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Schrödinger's Cat and the Measurement Problem

Quantum mechanics is weird. Very weird. According to quantum theory, particles can be at two positions at the same time. Other particles tend to behave like waves, whereas waves, in turn, sometimes act like particles. Still other particles suddenly pop into existence, out of "nothingness" — an actual creatio ex nihilo

Not surprisingly then, even some of the greatest scientists, such as Niels Bohr and Richard Feynman, claimed that they did not understand quantum theory. Einstein believed that some of the implications of quantum mechanics were so strange, and counter-intuitive, he fiercely rejected them till the end of his life. Schrödinger finally, after devoting most of his life to the development of quantum mechanics, admitted: "I don't like it, and I'm sorry I ever had anything to do with it."

"Everyone who is not shocked by quantum theory has not understood it" — Niels Bohr

But in the 100 years since then, most scientists have learned to live with the mind bending, deeply disturbing claims of quantum mechanics. We somehow grew accustomed to the thought that the quantum world (or the world of the very small) is just hugely different from our familiar world (the world of the normally sized).

There are other problems with quantum theory though, which aren't simply a result of the theory's intrinsic weirdness. These problems aren't a consequence of our inability to imagine the whereabouts of the quantum world. They are as real as any other scientific problem, and in desperate need of a solution.

One of the most crucial problems of quantum mechanics is known as the measurement problem and will form the subject matter of this post. The measurement problem, in essence, boils down to the inevitable clash between the two dynamical laws of quantum mechanics: the linear, deterministic evolution, as described by Schrödinger's equation, and the non-linear, indeterministic (probabilistic) evolution, as described by the collapse postulate (see further).

There's probably no better way to illustrate this problem than by looking at Erwin Schrödinger's tasteless thought experiment of the cat in a box. Now, before delving into the details of Schrödinger's Gedankenexperiment, I first have to say something about radioactive atoms and their quantum behaviour.

 

Quantum superpositions

Half LifeRadioactive atoms are, by their very nature, highly unstable and they tend to disintegrate (decay) into smaller, more stable, fragments. Bi-212, for example, a radioactive isotope of bismuth, has a half-life of 60 minutes (60.55 minutes to be exact), which means that approximately half of the Bi-212 atoms will have decayed after a period of 60 minutes.

Now, imagine observing a single, radioactive Bi-212 atom which started in its undecayed form. After an hour, there will be a 50% chance that the atom has decayed, and another 50% chance that it is still intact. Let us denote the decayed and the undecayed form by the kets \left|decayed\right\rangle and \left|undecayed\right\rangle. In that case, the evolution of the Bi-212 atom during that one hour period can be represented as follows:

\begin{equation}\left|undecayed\right\rangle\rightarrow\frac{1}{\sqrt{2}}\left(\left|undecayed\right\rangle+\left|decayed\right\rangle\right).\label{A}\end{equation}

(Don't mind the \frac{1}{\sqrt{2}} in front of the brackets. I'll come back to this in a moment.) The above evolution is governed by the Schrödinger equation. One of the equation's most important properties is that it is perfectly deterministic. That is, given the initial state of the atom at time t=0 (i.e. \left|undecayed\right\rangle), one can deterministically predict (that is, with absolute certainty) what its final state will be at time t=60 (i.e. \left|undecayed\right\rangle + \left|decayed\right\rangle). This final state is also called a quantum superposition as the atom is in a superposition of being both decayed and undecayed at the same time.

 

Schrödinger's cat

Now, let's return to Schrödinger's devilish experiment. Imagine a cat in a box, along with one single Bi-212 atom, a Geiger counter, a release mechanism with hammer, and a vial of prussic acid (hydrogen cyanide, HCN). The box is sealed, and left untouched for exactly one hour.

