Last time I set the stage, the mathematical location for quantum mechanics, a complex vector space (Hilbert space) where the vectors represent quantum states. (A wave-function defines where the vector is in the space, but that’s a future topic.)
The next mile marker in the journey is the idea of a transformation of that space using operators. The topic is big enough to take two posts to cover in reasonable detail.
This first post introduces the idea of (linear) transformations.
Whether it’s to meet for dinner, attend a lecture, or play baseball, one of the first questions is “where?” Everything that takes place, takes place some place (and some time, but that’s another question).
Where quantum mechanics takes place is a challenging ontological issue, but the way we compute it is another matter. The math takes place in a complex vector space known as Hilbert space (“complex” here refers to the complex numbers, although the traditional sense does also apply a little bit).
Mathematically, a quantum state is a vector in Hilbert space.
I’d planned a different first post for May Mind Month, but a recent online conversation with JamesOfSeattle gave me two reasons to jump the gun a bit.
Firstly, my reply was getting long (what a surprise), and I thought a post would give me more elbow room (raising, obviously, the possibility of dualing posts). Secondly, I found the topic unusual enough to deserve its own thread.
Be advised this jumps into the middle of a conversation that may only be of interest to James and I. (But feel free to join in; the water’s fine.)
It’s Science Fiction Saturday, so today I want to consider a fairly common question a fan might encounter: “Science Fiction or Fantasy?” The implication is that one tends to exclude the other. In these polarized times, it can amount to a declaration of your tribe.
One problem is there’s a spectrum from hard SF to pure fantasy with everything in between. But let’s take them as two legitimate poles and consider the question in terms of configuration space. (See posts #1 and #2 if you need to catch up.)
I think you’ll see that using a space give us a new take on the question.
Last time we considered a cube-shaped room where we could indicate our opinion about Neapolitan ice cream with a single marker. That worked well because we were dealing with three flavors and the room has three dimensions: east-west, north-south, up-down.
Later I’ll explore other examples of a 3D “room” but while we’re talking ice cream, I want to give you an idea where this goes, I want to jump ahead for a moment and consider good old Baskin-Robbins, who famously featured “31 flavors!”
So now the question is, can we set a marker for all 31 flavors?
Have you ever had (or at least seen) Neapolitan ice cream? It’s the kind with chocolate, vanilla, and strawberry, usually as separate layers in one package. As a kid, I didn’t care for the strawberry. I loved the chocolate, and was fine with the vanilla (wouldn’t usually choose it, but don’t disdain it).
That’s just my take on it: one flavor liked, one not liked, and one that’s just okay. Someone else might have the same pattern with different flavors. Or love them all equally, or want just the strawberry. Some might not like ice cream at all — any combination is possible.
What if we wanted to describe our feeling about Neapolitan as a whole?