 1 ⟩ ≈ �M 2 ⟩ . Hence, the two entangled states approximate product states: The two approximately entangled states of single electrons have different spin and path states ( �1 � ⟩, �2 � ⟩;�1⟩, �2⟩) . More importantly, the two states are coherent since interference patterns can be detected by flipping one of the spin components when the two path states of each of the single electrons meet. Thus, the two states are different from the theoretical states and are dependent on different magnetic fields and interactions. In quantum information, if a qubit stays in 1 (�g ⟩ + �e ⟩) and has no different path states, then its state has no coherence. The coherent superposed pure states of a qubit must be approximate and be dependent on other objects, electromagnetic field, and interactions. This is the neglected difficulty in quantum information since many qubits may be dependent on many objects or electromagnetic fields. For a squeezed or coherent light state [62], the light and the light source apparatus exchange energy due to the interaction between them, and the AICQS evolves into an entangled state. The state of the light source apparatus remains approximately unchanged, and the entangled state approximates a product state. The light field approximates a superposed state of different photon number states and different relative phases, i.e., a squeezed state or coherent state under different conditions. The strange, entangled state �g⟩�0⟩ + �e⟩�1⟩ can be considered: where �E + 2⟩ ≈ �E⟩ is the state of a macroscopic object, and �E + 2⟩ has two more photons than �E⟩ . The entangled state, �g⟩�0⟩�E + 2⟩ + �e⟩�1⟩�E⟩ , is stricter, and the energies of the two superposed terms �g⟩�0⟩�E + 2⟩ and �e⟩�1⟩�E⟩ are equal in this state. Thus, this entangled state, �g⟩�0⟩ + �e⟩�1⟩ , is approximate and is not strange even though the energies of the two superposed terms, �g⟩�0⟩ and �e⟩�1⟩ , are not equal. The plane wave of an assumed free particle is expressed by the momentum eigenstate. The state can theoretically be a superposition of spherical wave states with various angular momenta. This is inaccessible. When a large mass object in the center of the sphere interacts with the particle, the AICQS evolves into an entangled state, which can be expanded spherical wave states. The motion state of the large mass object is approximately unchanged. The entangled state approximates a product state, and the particle approximately remains in a superposition of spherical wave states. If there is no such an object, then we cannot directly detect identical particles that remain in different angular momentum states with different probabilities.
For the spread of the wave packet of a free particle in some quantum mechanics books, the particle is actually not free and interacts with the environment, and thus, the AICQS of the particle and the environment evolves into an entangled state. Within a short period of time, the environment remains approximately unchanged, the entangled state approximates a product state, and the particle approximates a superposed state or a wave packet. The wave packet naturally spreads as the environment changes in a longer time.

Double-Slit Experiments
Recently, some diatomic molecules have been simulated as double slits [22][23][24][25]. The interaction between a particle and such a double slit and the Hamiltonian of the composite system are known. However, no one has explained the interference pattern by solving Schrödinger equation since the theoretical derivation is certainly complex. Thus, their explanations are semi-quantitative or semi-qualitative. In this section, we discuss a macro double-slit experiment. The incident particles approach the double slit one by one in the experiment. A particle and the double-slit object can be considered to be an AICQS. The interaction between the two subsystems always drives their state to evolve into the entangled state. Since the interaction energy is unknown, we can only qualitatively describe the entangled state.
Here, we compare the macro-double slit and macro-beam splitter. Some beam splitters have two separate inputs, a and b (Fig. 2). A particle does not enter a beam splitter from two inputs in wave form. It clearly enters the beam splitter from one certain input, and then, it exits from two outputs and stays in a superposition of the two path states. A similar phenomenon occurs for the other input. For a double slit, we regard slits 1 and 2 as being similar to the two inputs a and b of the beam splitter, respectively, and to the two outputs c and d , which will be explained below. We searched for publications with the key words double slit and beam splitter, for example, the research conducted by Sadana et al. [63], but none of the publications considered this comparison.