The idea is as simple as it is cruel: as long as the atom remains intact, nothing happens, and the cat lives. But as soon as the Bi-212 atom decays, the Geiger counter will detect the radioactive radiation and set the release mechanism in march, causing the hammer to fall and shatter the vial into pieces. The hydrogen cyanide gas is released, and the cat dies a gruesome death.

Schrödinger's Cat 3

Now, after one hour, the state of the atom has evolved to the right hand side of Eq. \eqref{A} — a superposition of being both decayed and undecayed. As a consequence, the Geiger counter, too, will find itself in a superposition of having both detected and not detected the decay, which in turn leaves the hammer in a superposition of having fallen and not fallen. This causes the vial to be in a superposition of being smashed to pieces and still intact, which finally leads to the strange and quite absurd result of the cat being in a superposition of being both dead and alive at the same time!

Schrödinger's Cat 1

This is clearly nonsense! After all, every sensible human being knows that when one would open the box, only one of two possibilities will occur:

1. The atom has not yet decayed, and the cat is still alive;
2. The atom did decay, and the cat is dead.

After one hour, both possibilities will occur with an equal chance of 50%. But, in any case, one will always see the cat either alive or dead, never both alive and dead. This, in essence, is the measurement problem of quantum mechanics. Somehow, quantum mechanics seems to force us into believing that cats can find themselves in zombie superpositions of being simultaneously dead and alive, which is in flat contradiction with our common sense. So what is going wrong here?

Before we continue, here's how Schrödinger first coined his paradox in 1935:

"A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat); in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The \psi-function [i.e. state] of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

Schrödinger devised this hypothetical experiment to illustrate some of the flaws and limits of the Copenhagen interpretation of quantum mechanics (see further).

 

Quantum entanglement

If we want to gain a deeper insight into the measurement problem, we will need some formalism. Now, don't click away! The only concepts we need are those of quantum superpositions (defined above) and quantum entanglement, both of which lie at the basis of the paradoxical outcome described above. Now, as the Geiger counter interacts with the Bi-212 atom, its state gets entangled with the state of the atom. This is how it works within the mathematical formalism of quantum mechanics:

Let the ket \left|ready\right\rangle represent the initial, ready-to-measure, state of the Geiger counter. The moment it detects radiation, an audible "click" is produced, and the \left|ready\right\rangle state evolves to the ket \left|``click. Alternatively, if no radiation is measured, no sound is produced, and the \left|ready\right\rangle state evolves to the ket \left|no\;``click. The measurement process of a decaying Bi-212 atom by the Geiger counter can then be written as:

\begin{equation}\left|decayed\right\rangle\left|ready\right\rangle\rightarrow\left|decayed\right\rangle\left|``click"\right\rangle,\label{B}\end{equation}

and similarly:

\begin{equation}\left|undecayed\right\rangle\left|ready\right\rangle\rightarrow\left|undecayed\right\rangle\left|no\;``click"\right\rangle.\label{C}\end{equation}

Now consider what would happen if the Geiger counter would interact with the superposed state on the RHS of Eq. \eqref{A}. The pre-measurement state is as follows:

\begin{equation}\frac{1}{\sqrt{2}}\left(\left|undecayed\right\rangle+\left|decayed\right\rangle\right)\left|ready\right\rangle.\end{equation}

Working out the brackets, we obtain:

\begin{equation}\frac{1}{\sqrt{2}}\left|undecayed\right\rangle\left|ready\right\rangle+\frac{1}{\sqrt{2}}\left|decayed\right\rangle\left|ready\right\rangle.\end{equation}

Due to the linearity of the Schrödinger equation, each of these two terms will evolve according to the above equations \eqref{B} and \eqref{C}, and the resulting, final, state becomes:

\begin{equation}\frac{1}{\sqrt{2}}\left|undecayed\right\rangle\left|no\;``click"\right\rangle+\frac{1}{\sqrt{2}}\left|decayed\right\rangle\left|``click"\right\rangle.\end{equation}

The first term represents the situation where the Bi-212 atom did not decay, and the Geiger counter did not record anything, whereas the second term represents the situation in which the Bi-212 atom did decay, and the Geiger counter produced an audible click.

