**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 splitters [1–5] and the quantum tunneling effect [6–11]. 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 [12–25]. 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 double-slit 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,

$${P}^{2}+{W}^{2}\le 1$$

1

where \(P\) is the probability information about a particle passing through a slit, and \(W\) is the fringe visibility [26]. 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 being independent of other objects. Thus, this independence is the main reason inducing the counterintuitive explanations (see **Supplementary Section S1**). To explain many quantum phenomena more intuitively, we can postpone conducting the approximation. 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: 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 [27], and it does not contain a discussion similar to ours.

## A Common Sense Example

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 guess 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 [28] 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.

## 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 \(\frac{1}{\sqrt{2}}(\left|g\right.⟩\left|{\gamma }_{g}\right.⟩+\left|e\right.⟩\left|{\gamma }_{e}\right.⟩)\). States \(|{\gamma }_{g}⟩\) and \(|{\gamma }_{e}⟩\) represent the beam splitter states with mean photon numbers of \(N+1\) and \(N\), respectively. The entangled state means that the atom and the splitter exchange the energy of a photon. The combined atom + beam splitter can be assumed to constitute an approximately isolated composite quantum system (AICQS). The interactions between the subsystems in the AICQS are stronger than that between the AICQS and the environment, so the latter is omitted. For example, an isolated atom is an AICQS. Regardless of the state the atom or the 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 \(N\approx N+1\gg 1\)(the case \(N=0\) is the quantum limit), and Haroche’s group proposed that \(|{\gamma }_{g}⟩\approx |{\gamma }_{e}⟩\). Naturally, the entangled state approximates the product state:

$$\frac{1}{\sqrt{2}}(\left|g\right.⟩\left|{\gamma }_{g}\right.⟩+\left|e\right.⟩\left|{\gamma }_{e}\right.⟩)\approx \frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\left|{\gamma }_{e}\right.⟩$$

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However, they did not write out this approximation equation. In 2013, Zeng [29] proposed the approximation equation. In the same year, Fayngold et al. [30] also proposed a similar approximation equation. They discussed the momentum conservation of a particle and its passage through a 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, \(\frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\), and the state is actually dependent on the 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, \(\frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\)) of different energy levels according to the principle of the superposition of states. Measuring such a superposed state can provide an energy eigenstate. However, based on the free Schrödinger equation for a single atom, it is impossible that an energy eigenstate \(\left|g\right.⟩\) (\(\left|e\right.⟩\)) evolves into the superposed pure state \(\frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\). Thus, the atom has to interact with at least one object or electromagnetic field. The AICQS evolves into an entangled state, and the atom stays in a mixed state, not a pure state. The above approximations may be the only way to obtain \(\frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\). Thus, there is no such strict pure state \(\frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\) and maintaining this state still requires the exchange of energy with at least one object or electromagnetic field.

The state \(\frac{1}{\sqrt{2}}(\left|g\right.⟩+\left|e\right.⟩)\) 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 splitter or double slit, to achieve the superposed state of two or more paths.

In Haroche et al.’s Fig. 2 [3], after an atom leaves the coherent light field beam splitter (\(BS1\)), the atom passes along two paths simultaneously. The quantum phase difference \(\theta\) of the two path states can be tuned. A second 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 splitters are exaggerated. Thus, this part of the experimental apparatus is really small, and thus, the interactions between the atom inside and the two 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:

$$\frac{1}{\sqrt{2}}(\left|g\right.⟩\left|1\right.⟩\left|{\gamma }_{g}\right.⟩+{e}^{i\theta }\left|e\right.⟩\left|2\right.⟩\left|{\gamma }_{e}\right.⟩)$$

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where \(\left|1\right.⟩\) and \(\left|2\right.⟩\) are the path states of the atom, and their phase difference \(\theta\) can be tuned. The atom stays in a mixed state. At output \(D\) of \(BS2\), the path \(\left|g\right.⟩\left|1\right.⟩\) remains unchanged and \(\left|e\right.⟩\left|2\right.⟩\) becomes \(\left|g\right.⟩\left|1\right.⟩\). The interference pattern at output \(D\) means that the original states \(\left|g\right.⟩\left|1\right.⟩\) and \(\left|e\right.⟩\left|2\right.⟩\) in Eq. (3) 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 \(\left|g\right.⟩\left|1\right.⟩\) and \(\left|e\right.⟩\left|2\right.⟩\) increases as \(N\) increases. Thus, the coherence is still dependent on \(BS1\) (see **Supplementary Section S2**). This differs from Dirac’s viewpoint that each photon interferes only with itself [31]. 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 **Supplementary Section S2**. In **Supplementary Section s S3 and S4**, we provide explanations of the quantum tunneling effect and other quantum phenomena.