The AICQS state of a particle and the macroscopic double-slit object evolves into the entangled state, Fig. 2 A particle enters beam splitter BS1 from input a and exits from the two outputs, and it remains in an approximately coherent superposed pure state of the two path states. By adjusting the relative phase ϕ, one can use detector D to obtain an interference pattern from one output of BS2 . For input b of BS1 , one can obtain the other coherent superposed pure state and the corresponding interference pattern before the particle passes through the double slit (omitting the probability amplitudes in some cases in this paper). The particle path states are � 1 ⟩ approaching slit 1, � 2 ⟩ approaching slit 2, and � 3 ⟩ when blocked by the double slit. The state of the macroscopic double-slit object remains approximately unchanged, �S 10 ⟩ ≈ �S 20 ⟩ ≈ �S 30 ⟩ . Thus, the entangled state approximates the product state (� 1 ⟩ + � 2 ⟩ + � 3 ⟩)�S 10 ⟩ . The particle is approximately in the coherent superposed pure state � 1 ⟩ + � 2 ⟩ + � 3 ⟩.
If a particle is detected before slit 1 (2), then the particle is usually believed to pass through this slit. If a particle is detected after slit 1 (2), then the particle can be said to have passed through this slit. These detections make the interference pattern disappear. Our explanations for these are provided in Appendix B. Now, there is no detector before and after the double slit. The double slit can be considered to be a measuring apparatus. The process of a particle passing through the double slit is equivalent to the particle being measured by the double slit. A particle just reaching one of the slits can be described as the particle superposed state collapsing to state � 1 ⟩ or � 2 ⟩ at the slit after being measured. This is similar to a particle entering one input of a beam splitter. However, this does not mean that the measured result has been read out, so the experimenters still do not know the which-path information. Assuming that the state of the particle just exiting from slit 1 or 2 is � 10 ⟩ or � 20 ⟩ , the particle state can be described as a mixed state, the density operator form (omitting probability) of which is In the process of the particle moving from a slit position to the screen, the kinetic energy operator of the macroscopic double-slit object and the interaction between it and the particle are still unknown. We still describe the process qualitatively and assume one of the two new initial states to be � 10 ⟩�S 10 ⟩ or � 20 ⟩�S 20 ⟩ . According to Feynman's path integrals [35], the particle has many possible paths from a slit position to a point on the screen. We believe that the reason for producing many paths is also due to the interaction between the particle and the double-slit object. To produce an interference pattern, a particle must have at least two path states that meet at different points on the screen. For simplicity and to inherit the past interpretation, we assume that a particle only has two such path states. Actually, the particle path states are dependent on the double-slit object and their interaction. Their state evolves into one of the following entangled states: where � 11 ⟩ and � 12 ⟩ can be considered to be the two path states after the particle exits from slit 1 and 2, and �S 11 ⟩ and �S 12 ⟩ are the two corresponding states of the doubleslit object, respectively. This is similar to a particle exiting from the two outputs of a beam splitter. When a particle has entered slit 1 and has just exited from the two slits, The farther the particle is from the double slit, the closer the two probabilities are. A similar phenomenon occurs for slit 2. The states of the macroscopic double-slit object are �S 11 ⟩ ≈ �S 12 ⟩ and �S 21 ⟩ ≈ �S 22 ⟩ . Thus, the two entangled states approximate two product states. The particle is approximately in one of the two coherent superposed pure states, i.e., � 1 � 11 ⟩ + 2 � 12 ⟩ � ≡ �Φ 1 ⟩ and 1 � 21 ⟩ + 2 � 22 ⟩ ≡ �Φ 2 ⟩ , or the particle state can be expressed as the mixed state �Φ 1 ⟩ ⟨Φ 1 � � + �Φ 2 ⟩ ⟨Φ 2 � � . Two interference patterns can be obtained from �Φ 1 (t)⟩ and �Φ 2 (t)⟩ , which are similar to those corresponding to the two inputs in a beam splitter experiment. Thus, the particles passing through one slit produce an interference pattern. The sum of the two interference patterns is the general interference pattern. Equations (14) and (15) can be used to discuss not only symmetric double-slit experiments but also asymmetric double-slit experiments. In next section, we provide a double-slit thought experiment that could detect the which-slit (which-path) information for the particles that passed through after obtaining the two interference patterns corresponding to the two slits.