In a similar vein, the release mechanism, the hammer, the vial, and the cat will all become entangled with the atom and the Geiger counter. The initial state at t=0 could then be written as:

\begin{equation}\left|undecayed\right\rangle_{atom}\left|ready\right\rangle_{Geiger}\left|ready\right\rangle_{hammer}\left|ready\right\rangle_{vial}\left|ready\right\rangle_{cat}.\end{equation}

Following Eq. \eqref{A}, the final state at t=60 becomes:

\begin{equation}\frac{1}{\sqrt{2}}\left|undecayed\right\rangle_{atom}\left|no\;``click"\right\rangle_{Geiger}\left|not\;fallen\right\rangle_{hammer}\left|intact\right\rangle_{vial}\left|alive\right\rangle_{cat}\\ +\frac{1}{\sqrt{2}}\left|decayed\right\rangle_{atom}\left|``click"\right\rangle_{Geiger}\left|fallen\right\rangle_{hammer}\left|shattered\right\rangle_{vial}\left|dead\right\rangle_{cat}.\label{D}\end{equation}
 

The collapse postulate

Once again, we end up with a non-classical superposition of two classical states — one where the cat is still alive, and the other where the cat has died. Since one never observes such superposed states, the idea was introduced by Dirac and von Neumann that the quantum superposition in Eq. \eqref{D} would collapse onto one of the two classical terms whenever an observation was made (that is, whenever one opens the box). This, in short, is the collapse postulate — an essential ingredient of the (orthodox) Copenhagen interpretation of quantum mechanics.

Quantum collapses happen with a probability \mathfrak{P} given by the square of the coefficient (or probability amplitude) that precedes the term onto which the state collapses. Since \left|\frac{1}{\sqrt{2}}\right|^2=\frac{1}{2}, each term will occur with equal probability \mathfrak{P}=\frac{1}{2}, and this is exactly what we predicted above for the two possible determinate outcomes.

Notice that the collapse postulate is non-linear and indeterministic (that is, probabilistic). The typically random behaviour of quantum events thus finds its origin in this second dynamical law of quantum theory. As long as no observation is made, the system's state evolves in a deterministic fashion according to Schrödinger's equation (the first dynamical law). It is the act of observation which forces the system to take up one of the two possible classical outcomes.

 

The measurement problem

At first sight, the above "solution" in terms of quantum collapses seems to correctly account for our determinate observations. But it also raises many deep questions which have haunted scientists and philosophers alike for the last century:

1. First of all, there is the peculiar role of the conscious observer in quantum mechanics. After all, it seems that superpositions can exist as long as we do not observe them. A cat can be in a superposition of being dead and alive, as long as we don't look. But as Einstein said in a letter to Schrödinger: "Nobody really doubts that the presence or absence of the cat is something independent of the act of observation."

In a similar vein, Einstein once asked the question: "Is the moon really there when we don't observe it?" Although the Copernican revolution distanced itself from any belief in a geocentric, human-centered universe, it seems the quantum revolution has placed humans back at the center of everything.

Moon 2

"I like to think that the moon is there even if I am not looking at it." — Albert Einstein

2. This naturally brings us to the second question: What counts as an observation/measurement? Or, to put it differently, when exactly does a quantum superposition collapse? Is the measurement by the Geiger counter enough to collapse the superposition of the Bi-212 atom in Eq. \eqref{A}? Or does the collapse of the wavefunction only occur when the cat becomes entangled with the entire system as in Eq. \eqref{D}?

Perhaps quantum collapses only occur at the level of the human observer? In which case one could ask ourselves the question whether the state collapses as soon as the first photon reaches our retina, or only at the point where we become consciously aware of the state of the cat.

3. A final and closely related problem concerns the microscopic to macroscopic transition, or the transition from the quantum to the classical. Once again, it is not clear where the boundary lies.

