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Preparing a particle in a superposition or a wave packet of eigenstates of a physical quantity is to let it interact with a large object. The composite system composed of the particle and the large object evolves into an entangled state. When the state of the large object is considered to be approximately unchanged, the entangled state can be approximately considered as a product state, and then the particle is prepared in an approximate superposed state. We consider the Schrodinger equation for a composite system with interactions between subsystems as a fundamental postulate and a single particle’s Schrodinger equation must be approximately obtained from it. We argue that superposition of states exists only in composite systems. Interaction exchanging some quantities between subsystems makes conservation laws strictly hold, and no wave packet of a free particle yields. With this point, we can also understand the double-slit experiment and the tunnel phenomenon.

According to the measurement postulate in quantum mechanics [

For a pair of incompatible observables for a free particle, the corresponding operators do not commute with each other and have not shared eigenstates (or eigenfunctions). An eigenstate of one observable seems naturally to be the SE of the other since it is easily expanded in other eigenstates mathematically, but the general way in physics is to let it interact with other large object (LO, explained in Section 2), for example, a particle interacting with a double-slit, a photon interacting with a beam splitter, an electron interacting with a Stern-Gerlach apparatus. According to the viewpoint of Einstein et al. [

The example of quantities momentum

We ponder how a state of a free particle can return to its initial SE of a physical quantity after the SE has collapsed into an eigenstate following a measurement. To our knowledge, there is little information available in literature about this. Here, we discuss this issue. We think when the state of the LO is considered in some approximation to be unchanged, the entangled state can be approximately considered as a product state, and then the particle is prepared in an approximate superposed state. We explain this in

In Section 2, we introduce two examples to explain the way to prepare a superposition of atomic ground and an excited states starting from one of them. Conservation laws are explained in the Section. In Section 3, we consider the Schrödinger equation for a composite system as a fundamental postulate and approximately obtain a single particle’s Schrödinger equation. Section 4 gives out our understandings about some quantum phenomena. Conclusions are in the Section 5.

Haroche and his colleagues [

beam splitter. Specifically, this entangled state is expressed by

of photons for the two coherent states satisfy

The entangled state can be approximately considered as a product state and thereby the approximate SE or the

simplest wave packet of the levels,

In quantum optics, when an atom initially in the ground state

means the atom exchanges a photon with the single-mode field by the INT. When n is large enough, then

The entangled state can also be approximately considered as a product state and the atom is said to be in an approximate SE and is not free either.

If the atomic state collapses into the state

energies are different from each other. Therefore, energy conservation does not hold for this atomic process and the state is only considered to carry statistical information. When we consider that the entangled state of the CS collapses into the state

We consider the INT between a particle and a LO (strong field, a large or macro apparatus) is a reasonable and helpful way to understand why particles may be in SEs. This is familiar in two instances; one is momentum exchange, considered by Bohr [

We consider the Schrödinger equation for a CS with INTs between subsystems as a fundamental postulate. There always exists the INT between a considered particle and other object. A free particle does not exist. Single particle’s Schrödinger equation must be approximately obtained from that of a CS composed of the particle and other LO with INT between them. The Schrödinger equation of the CS is

where

(potential) energy belonging to the CS. The CS state

We can use kinetic energy eigenstates of the LO to expand the CS state

When an electron in a spin state

where

A similar inference can be said for a photon polarized state or a photon path state. When a photon interacts with a beam splitter and the state of the beam splitter is approximately considered to be unchanged, its state may be resolved approximately into a superposition of polarized states or path states.

The most typical of quantum phenomena is considered to be the double-slit interference of material particles. Feynman [

where

In the symmetric case,

where two coefficients (1/2) express a particle passing through each slit with a half probability. Since

evolve approximately into the same superposed state

the particle source or double-slit perpendicular to the slits and the line of the source to the “center” of slits, we guess that the density operator should be as

where

In [

the single superposition

and is used to explain double-slit interference patterns in general, because the latter cannot be used to explain well the interference pattern (not appearing one bright fringe in center, but appearing two bright fringes symmetrically) near double-slit.

If the path of a particle passing through a slit is measured, the initial path state of a particle passing through slit 1 (2) is not

In recent papers [

The tunnel phenomenon can be explained as following. The barrier potential is actually the INT energy between a particle and a large object, the CS evolves into an entangled state, the energy of a particle may surpass with a probability the barrier potential due to INT exchanging energy between the particle and the large object. Therefore we can understand that the particle passes over with a part of the probability, the particle maintaining the incident direction, the barrier naturally, and does not penetrate the barrier.

The principle of superposition of states (one state may be an eigenstate or a SE) in quantum mechanics [

The SE may be considered as a core feature of the wave. If there is no INT, a free particle will not be in a SE, and hence will not display wave-like properties. A de Broglie wave-vector and frequency can only be considered as parameters corresponding to a definite momentum and energy of a free particle, respectively. In the original concept of wave-particle duality in quantum mechanics, the wave and particle properties of a particle are intrinsic and the wave property is not related to any INT. Because INTs exist everywhere, our proposed concept¾that the wave property of a particle and the principle of superposition of states are related to INT¾may be a better choice than that encountered in conventional quantum mechanics.

I thank Shou-Yong Pei, Jian Zou, Bin Shao, Feng Wang, Xiu-San Xing, Jun-Gang Li, Xiang-Dong Zhang, Yu-Gui Yao, Jin-Fang Cai, Rui Wang, Chang-Hong Lu, Wen-Yong Su and Fan Yang for enlightening discussions and comments, Hao Wei, Li-Fan Ying, Gui-Qin Li and Yong-Jun Lu for help, and Pei-Zhu Ding and Shou-Fu Pan for encouragement. The work was partly supported by the National Natural Science Foundation of China (Grant Nos. 11075013, and 11375025).