A Double-Slit Thought Experiment
In this section, we design and discuss a double-slit thought experiment for testing the above explanation. By adding the Stern-Gerlach experiment apparatus into the double-slit experiment, we can obtain not only the usual interference pattern but also the two patterns corresponding to the two slits and which-path information with a high probability. The incident particles are spin-1/2 neutrons. Without the Stern-Gerlach apparatus, the double-slit experiment is a usual one for neutrons. Single neutrons with an initial spin state � →⟩ parallel to the horizontal slits pass through the first Stern-Gerlach nonuniform magnetic field, and some of them pass through the up (down) slit in the spin-up (spin-down) state � ↑⟩ ( � ↓⟩ ). Wu [40] proposed that an interference pattern could not be obtained for such an experiment based on the usual explanation for the interference in the double-slit experiment. He believed that a neutron in wave form passes through the double slit and enters the entangled state �↑ ⟩�u ⟩ + �↓ ⟩�d ⟩ , where �u⟩ and �d⟩ are the up and down path states, respectively. When the two path states meet at a position, there is no interference because ⟨↑ � ↓⟩ = 0 , i.e., the interference terms are zero. The following analysis shows that the entangled state is wrong.
The first Stern-Gerlach apparatus (S-G1) with magnetic field B 1 is placed between a neutron source and a double-slit apparatus with symmetric up and down slits (Fig. 3). Adjusting B 1 allows spin-up (down) neutrons, which reach the screen, to pass through the up (down) slit with a probability of at least 99%. This can be done by counting the neutrons, without knowing their spin states, just behind the up (down) slit per unit time. Then, a second Stern-Gerlach apparatus (S-G2) is placed just behind the up (down) slit, and the spin-up (down) neutrons are counted at the same time. Thus, the probability of a spin-up (down) neutron passing through the up (down) slit can be obtained. To increase the probability, the B 1 gradient must be increased. After obtaining the probability, S-G2 is removed. This will be used below.
According to the analysis in Sect. 7, a neutron entering a double slit is measured. This does not mean that we read out the measured result, and thus, the which-way information is still unknown. A neutron stays in a mixed state, If it is in the up slit, then the initial state of the AICQS of the neutron and the doubleslit object is in the product state �↑ ⟩�u ⟩�S 10 ⟩ . From the up slit to the screen, the interaction drives the AICQS state to evolve into the entangled state ( 1 � � u 11 ⟩�S 11 ⟩ + 2 � � u 12 ⟩�S 12 ⟩)�↑ ⟩ , where �u 11 ⟩ and �u 12 ⟩ are the two path states, and �S 11 ⟩ and �S 12 ⟩ are the corresponding two states of the double slit. Similarly, for a macroscopic double-slit object, �S 11 ⟩ ≈ �S 12 ⟩, and the neutron stays in the approximately coherent superposed pure state (16) �↑ ⟩�u ⟩ ⟨↑� ⟨u� + �↓ ⟩�d ⟩ ⟨↓� ⟨d� , then the spin-up and spin-down neutrons are easily omitting those not to pass through the slits. When the neutron reaches the double slit, it is measured, and the state of the AICQS is either �↑ ⟩�u ⟩�S 10 ⟩ or �↓ ⟩�d ⟩�S 20 ⟩ . Starting at this time, the state �↑ ⟩�u ⟩�S 10 ⟩ of the AICQS evolves into a new entangled state ( 1 � � u 11 ⟩�S 11 ⟩ + 2 � � u 12 ⟩�S 12 ⟩)�↑ ⟩ . For a macroscopic double-slit object �S 11 ⟩ ≈ �S 12 ⟩ , the entangled state approximates the product state, and the neutron stays in the approximately coherent superposed pure state, ( 1 � � u 11 ⟩ + 2 � � u 12 ⟩)�↑ ⟩ . This is similar for the other state �↓ ⟩�d ⟩�S 20 ⟩ . After many neutrons have passed through movable slit 3 and through S-G2, the spin-up and spin-down neutrons are counted separately. Thus, the two interference patterns can be obtained, and the sum of these two patterns is the typical pattern through the up (down) slit and slit 3 is close to the diffraction pattern for a single slit. Thus, two large wave peaks and some small peaks appear in the usual interference pattern. As the distance increases, then the two large wave peaks become smaller, and the small peaks become larger. This is similar to the results reported by Kocsis et al. [66]. According to Greenberger et al. [26], when 99% of the particles that reach the screen pass through slit 2 and only 1% pass through slit 1, the contrast of the interference pattern or fringe visibility is 20%. Similarly, when 99% of the spin-up neutrons that reach the screen pass through the up slit, then 1% pass through the down slit. If the contrast is greater than 20%, the duality equation (Eq. (1)) is violated, that is, P 2 + W 2 > 1 . We believe that this will be proven. Thus, these results indicate that the coherent superposed state of each neutron is dependent on the double slit and their interaction. This design can also be used to investigate the interference pattern of an asymmetric double-slit experiment. Our design is expected to draw immediate experimental attention, and can be tried to do the experiment of single photons passing through two slits with different polarization.