## Double-Slit Experiment

Recently, some diatomic molecules have been simulated as double slits [22–25]. 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 splitters have two separate inputs, \(a\) and \(b\) (Fig. 1). A particle does not enter a splitter from two inputs in wave form. It clearly enters the 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 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 splitter, for example, the research conducted by Sadana et al. [32], 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,

$$|{\phi }_{1}⟩|{S}_{10}⟩+|{\phi }_{2}⟩|{S}_{20}⟩+|{\phi }_{3}⟩|{S}_{30}⟩$$

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before the particle passes through the double slit (omitting the probability amplitudes in some cases in this paper). The particle path states are \(|{\phi }_{1}⟩\) approaching slit 1, \(|{\phi }_{2}⟩\) approaching slit 2, and \(|{\phi }_{3}⟩\) when blocked by the double slit. The state of the macroscopic double-slit object remains approximately unchanged, \(|{S}_{10}⟩\approx |{S}_{20}⟩\approx |{S}_{30}⟩\). Thus, the entangled state approximates the product state \((|{\phi }_{1}⟩+|{\phi }_{2}⟩+|{\phi }_{3}⟩)|{S}_{10}⟩\). The particle is approximately in the coherent superposed pure state \(|{\phi }_{1}⟩+|{\phi }_{2}⟩+|{\phi }_{3}⟩\).

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 **Supplementary Section S5**. 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 \(|{\phi }_{1}⟩\) or \(|{\phi }_{2}⟩\) at the slit after being measured. This is similar to a particle entering one input of a 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 \(|{\phi }_{10}⟩\) or \(|{\phi }_{20}⟩\), the particle state can be described as a mixed state, the density operator form (omitting probability) of which is

$$|{\phi }_{10}⟩\left.⟨{\phi }_{10}\right|+|{\phi }_{20}⟩\left.⟨{\phi }_{20}\right|$$

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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 \(|{\phi }_{10}⟩|{S}_{10}⟩\) or \(|{\phi }_{20}⟩|{S}_{20}⟩\). According to Feynman’s path integrals [33], 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:

\(|{\phi }_{10}⟩|{S}_{10}⟩\to {\alpha }_{1}|{\psi }_{11}⟩|{S}_{11}⟩+{\alpha }_{2}|{\psi }_{12}⟩|{S}_{12}⟩\) ,\(|{\phi }_{20}⟩|{S}_{20}⟩\to {\beta }_{1}|{\psi }_{21}⟩|{S}_{21}⟩+{\beta }_{2}|{\psi }_{22}⟩|{S}_{22}⟩\)(6)

where \(|{\psi }_{11}⟩\) and \(|{\psi }_{12}⟩\) 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 double-slit object, respectively. This is similar to a particle exiting from the two outputs of a splitter. When a particle has entered slit 1 and has just exited from the two slits, \({\left|{\alpha }_{1}\right|}^{2}\gg {\left|{\alpha }_{2}\right|}^{2}\). 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 \(|{S}_{11}⟩\approx |{S}_{12}⟩\) and \(|{S}_{21}⟩\approx |{S}_{22}⟩\). Thus, the two entangled states approximate two product states. The particle is approximately in one of the two coherent superposed pure states, i.e., \(\left({\alpha }_{1}|{\psi }_{11}⟩+{\alpha }_{2}|{\psi }_{12}⟩\right)\equiv |{\varPhi }_{1}⟩\) and \({\beta }_{1}|{\psi }_{21}⟩+{\beta }_{2}|{\psi }_{22}⟩\equiv |{\varPhi }_{2}⟩\), or the particle state can be expressed as the mixed state \(|{\varPhi }_{1}⟩\left.⟨{\varPhi }_{1}\right|+|{\varPhi }_{2}⟩\left.⟨{\varPhi }_{2}\right|\). Two interference patterns can be obtained from \(\left|{\varPhi }_{1}\right(t)⟩\) and \(\left|{\varPhi }_{2}\right(t)⟩\), which are similar to those corresponding to the two inputs in a 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. Eq. (6) 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 three interference patterns but also which-path information with a high probability. The incident particles are spin-1/2 neutrons. Single neutrons with an initial spin state \(|\to \rangle\) 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 \(|\uparrow \rangle\) (\(|\downarrow \rangle\)). Wu [34] 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 \(\left|\uparrow \right.⟩\left|u \right.⟩+\left|\downarrow \right.⟩\left|d \right.⟩\), where \(|u \rangle\) and \(|d \rangle\) are the up and down path states, respectively. When the two path states meet at a position, there is no interference because \(⟨\uparrow |\downarrow ⟩=0\), *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. 2). 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 above analysis of the double-slit experiment, 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,

$$\left|\uparrow \right.⟩\left|u\right.⟩\left.⟨\uparrow \right|\left.⟨u\right|+\left|\downarrow \right.⟩\left|d\right.⟩\left.⟨\downarrow \right|\left.⟨d\right|$$

