| 彭小富1，罗宇航2，朱 江3，华邦辉1，陈雪楠1，
连丹丹2，陈子维2，陈相松1,2∗.重复可读态、自发塌缩和量子/经典界限[J].海南师范大学学报自科版,2026,(1):1-12 |
| 重复可读态、自发塌缩和量子/经典界限 |
| Repeatedly Readable State, Spontaneous Collapse, and Quantum/Classical Boundary |
| |
| DOI：10.12051/j.issn.1674-4942.2024.04.001 |
| 中文关键词: 量子测量问题 自发塌缩模型 量子/经典界限 重复可读性 |
| 英文关键词: quantum measurement problem; spontaneous collapse model; quantum/classical boundary; repeatable readability |
| 基金项目:国家自然科学基金项目（11535005） |
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| 中文摘要: |
| 本文提出了一个模型来确定量子/经典界限。该模型引入了态叠加的自发塌缩：
d
d t
ρij = −
i
ℏ
[H, ρ]ij −
ρij
τij
。与其他塌缩模型不同，本模型的塌缩尺度 τij 不包含任何普适参数，
而是由 2 个态 |i⟩ 和 |j⟩ 本身来指定。如果原则上每个态都是可重复读取的 [通常通过量子非破坏
（QND）测量]，则 τij 就是区分这 2 个态所需的测量时间，并且该塌缩自发发生，无需任何实际监
控。若非如此，则 τij = ∞，意味着态之间将不会塌缩并永远处于叠加，这种情况会在一个态无法
被重复读取或者在特定情况下无法区分 2 个态（例如在 Rabi 振荡中）时出现。详细分析表明，对
于“阱中的薛定谔猫”，如果 ED ≫ 4πℏc，则 |here⟩ 和 |there⟩ 的叠加是被禁止的；如果 ED ≤ 4πℏc，
则叠加被允许，其中 D 是阱分离的距离，E 是阱的能隙且可以用 Mv2 估算。例如，假设 D ∼ 10
µm 和 v ∼ 102 m/s，则量子/经典边界位于 M ∼ 2 × 103 GeV/c
2 处。该模型还限制“自由的薛定谔
猫”在 pθd ≥ 8ℏ 时不显示干涉条纹，其中 p = Mv，θ 是 2 条轨迹所形成的角度，d 是狭缝宽度。
对于典型值 v ∼ 103 m/s、θ ∼ 10−5
rad 和 d ∼ 10 µm，量子/经典边界位于 M ∼ 5 GeV/c
2 处。令
人印象深刻的是，用氦原子进行物质波干涉的实验恰好处在该边缘。另外，该模型对无质量光子
的相干长度没有限制，因此迈克尔逊干涉仪的臂可以任意长。本文提出的自发塌缩可能发生在孤
立系统中，并且与环境相互作用引起的退相干效应平行存在。 |
| 英文摘要: |
| This paper proposes a model to determine the quantum/classical boundary. The model introduces a spontaneous
collapse of state superposition: d ρij
d t
= −
i
ℏ
[H, ρ]ij −
ρij
τij
. Different from other collapse models, the collapsing scale τij here does not contain a universal parameter, but is specified by the two states |i⟩ and |j⟩. If each state is in principle
repeatedly readable [typically by a quantum non-demolition (QND) measurement], then τij is the needed measuring time
to discriminate the two states, and the collapse occurs spontaneously without any actual monitoring. Otherwise, τij = ∞,
which means no collapse and everlasting superposition. This happens if one state is not repeatedly readable, or if the two
states cannot possibly be discriminated in a particular circumstance (for example, in the Rabi oscillation). Detailed analysis
shows that for a "trapped Schrödinger’s cat", the superposition of |here⟩ and |there⟩ is forbidden if ED ≫ 4πℏc, and
allowed if ED ≤ 4πℏc, where D is the trap separation and E is the energy gap, which can be estimated with Mv2
. For
example, if D ∼ 10 µm and v ∼ 102 m/s, then the quantum/classical boundary is at M ∼ 2 × 103 GeV/c
2
. The model
also constrains a "free Schrödinger’s cat" to display double-slit interference if pθd ≥ 8ℏ, where p = Mv, θ is the angle
spanned by the two trajectories, and d is the slit width. For typical values v ∼ 103 m/s, θ ∼ 10−5
rad, and d ∼ 10 µm,
the quantum/classical boundary is at M ∼ 5 GeV/c
2
, which, impressively, is just marginal for the performed experiments
with Helium. In contrast, this model sets no limit on the coherent length of massless photon, thus the arm of a Michelson
interferometer can be arbitrarily long. The spontaneous collapse which we propose can occur for an isolated system, and
parallels the decoherence induced by interaction with the environment. |
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