The Zeroth Postulate
No material particle is free. We call this statement the zeroth postulate, and it is referred to as the zeroth law of thermodynamics. Adding this postulate to the content of the existing postulates [33,36] of quantum mechanics may be necessary because it helps us to explain some quantum phenomena more intuitively. All of these postulates are similar to the two famous postulates of special relativity, which are the starting points of the theory. This zeroth postulate is different from "postulate 0" [67] or the "zeroth axiom" [68], that is, the Hilbert space of an AICQS is the tensor product of the Hilbert spaces of the components. If one solves a dynamic equation for an AICQS, then the calculation will be longer and more complex than that of the approximate dynamic equation for a particle. For example, for an approximately isolated hydrogen atom is a two-body AICQS, the calculation of its wave function is slightly more complex than that of its electron and an approximately static nucleus. Therefore, the zeroth postulate is not used for the calculation, but it does hint at explaining the coherent superposed pure states of a particle, aiming at counterintuitive quantum phenomena. By accepting the zeroth postulate, one can naturally consider a particle to be a subsystem of an AICQS, the states of which are all entangled states, except for the moment after a measurement.
For the motion of a particle in an external field, the postulate reminds one that any external field is an approximation of the interaction energy between the particle and some objects or electromagnetic field. Thus, for a particle, all of the dynamic equations, such as the Schrödinger, Heisenberg, and Feynman path integral equations, are approximate, and a pure superposition of the energy eigenstates or path states of a particle is also approximate. Figure 4 is the inference derived from the zeroth postulate. Since such superposed states are not only the solutions of the Schrödinger equation or Feynman path integral equation, but they can also be obtained using the principle of superposition of states, these equations and the principle have led us to believe that the states are strict. Thus, neglecting these approximations induces that the coherent superposed states of a particle is considered as its intrinsic or independent property. This is the main reason inducing the counterintuitive explanations of the quantum phenomena here. From the zeroth postulate and the Schrödinger equation, one can deduce the purely coherent superposition in the wave property of single particles in limit cases. But we cannot deduce the zeroth postulate from the purely coherent superposition in the wave property and the Schrödinger equation.

