The popular culture picture of quantum mechanics is highly inaccurate. The word “quantum” has become synonymous with weird and wacky physics. “In quantum physics, a cat can be dead and alive at the same time!”, people say. In this article I will attempt to explain QM in a way that people can understand as well as shed light on the infamous cat, while utilising minimal notation and mathematics.
The famous “Schrödinger’s cat” thought experiment is misunderstood. For those who don’t know, the idea is that there is a cat in a box in which there is a radioactive isotope which, when it decays, releases some sort of instrument which (let’s say) puts the cat to sleep1. There are two possible outcomes once we make a measurement (open the box): the cat is awake or the cat is asleep. The probability of each state occurring depends on the probability of the decay of the isotope.
The probabilities of decay are encoded in what is called a superposition of (eigen-) states. The traditional (Copenhagen2) interpretation is that once a measurement on a quantum system is performed, the system collapses into one of these states. This begs us to ask the question of what happens before the measurement. And the answer to that has traditionally been that the system is in all of those states at once.
Now, those stuck in abstractions might accept this type of description for a quantum system since it typically occurs at a scale that we are not able to perceive with our senses. But what if we somehow link a quantum system with an everyday phenomenon? What results is a seeming ‘paradox’.
In our thought experiment, the state of the cat is causally linked with the decay of the isotope. Therefore, it too must acquire these quantum properties3. If the isotope is in a superposition of decayed and undecayed states, the cat must also be in a superposition of awake and asleep! This is obviously absurd when treated with common sense! Unfortunately, this is a yet unsolved problem in the foundations of QM. Later in this post I will briefly discuss the opponents of the traditional view.
Now, for the math. I know I promised minimal jargon, but I really should explain what some of it means such that you get a viable understanding. I talked all this time about superpositions of states. But what is this really?
A state is a mathematical object (a vector in a separable Hilbert space), that contains all the information about the system4. It is denoted: \(\left\lvert \psi \right\rangle\). For instance, a state that represents that the cat is awake (and no other information) may be written as: \(\left\lvert \text{😺} \right\rangle\), conversely: \(\left\lvert \text{😿} \right\rangle\).
We get the information “out” by (self-adjoint) operators which correspond to obvervables or, loosely-speaking, measurements. These are functions (linear maps) that take as input a given state and output another. We write this as: \(\hat{O} \left\lvert \psi \right\rangle\)5.
There are certain special states, known as eigenstates or characteristic states of a given operator which yield: \(\hat{O} \left\lvert \psi \right\rangle = \lambda \left\lvert \psi \right\rangle\), for some \(\lambda \in \mathbb{R}\) (which is just a real number, for those of you who do not know mathematical notation)6. These are associated with the possible states that the observable gets out. \(\lambda\) are the corresponding eigenvalues; these are the values that you can actually get upon measurement. Let’s say that our observable was checking whether the cat was awake.
To illustrate this, let us assume that the cat is awake and put into a box where the probability of the isotope decay is 0%, then the state must be \(\left\lvert \psi \right\rangle = \left\lvert 😺 \right\rangle\), i.e. \(\hat{A} \left\lvert \psi \right\rangle = 1 \cdot \left\lvert 😺 \right\rangle\), conversely if it is 100%, then \(\hat{A} \left\lvert \psi \right\rangle = 0 \cdot \left\lvert 😿 \right\rangle = \left\lvert 0 \right\rangle\). These are eigenstates of the awake operator, \(\hat{A}\) with eigenvalues \(0\), \(1\). I have chosen \(1\) to mean awake and \(0\) to mean asleep.
Consider an eigenstate of an operator \(\left\lvert \psi \right\rangle\) and some arbitrary state (which could be an eigenstate) \(\left\lvert \phi \right\rangle\). The probability of the \(\left\lvert \phi \right\rangle\) state of collapsing into state \(\left\lvert \psi \right\rangle\) is given by7:
\[P_{\phi \leftrightarrow \psi} = \left\lvert \left\langle \psi \mid \phi \right\rangle \right\rvert^2\]
This is known as the Born rule8.
For instance, the probability of \(\left\lvert 😿 \right\rangle\) going to state \(\left\lvert 😺 \right\rangle\) is obviously zero9. Therefore: \[\left\lvert \left\langle 😺 \mid 😿 \right\rangle \right\rvert^2 = P_{😿 \leftrightarrow 😺} \overset{!}{=} 0\]
On the other hand: \[\left\lvert \left\langle 😺 \mid 😺 \right\rangle \right\rvert^2 = \left\lvert \left\langle 😿 \mid 😿 \right\rangle \right\rvert^2 = 1\]
Now that we have that lingo out of the way, we can finally tackle superposition. Superposition is simply the addition10 of an arbitrary number of (normalised11) eigenstates of a given operator, like: \[\left\lvert \phi \right\rangle = c_1 \left\lvert \psi_1 \right\rangle + c_2 \left\lvert \psi_2 \right\rangle\]
It might seem scary, but the only thing that this says is that the probability to get state \(\left\lvert \psi_1 \right\rangle\) is \(\left\lvert c_1 \right\rvert^2\) when one does the measurement/observation that corresponds to the given operator. This is assuming that \(\left\lvert \left\langle \phi \mid \phi \right\rangle \right\rvert^2 = 1\); i.e. that \(\left\lvert \phi \right\rangle\) is also normalised.
