Mathematical Field Notes

Archive for the ‘Quantum mechanics’ Category

Resonances

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How could you “detect” a new subatomic particle, given that it’s so small you can’t see it and (often) so short-lived that you’d miss it even if you didn’t blink?

Let’s suppose you have a process which you think might involve your tiny, short-lived new friends. Maybe you suspect that if you smash protons together, you might produce some particles… which decay into particles…. which decay into the particles you’re looking for… which then themselves decay almost immediately into something else, something that is more stable and easier to detect (something like a pair of photons). How could you check that you were right?

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Written by Jonny Evans

October 30, 2016 at 12:02 pm

Why Schrödinger’s equation?

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“Why this equation?”

I recently overheard someone ask this about Schrödinger’s equation. The answer they received was, for me, unsatisfying. “Because it agrees with experiment.” Of course, that answers perfectly why the equation was adopted by future generations of physicists and indeed the calculation of the spectrum of atomic hydrogen from the energy eigenvalues of the Schrödinger operator is one of the most convincing and wholesome computations a young physicist can do. But the question that was left unanswered, the question I believe was being asked, was: “Why did Schrödinger write this equation down? Why not something else?” I don’t believe for a second that Schrödinger sat down with an array of different equations and worked out what each of them predicted about hydrogen before he found the one that fit…

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Written by Jonny Evans

November 5, 2012 at 10:18 pm

Quantum computers: Grover’s algorithm

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I’ve been meaning to get my head around the idea of a quantum computer for a while now and, since my mathematical energy is currently reduced to clearing out my ETH office, I thought I’d do some reading and find out more. I leafed through my dusty copy of Wikipedia [Wik], picked up Kitaev et al [KSV] from the library and turned to chapter two…

I think the easiest way to illustrate how a quantum computer differs (functionally) from a classical computer is by explaining an algorithm which only makes sense for quantum computers and which really outperforms a classical algorithm for the same task (that is the point of quantum computers, after all). The first algorithm they explain in [KSV] is called Grover’s algorithm and it performs the task of searching a database. Wikipedia [Wik] has a really nice exposition too, but I was initially confused by what they both call a “quantum oracle” (sounds like something from Star Trek TNG). I tried to explicitly avoid that in what I said below.

TLDR: Classically you have to look through all N elements of a database until you find the right one (so runtime increases linearly in N); Grover’s algorithm has a surprising runtime of order \sqrt{N} to do the same thing, using clever ideas from quantum mechanics.

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Written by Jonny Evans

August 13, 2012 at 10:13 pm