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?
 A: 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:

*

*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)

*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.
A: 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.
A: 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!).
A: 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.
A: 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.