 

Interpreting the quantum world

Over the years, many different interpretations of quantum mechanics have been proposed in an attempt to answer some of the above-mentioned questions. Besides 1. the Copenhagen interpretation, there are 2. the hidden variable interpretations by de Broglie and Bohm, 3. the dynamical collapse interpretations such as the GRW interpretation by Ghirardi, Rimini and Weber, and 4. the no-collapse interpretations which reject von Neumann's collapse postulate such as Everett's many-worlds interpretation or the many-minds interpretation.

One of the craziest of interpretations is definitely the many-worlds interpretation which posits that upon opening the box, the universe splits in two worlds corresponding to the two terms in Eq. \eqref{D}. In the first world, the Bi-212 atom remained intact and the cat lives; in the second world, the atom decayed and the cat is dead.

Schrödinger's Cat 4

In this interpretation, everything that is physically possible, also happens. But every particular outcome happens in a different world (or branch), which necessarily leads to an infinitude of worlds, or parallel universes. However, although all branches are equally real, they cannot interfere with each other. So an observer in one world cannot know what is happening in the other worlds.

But these are stories for another post! Indeed, one of the aims of this blog is to pass all of the above-mentioned interpretations in review during future posts, along with a careful and critical examination of their strengths and weaknesses. So stay tuned!

Please, do not hesitate to comment on the above post by leaving a reply below. Feel free also to subscribe to thelifeofpsi.com by entering your email in the top-right corner in order to receive notifications of new posts by email.

Pieter Thyssen

Whereas his left brain was trained as a theoretical scientist, his right brain prefers the piano. At work, Pieter builds time machines (on paper) and loves to dabble in the history and philosophy of science. He often gets stuck in another dimension, contemplating time travel and parallel universes, or thinking about ways to save Schrödinger's cat (maybe). He explores the world on foot, and takes life one cup of (Arabica) coffee at a time. Follow him on Twitter @PieterThyssen or at thelifeofpsi.com. You can reach Pieter via email at pieterthyssen@gmail.com.

  8 comments for “Schrödinger's Cat and the Measurement Problem

  1. Dave Tett
    September 12, 2013 at 2:03 pm

    Great stuff Pieter, clear concise and some brilliant illustrations!

    • September 12, 2013 at 4:51 pm

      Thanks Dave! I'm really happy you're the first person to comment on my blog.

      By the way, I'm already preparing my second post, which will be about nothing less than ... Mr. Bertlmann's socks 😉

      PS: By subscribing to The Life of Psi, you will be automatically notified about future posts by email.

  2. September 16, 2013 at 8:00 am

    Leuk, Pieter! :-)

    Raph

    • Lukas Vandersteene
      December 16, 2013 at 4:54 pm

      hmmm ...
      In Latin I would say:

      "nihil est tam absurdum vel ridiculum ut philosophus quidam non dixerit"

      free translation:
      Er is niets zo ongerijmd of belachelijk of het is nog niet beweerd door een filosoof.

      Nothing is so absurd or ridiculous to not find a philosopher who claimed it.

      or in the original :
      "Nihil tam absurde dici potest quod non dicatur ab aliquo philosophorum"
      Cicero, in: De Divinatione 2.58.

  3. Tori
    May 6, 2015 at 5:59 pm

    This is absolutely fantastic- this whole article explains things in a tangible way and I am quite fond of your writing style. I am getting a Schrodinger's cat tattoo- the dead cat with its corresponding equation on my left hip and the cat that is alive, along with its corresponding equation on my left hip. I am very fond of this concept but you took it above and beyond and I shall be continually fascinated by everything physics. Thank you for taking the time to write this!

  4. Kofi Agyemang
    December 6, 2015 at 1:31 am

    Hi, I am replying to this article two years later but I was wondering if you could explain to me quantum teleportation and why it's not an example of "spooky action at a distance". Thanks

Comments are closed.