# Why is Schrödinger's cat in a superposition and not a mixture if you model decay with Fermi's golden rule?

I am teaching quantum information for undergraduate math students and as a perspective I thought it would be cool for them to discuss Schrödinger's cat a bit.

More precisely I'd like to come up with some explaination of how the cat gets to the superposition: \begin{align*} \frac{\vert {\text{no decay, cat alive} \rangle } + \vert {\text{decay, cat dead} \rangle }}{\sqrt{2}}. \end{align*}

If I look at a standard explanation of radioactive decay using Fermi's golden rule the decay happens if you have a pertubation $$H'$$ which enables tunneling to a lower energy eigenstate. If you are in an eigenstate of a Hamiltonian then you stay in an eigenstate.

But doesn't that mean that the cat does not end up in a superposition, but it will be in a statistical mixture between alive and dead? Do you have some explanation of how one could achieve the superposition?

• You are missing the point of the thought experiment. The question is to illustrate the conceptual difficulty with why and how state vector collapse happens. Why is there ever a mixed state instead of a coherent superposition?
– Buzz
Feb 22, 2021 at 22:51
• The description of the cat as a coherent superposition of dead and alive is wrong. It should be a very large density matrix involving all possible "dead" and "live" coherent states of the system. The statistical interpretation then says that this represents an ensemble of dead and live cats. Feb 22, 2021 at 23:46
• In reality it is, remember this thought experiment was devised to explain the issues with superposition states. Notions of decoherence, etc... were not yet developed Feb 23, 2021 at 0:25

I think that using Fermi's golden rule here is going a bit too far. If you look at a standard derivation of the rule, you will see that, roughly speaking, they calculate a transition amplitude $$\langle f| e^{-iHt} |i\rangle$$ (equivalently, they calculate the time evolution of the initial state $$|i\rangle$$). Then they take the magnitude squared, $$\left|\langle f| e^{-iHt} |i\rangle\right|^2$$ and differentiate this with respect to time to get the transition rate. But already in taking the magnitude squared, you are throwing away your information about the full quantum state, only caring about the probabilities of measurement results in a single basis. This effectively gives you a statistical mixture.

In order to get the Schrödinger's cat scenario, you just need to make sure your Hamiltonian and time are chosen such that $$e^{-iHt} |\text{no decay, cat alive}\rangle = \frac{|\text{not decayed, cat alive}\rangle + |\text{decayed, cat dead}\rangle}{\sqrt{2}}$$, for example, if $$U = e^{-iHt}$$ is a Hadamard gate or a 90-degree rotation gate. Or you could even model the situation as two time evolutions, one $$U_1$$ that puts the particle in $$\frac{|\text{not decayed}\rangle + |\text{decayed}\rangle}{\sqrt{2}}$$, i.e. the whole system in $$\frac{|\text{not decayed}\rangle + |\text{decayed}\rangle}{\sqrt{2}} \otimes |\text{alive}\rangle$$, and another $$U_2$$ that entangles the particle with the cat, i.e. a CNOT gate "kill the cat if there was a decay" (this also necessarily resurrects the cat if it was already dead, which could be a good starting point to discuss unitarity!).

You can make sure your $$e^{-iHt}$$ is a rotation gate $$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$$ if you choose the Hamiltonian $$H = -\sigma_y$$ (the Pauli matrix) and the duration $$t = \pi/4$$. Or if you want $$e^{-iHt}$$ to be a Hadamard gate, you can choose $$H = \mathrm{HG} - 2$$, a Hadamard gate minus two times the identity matrix, and $$t = \pi/2$$ (thanks Mathematica!).

• The description of the cat as a coherent superposition of dead and alive is wrong. Feb 23, 2021 at 11:15
• How does this adress Schroedingers cat or the decay of an atom. It seems like you are changing mid-sentence to a generic 2-state system, where you choose the time-evolution such that it rotates your state by $90^\circ$. But this is a physical system. You have to derive the time-evolution from the Hamiltionian. (And the problem is, that the setup cannot be described by quantum mechanics) Feb 23, 2021 at 12:01
• Yes, I am taking the cat as just a colourful illustration of a two-level system; I think that was in the spirit of the question? The issue about spontaneous decay, while valid, sounds a little bit beside the point to me... Whether $H$ is a semiclassical QM Hamiltonian or a QFT one shouldn't change the fact that its time evolution can take your initial state into a superposition of states you are interested in, right? Feb 23, 2021 at 19:18
• As I understand it, the cat paradox is about whether superpositions can become macroscopic, which is still a contentious thing (at least the many-worlds people believe they can...). So yes, perhaps best to add a disclaimer that cat superpositions are not universally accepted physics. Feb 23, 2021 at 19:21
• Or am I missing something more fundamental? Why can this setup not be described by this sort of quantum mechanics? Feb 23, 2021 at 19:28