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If it is in the up slit, then the initial state of the AICQS of the neutron and the double-slit object is in the product state \(\left|\uparrow \right.⟩\left|u \right.⟩|{S}_{10}⟩\). From the up slit to the screen, the interaction drives the AICQS state to evolve into the entangled state\(({\alpha }_{1}\left|{u}_{11}\right.⟩|{S}_{11}⟩+{\alpha }_{2}\left|{u}_{12}\right.⟩|{S}_{12}⟩)\left|\uparrow \right.⟩\), 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, \(|{S}_{11}⟩\approx |{S}_{12}⟩\),

$$({\alpha }_{1}\left|{u}_{11}\right.⟩|{S}_{11}⟩+{\alpha }_{2}\left|{u}_{12}\right.⟩|{S}_{12}⟩)\left|\uparrow \right.⟩\approx ({\alpha }_{1}\left|{u}_{11}\right.⟩+{\alpha }_{2}\left|{u}_{12}\right.⟩)\left|\uparrow \right.⟩|{S}_{11}⟩$$

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and the neutron stays in the approximately coherent superposed pure state \(({\alpha }_{1}\left|{u}_{11}\right.⟩+{\alpha }_{2}\left|{u}_{12}\right.⟩)\left|\uparrow \right.⟩\). The neutrons in this state reach the screen and form an interference pattern. A similar phenomenon occurs for the down slit.

We recommend setting up a movable slit 3 on the screen to replace the usual screen (Fig. 2). The distance between the screen and the double slit can also be changed. First, when the distance is fixed and slit 3 is at some position on the screen, a detector behind slit 3 counts the neutrons per unit time. Second, slit 3 is moved by a step-size equal to or smaller than the width of the slit, and the counting is repeated; thus, the distribution of the neutrons or the interference pattern can be obtained.

Our way of obtaining which-path information is different from the delayed-choice [35, 36] and post-selected [32] methods, which use groups of two or more particles and the coincident measurement. However, many particles are not counted in the coincident measurement, which may result in a lack of information. We can obtain which-path information **after** obtaining the neutrons distribution (interference pattern) behind slit 3 without entanglement and coincident measurement. S-G2 is placed behind slit 3, and the direction of the gradient of the magnetic field \({B}_{2}\) is opposite to that of \({B}_{1}\). Assuming that the force is \({F}_{1}=-\mu \frac{\partial {B}_{1}}{\partial z}>0\), the direction in which \({F}_{1}\) acts upon the spin-up neutrons is up, so \({F}_{2}=-\mu \frac{\partial {B}_{2}}{\partial z}<0\). The direction in which \({F}_{2}\) acts upon the spin-up neutrons is down, and vice versa for spin-down neutrons. If \(\left|\frac{\partial {B}_{2}}{\partial z}\right|>\left|\frac{\partial {B}_{1}}{\partial z}\right|\), then the spin-up and spin-down neutrons are easily counted separately. The addition of the spin-up and spin-down neutrons is close to the number of neutrons counted just behind slit 3. By changing the position of slit 3 and repeating the measurement, the two interference patterns corresponding to the spin-up and spin-down neutrons can be obtained. Because 99% of the detected spin-up (down) neutrons come from the up (down) slit, not only the interference pattern but also which-path information with a large probability can be obtained. The addition of the two interference patterns produces a pattern close to the usual interference pattern. If slit 3 is near the double slit, \({\left|{\alpha }_{1}\right|}^{2}\gg {\left|{\alpha }_{2}\right|}^{2}\) and \({\left|{\beta }_{2}\right|}^{2}\gg {\left|{\beta }_{1}\right|}^{2}\), then the interference pattern of the neutrons that pass 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. [37]. 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 [31, 38] 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” [39] or the “zeroth axiom” [40], 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 3 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 the superposition of states, these equations have led us to believe that the states are strict. Thus, neglecting these approximations is the main reason inducing the counterintuitive explanations of some quantum phenomena.