**...which Schwinger never used in the class, as Erwin noticed...**

Willie Soon sent me a link to a cute 3-weeks-old article,

For Some Reason, These Quantum Mechanics Toys Didn’t Catch On (IEEE.org)Allison Marsh described metallic objects that are probably the only objects in the world's museums related to quantum mechanics!

It's a collection of 21 aluminum cubes – some of them have a 1-inch side (boxes to teach quantum mechanics to newborns), others have a 4-inch side (those are for the kindergartens) – which are labeled by symbols such as \(\sigma_x\) and other operators on a two-dimensional Hilbert space.

With a time machine, Julian Schwinger could have used these toys while teaching the course 251a Quantum Mechanics at Harvard because someone created them for him after he stopped teaching the course. I just checked my mailbox – I wasn't ever teaching this particular course (Hau and Halperin did it when I was there) but I taught a similar undergraduate course 143b and if these cubes were easy to access, I would have used them.

When I saw the article sent by Willie Soon for the first time, I was unimpressed. Why? Because I thought that the boxes were simply dull, passive pieces of a metal that you may use instead of

*writing*the operators on the two-dimensional Hilbert space such as \(\ket{+}\bra{-}\) and their linear combinations.

However, at some moment, I finally read the article somewhat carefully and saw what was going on. These aluminum boxes were actually

*active*and manipulated the 2D states in a way that was equivalent to quantum mechanics on the 2-dimensional Hilbert space.

How did it work? Polarized light was generally going through the boxes and there were Polaroid polarizers inside. The photon's polarization states \(\ket x\) and \(\ket y\) generate another 2-dimensional Hilbert space. Photons are really sent through these toys. Well, many photons do the same thing but the mathematics remains the same. Instead of measuring probabilities, you may measure the intensity of the light at the end. That's the simplest example of a classical limit.

There is a technical paper in Physics in Perspective that describes these toys.

These toys never became popular. Sadly, we might say. But even though he hasn't used them, either, they represent Schwinger's school's different approach to teaching of quantum mechanics. He really wanted some quantum mechanical intuition to be the primary thing that babies (such as Harvard students) learn that is related to quantum mechanics. Conventional courses don't paint quantum mechanics as the "first insight in a chain" but as a "second" or "later insight in a chain" – as an extension of classical physics.

If people were starting to learn quantum mechanics from scratch, as a standalone theory, they could have a better chance to genuinely understand the 20th century theory. But they're always learning quantum mechanics as if it were something "derived from classical physics" or "built on classical physics", some "classical physics plus" – and in this picture, they almost universally have a psychological problem with the "plus" because they're encouraged in their view to interpret classical physics as a "golden standard" that better theories should "match". But quantum mechanics doesn't depend on any classical theory and it surpasses and partly invalidates, not only matches, the very framework of classical physics.

Homework exercise for you: Can you figure out why you can't easily build a simple quantum computer out of slightly larger aluminum boxes of Schwinger's type? ;-)

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