Conclusions
For the superposed state of the different energy eigenstates of a particle, the interaction between the particle with at least one object or one electromagnetic field that leads to the exchange of energy must also exist, even when the coherence is not considered. The coherence or interference of a material particle requires the superposition of different path or momentum states. Such a superposed state of a particle requires interactions with other objects (and/or an electromagnetic field) to achieve the consecutive exchange of momentum. Therefore, the coherent superposed state or the wave property of a particle comes from interactions with other objects. This is the unified, more intuitive explanation for many quantum phenomena. Based on this, we designed a double-slit thought experiment, and we believe that the results of this experiment will be proven. Thus, the wave property of a particle always accompanies other objects (and/or an electromagnetic field) and their interactions, and it is not independent or intrinsic. We present the zeroth postulate that no material particle is free for the better association of coherent superposed states with common sense. Considering the postulate could keep one avoid counterintuitive explanations of some quantum phenomena. The coherent superposed state of a qubit is also dependent on other objects and their interactions. This state is also approximately obtained. The preparation of the coherent superposed states of several qubits may be dependent on several objects or fields and their interactions. This is probably an overlooked difficulty in quantum information processing. We explain the four interference patterns of N = 0.31, 2, 12.8, 0 in Figure 2 in the paper published by Haroche's group [3]. This figure is redrawn in Fig. 5. An atom initially in �e ⟩ enters one input of BS1 . After exiting from BS1 , the atom and BS1 are still in an entangled state 1 , where �1⟩ and �2⟩ represent the atom's two path states, and is the adjustable relative phase. An interference pattern that varies with can be obtained at one output of BS2 . The atom stays in a mixed state. We note the coherent state of BS1 by the mean photon number, � g ⟩ ≡ �N + 1⟩ and � e ⟩ ≡ �N⟩ , which is different from the photon number state. When just entering BS2 (initially in state �N ′ ⟩ ) and omitting a global phase, the total state of the atom and the two beam splitters can be expressed as Upon exiting from BS2 , the total state of the atom and the two beam splitters is The atom stays in state �g ⟩�3⟩ at output 3 of BS2 , and its appearance probability is When BS2 stays in the classical limit state, N ′ ≫ 1 and �N � ⟩ ≈ �N � + 1⟩ , so When N = 0.31, 2, 12.8 in BS1 , all of the experiment results obtain interference patterns. The larger N is, the larger the fringe visibility is, or the clearer the fringe is. This indirectly reflects the degree of coherence of states �g ⟩�1⟩�N + 1⟩ and �e ⟩�2⟩�N⟩ in the entangled state and reflects the coherence of �g ⟩�1⟩ and �e ⟩�2⟩ in the atomic mixed state. Theoretically, the larger N is, the closer the entangled state of the atom and BS1 is to the product state, and the (A1) 5 Atoms in state �e ⟩ enter an input of the first beam splitter BS1 one by one, and the relative phase is adjusted in path 2. After exiting from BS1 , each atom enters the second beam splitter BS2 . By detecting the atomic probability, which varies with at output 3, an interference pattern varying with can be obtained closer the atomic mixed state is to the pure state. All coherent mixed states of such an atom are dependent on BS1 . Since the coherent superposed pure state is approximated from a mixed state, the pure state is naturally dependent on BS1 . When N = 0 in BS1 , it stays in the quantum limit state, so � e ⟩ = �0⟩ , � g ⟩ = �1⟩ , and ⟨ g � e ⟩ = 0 . The atom and BS1 stay in the entangled state As can be seen from Equation (S4), P = 1 2 , which is a constant, and there is no interference fringe. This also indirectly reflects that the degree of coherence of �g ⟩�1⟩ and �e ⟩�2⟩ in the atomic mixed state is zero.
The two beam splitters are set in the same cavity, i.e., BS1 and BS2 can correspond to two pulses with a short interval time in the same cavity. When both BS1 and BS2 remain in the quantum limit state, i.e., N = N � = 0 , an interference fringe is reobtained in the experiment. Since and if an atom is detected at output 3 of BS2 , then the states of the cavity make no difference. If an atom is detected at output 4 of BS2 , then and A photon is left after the pulse corresponding to BS1 has acted on the atom, and the state � − 1⟩ of BS2 actually expresses that the atom absorbs the photon from the cavity. Thus, the mean photon number in �1⟩� − 1⟩ in the cavity expresses zero. Upon exiting from BS2 , the total state of the atom and cavity can be expressed as At output 3 of BS2 , the probability of detecting an atom is and an interference fringe may be detected. This also indicates that the atom's state is dependent on the two beam splitters.
then, the atom, which is in different states with different probabilities, interacts with the double-slit object. This is similar to the above case of the measurement before the atom enters a slit, in which the interference pattern disappears. In the case of detection using a classical field, the fields in the two cavities remain approximately unchanged, the entangled state approximates a product state, and the atomic state approximately remains in a pure state. Thus, the interference pattern will appear. If N 1 and N 2 change from small to large numbers, then the atomic mixed state tends to the pure state, and the contrast of the interference pattern increases. In addition, after detecting the path state of an atom approaching the slits using a quantum field and obtaining the which-way information, Scully et al. reobtained the interference pattern by the eraser and explained that the pattern's reappearance is due to the which-way information deletion. We have explained pattern reappearance in another way in the experiment in Appendix A.