As I see it, Schrödingers cat is really just a metaphor for a state being in a superposition, not something where it would make actual sense to describe in quantum mechanical terms. I think this because:

1. Spontanous decay is not actually part of quantum mechanics. As you said, you would need a perturbation, as in induced decay, or insert it per hand. A superposition of eigenstates only ever appears after a measurement at a fixed time (e.g. Stern-Gerlach or a photon passing through a polarizer)
2. A cat is really not a quantum mechanical object. The idea of superposition is related to the ability to change basis. Something that is in an eigenstate in one basis is in a superposition in another basis (think again of a photon with polarization). A cat is either dead or alive. There is no change of basis to accomodate both at once (except for zombie, maybe). The same holds for a spontaneous emission. It either happens or not. There is no superposition of an atom before and after emission of a photon.

Let me know if I misunderstood anything or there are questions!

Edit:
To clarify my point: Neither Schrödingers cat nor the setup for Fermi's goldon rule are good examples of a superposition in QM. If OP wants to teach it, it is important to know why exactly it is problematic.

• @Cream If spontaneous emission couldn't be described in QM, that would be a massive hole in QM – but I haven't seen any pop-science articles saying “physics can't explain radioactivity!”, so I doubt that. Feb 22, 2021 at 22:00
• @wizzwizz4 The Wikipedia article on spontaneous emission explains that quite well. Spontaneous emission cannot be explained by the Schrödinger equation, it needs the quantization of the photon. Fermi's goldon rule is essentially a trick that allows you to compute a transition rate by inserting an artificial perturbation into the Hamiltonian,. but cannot yield decay rates. It can describe induced emission, though. Feb 22, 2021 at 23:15
• "A cat is either dead or alive. There is no change of basis to accommodate both at once" - That's the whole point of the thought experiment, isn't it? He was trying to show that the idea of "particles in a superposition" is a ridiculous concept. He was arguing the Copenhagen Interpretation must be wrong. Feb 22, 2021 at 23:56
• @BlueRaja - Danny Pflughoeft Right, I forgot that thanks for pointing it out! Makes sense that it's a poor analogy, then. Feb 23, 2021 at 7:07
• @BlueRaja-DannyPflughoeft "He was trying to show that the idea of "particles in a superposition" is a ridiculous concept." I don't think that's quite right. My understanding is that he was trying to demonstrate that the need for an observer to collapse the waveform is ridiculous. It seems to have failed, at least in with the general public. Wigner's friend and the derivatives are more effective, I think because we don't have to argue about whether a cat is an observer (it is BTW, which is obvious if you've known any cat.) Feb 23, 2021 at 16:36

Fermi Golden rule is not suitable for describing a coherent superposition, as it implies decoherence and localization in the final state (i.e., the wave function collapse). This is usually hidden the density of the final states, which si either written explicitly or appears when the delta-function is integrated.

• Fermi's golden rule is just the long wave length approximation to first order perturbation theory for an electric dipole transition. It does not imply decoherence and collapse. It doesn't say anything about the phase space for example. Feb 23, 2021 at 11:17
• @my2cts As I said: it is hidden. Books deriving Fermi Golden rule never discuss decoherence and collapse, because it is introduced at an elementary level. But they are there, if you think about it: either via the density of states or via trancating time or otherwise - there is always a transition to continuum limit. WIthout that, e.g., for a two-level system, one obtains coherent Rabi oscillations. Feb 23, 2021 at 11:21
• The Fermi golden rule is just a matrix element, nothing more. It doesn't say that photons are actually emitted. Feb 23, 2021 at 11:28
• @my2cts It is quite more than just a matrix element, and there are strong assumptions in deriving and applying it. That the amplitude is squared and that we talk about "transitions" (by definition withd efined final state) are good hints. Feb 23, 2021 at 12:30
• I repeat, the asssumptions are: minimal coupling (more a fact than an assumption) , large wavelength, first order perturbation theory. That's all. Feb 23, 2021 at 12:39

The description of the cat as a coherent superposition of dead and alive is indeed wrong. The 'statistical mixture' that you are looking for is von Neumann's density matrix formalism for incoherent superposition. The state is a very large density matrix involving all possible "dead" and "live" coherent states of the system. What does such a state mean? Following the statistical interpretation of quantum mechanics this represents an ensemble of dead and live cats. No single cat is half alive and half dead.

He will always be superimposed as opposed to imposed because the anomaly of his existence does not allow for permanency of his existence. Were Schrödinger's cat a permanent fixture such as a tree with roots, its tangibility would be rooted in the exactness of its existence.

This is why the scenario is more than plausible. The transitory nature of time shifting paradoxes lend themselves to their own fluidity.

• If I repeat Schrödinger's cat but with a tree and a chainsaw instead of a cat and poison, what then? Feb 23, 2021 at 8:33
• This is word salad, but very poetic. Feb 23, 2021 at 9:25