Schrödinger’s Cat

Schrödinger’s cat amplifies a microscopic quantum alternative into macroscopically different records. Its purpose is to expose what a universally applied quantum state description implies, not to claim that ordinary cats exhibit observable interference or that human looking is a special physical operation. The historical treatment discusses the original context; this page analyzes the logical structure.

Helpful background. Measurement Problem explains the universal-linearity tension.

Amplification creates correlated alternatives

In a simplified model, a microscopic trigger state becomes correlated with two distinct apparatus-and-cat records:

$$|\Psi\rangle=\alpha|u\rangle|L\rangle +\beta|d\rangle|D\rangle.$$

The labels $L,D$ denote coarse macroscopic alternatives, not a biological model. Their orthogonality is an idealization of distinguishable records. A realistic living system occupies an enormous subspace with internal thermal and environmental degrees of freedom; a pair of pure record vectors isolates the argument without pretending to describe those details.

Linearity transfers microscopic superposition to the joint state. The outcome problem asks how a theory represents the definite macroscopic record experienced in one run. Adding the word “macroscopic” does not supply a collapse mechanism.

Superposition is distinguished by coherence

A coherent state and an ignorance mixture can give the same probabilities for the coarse alternatives. To distinguish them, one needs an observable sensitive to the cross term between branches. In an ideal two-branch subspace,

$$W=|u,L\rangle\langle d,D|+|d,D\rangle\langle u,L|$$

has expectation $2\operatorname{Re}(\alpha^*\beta)$ in the coherent state and zero in the corresponding mixture. Measuring only whether the trigger fired, or whether a macroscopic record is present, cannot reveal this cross term.

An environmental record suppresses coherence accessible to the reduced system, but unitary theory stores the correlations in a larger state. A claim of a macroscopic superposition therefore needs more than a bimodal histogram: it needs interference or a suitable coherence witness, plus an account of the separation and size of the alternatives. Macroscopic Superpositions develops that experimental distinction.

Why opening the box is not the central event

Air molecules, thermal radiation and apparatus couplings can monitor the record before a person reads it. The physical dynamics do not wait for visual attention. Reading an already stable record conditions the observer’s predictions; whether an objective reduction occurred earlier depends on the proposed physical theory.

The thought experiment also does not force a contradiction in operational probability calculations. Standard instruments predict correlated records consistently. Its challenge concerns the completeness and interpretation of that description. Collapse, additional-variable and relative-state accounts give different answers to that challenge and should be compared on their explicit assumptions.

Exercise: a phase cannot be read from populations

Compare $(|u,L\rangle+|d,D\rangle)/\sqrt2$ with $(|u,L\rangle-|d,D\rangle)/\sqrt2$. What measurement distinguishes them in the ideal model, and why does counting the two record labels not suffice?

Solution

The two vectors are orthogonal eigenstates of $W$ with eigenvalues $+1$ and $-1$. Both assign probability $1/2$ to each coarse record. Population measurements therefore erase the distinction in their statistics. A coherent joint measurement of the branch subspace would reveal it, but implementing that operation on a thermally open macroscopic system is a separate physical obstacle.

References

• Schrödinger, Erwin, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812, 823–828, 844–849 (1935), doi:10.1007/BF01491891 — original argument, first installment.
• Schlosshauer, Maximilian, Decoherence and the Quantum-to-Classical Transition, Springer (2007), doi:10.1007/978-3-540-35775-9 — environmental records and interference.