In the case of the cat, we can for example have: \[\left\lvert \psi \right\rangle = \frac{1}{\sqrt{2}}\left\lvert 😺 \right\rangle + \frac{1}{\sqrt{2}} \left\lvert 😿 \right\rangle\]
The \(\frac{1}{\sqrt{2}}\) factors encode the probability is 50% to get each state. Furthermore, we retain \(\left\lvert \left\langle \psi \mid \psi \right\rangle \right\rvert^2 = 1\).
Let us make the measurement! Is the cat awake? \[\hat{A} \left\lvert \psi \right\rangle = \hat{A} \frac{1}{\sqrt{2}}\left\lvert 😺 \right\rangle + \hat{A} \frac{1}{\sqrt{2}} \left\lvert 😿 \right\rangle = \frac{1}{\sqrt{2}} \left\lvert 😺 \right\rangle\]
Now, let us check the probability of this new state of collapsing into the alive state: \[\left\lvert \left\langle 😺 \mid \frac{1}{\sqrt{2}} 😺 \right\rangle \right\rvert^2 = \left\lvert \frac{1}{\sqrt{2}} \left\langle 😺 \mid 😺 \right\rangle \right\rvert^2 = \frac{1}{2} \cdot 1 = \frac{1}{2}\]
We see that the probability is 50%, as we defined!
This is why people say the cat is ‘alive and dead’ at the same time. Our theory permits combining physical states into a new physical state which can be measured to be either of the combined states.
So, at this point, you should have understood well the basics of quantum mechanics. The only issue is that everything that we have done makes no sense to our everyday intuition. What does it mean to be in a superposition? We do not know.
This is all to say that quantum mechanics, going from the deterministic theory of Newton to the inherently probabilistic theory of Bohr & company, is a huge philosophical leap.
In fact, people like Einstein and others have taken issue with this:
[…] These functions are only supposed to determine in a mathematical way the probabilities of encountering those objects in a particular place or in a particular state of motion, if we make a measurement. This conception is logically unexceptionable, and has led to important successes. […] I cannot help confessing that I myself accord to this interpretation no more than a transitory significance. I still believe in the possibility of giving a model of reality, a theory, that is to say, which shall represent events themselves and not merely the probability of their occurrence. […]
(Albert Einstein, 1934)
To paraphrase: Einstein’s point is that the description of the probabilities of the system is not the same as the description of the system itself. Einstein raises other issues with quantum mechanics in the EPR paradox12 which I will not cover in this article.
Fundamentally, the collapse of the wave-function has not been understood. Perhaps I will write another article about this, but this whole area of study is very speculative even almost 100 years later…
Hopefully, this article has shed some light on quantum mechanics through the lens of Schrödinger’s cat and explained the fundamentals of its formalism. I will admit that it is hard to get into at first, but a lot of it is not as unintuitve as people say it is.
Albert Einstein (1934). On the Method of Theoretical Physics, [The University of Chicago Press, Philosophy of Science Association].
For the more historical description see Erwin Schrödinger – Schrödinger’s Cat Thought Experiment (Britannica) ↩︎
Note that I am foregoing talking about the fact that the cat ‘checking’ whether the isotope is decayed or not is also a measurement, as in so doing I am avoiding talking about entanglement. For pedagogical reasons, I think this approach is better. ↩︎
This is the position of the so-called Copenhagen interpretation; which considers the wave-function or state of the system as the complete description thereof and measurement/observation is given a statistical interpretation. That is, we cannot predict what value we will get (unless it is an eigenstate of the operator), but only the probability thereof or the expected value ↩︎
Note that we suppress the notation \(\hat{O}(\left\lvert \psi \right\rangle)\) precisely because of this linearity ↩︎
It is worth noting that the reason that these are real is because the operators are self-adjoint. Let \(v\) be an eigenvector of \(\hat{O}\) with eigenvalue \(\lambda\). Since \(\hat{O}\) is self-adjoint, we have \(\forall u, w \in \mathcal{H} : \left\langle \hat{O} u \mid w \right\rangle = \left\langle u \mid \hat{O} w \right\rangle\). In particular, we have \(\left\langle v \mid \hat{O}v \right\rangle = \left\langle \hat{O}v \mid v \right\rangle\). Therefore, we must have \(\left\langle \lambda v \mid v \right\rangle = \left\langle v \mid \lambda v \right\rangle\) by definition of \(v\) being an eigenvector with eigenvalue \(\lambda\). Therefore, by properties of the (Hermitian) inner product, we have \(\lambda^{\ast} \left\langle v \mid v \right\rangle = \lambda \left\langle v \mid v \right\rangle\). Since \(v\) is an eigenvector, it cannot be zero by definition, so the Hermitian product must be \(\left\langle v \mid v \right\rangle \gt 0\). Therefore, we get that \(\lambda = \lambda^{\ast}\) which means that \(\lambda \in \mathbb{R}\). Q.E.D ↩︎
See Feynman Lectures on Physics Vol. III, Ch. 3 – Probability Amplitudes if you would like to understand this better. ↩︎
Some information about it can be found in: Born Rule - an overview ↩︎
In our special case of the cat, the awake state cannot collapse into the asleep state either. We are barring, for the sake of pedagogy, any other tampering. ↩︎
Technically linear combination ↩︎
Normalised simply means that \(\left\lvert \left\langle \psi_i \mid \psi_i \right\rangle \right\rvert^2 = 1\) ↩︎
See Paradox of Einstein, Podolsky, and Rosen (Britannica) ↩︎