科学与数学scottaaronson.com
量子力学作为概率的推广（2007）
Quantum mechanics as a generalization of probability (2007)
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导读
这是 Scott Aaronson 著名量子计算讲义的第 9 讲：把量子力学理解为经典概率论的一种推广——把非负概率换成可带负号（复数）的振幅，再通过测量取模平方。文章论证这种推广在数学上非常自然，甚至维多利亚时代的数学家本可以推导出量子力学。
为什么经久不衰：它用最少的数学把量子力学还原为'带负号的概率'，为程序员和数学背景的读者提供了几十年不过时的直觉入口。
HN 评论精选
更准确的表述是量子力学是 L² 空间而非 L¹ 空间中的概率论
我对标题有点不同看法：与其说'允许负号的概率'，不如说是'把概率从 L¹ 空间换成 L² 空间'。两个连续概率分布之间的距离通常是 ∫|p1-p2|dx，而 L¹ 是个奇怪的空间，没有'两个向量夹角'这类概念。量子力学改用 ∫|p1-p2|²dx，一旦引入平方，内积、夹角等几何结构就立刻出现了。所以我认为量子力学就是 L² 中的概率论。
yummyfajitas查看英文原文
I'd disagree with the title of this a bit. Rather than describing it as "probability to allow minus signs", I'd describe it as "probability in L^2 instead of L^1". The author actually discusses this a bit towards the end. It makes sense to ask what is the distance between two continuous probability distributions. It's given by: \int | p1(x) - p2(x) | dx L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird space, and does not admit concepts like "what is the angle between two vectors". Quantum mechanics changes this to: \int |p1(x) - p2(x)|^2 dx Functions like this are called L^2. Once you put the square in, you can immediately derive a lot of geometry, inner products, angles between vectors, etc. So I'd argue that QM is probability in L^2.
量子力学本可更早被推导出，教学方式应与时俱进、更早普及这些思想
根据文末脚注，这建立在 Hardy 几年前的工作之上——他证明维多利亚时代的数学家本可以轻松把量子力学作为当时概率论的推广推导出来。这让我想起 Penrose 的《通向实在之路》。近几十年我们进步太快，常常忘了回头重新梳理教学方法、简化讲授、让这些思想更早被更多人理解，这很可惜。
richardjordan查看英文原文
Per the footnotes at the bottom this builds on the work of Hardy a few years ago, who demonstrated that QM could easily have been derived by Victorian era mathematicians as a generalization of probability theory available at the time. While not everything in the book is perfect, it also reminds me of Penrose's Road to Reality, which painstakingly takes the question "what is a number?" to build up a sophisticated understanding of what that means and then explains how this impacts Physics in a very real way. We've made so many advances in recent decades we sometimes forget to step back and reformulate our teaching methodologies to incorporate what we know, simplify the teaching, and make the ideas more accessible earlier, so we can put more minds to work on extending them. That's a shame.
L² 视角澄清了很多困惑，但 Born 概率的物理含义仍难有直觉
我很喜欢这个对量子的描述。Aaronson 链接的论文《Quantum Theory from Five Reasonable Axioms》也很值得一读。'L² 中的概率'为我澄清了很多困惑，不过我对 Born 概率到底是关于什么的概率仍缺乏直觉——如果接受多世界诠释，似乎是你'发现自己处在该结果发生的宇宙中'的概率，但这听起来也很奇怪。
Totient查看英文原文
I love this description of quantum. The paper Aaronson links to, "Quantum Theory from Five Reasonable Axioms" (http://arxiv.org/abs/quant-ph/0101012), is also a great read. The "probability in L^2" cleared up a lot of confusion for me, although I still have a very poor intuition for what the Born probabilities are probabilities of. If you believe the MWI story, it seems like it's the probability you will "find yourself in the universe where this outcome happened" but even that sounds odd to me.
量子力学教学太抽象，应紧扣具体物理系统和实验来建立直觉
我本科修过两门量子力学课。我最大的不满是量子力学的讲授悖论式地从不与物理挂钩。量子物理本质上是线性代数加统计，再扩展到复数，但讲起来就是又一门数学课，毫无物理。测量态、应用 Hadamard 门、在特征基下分解——这些抽象而无意义。一个具有某波函数的粒子看起来什么样？演化有多快，纳秒、秒还是小时？什么才算一个测量的实例？有太多能帮助直觉的具体问题从未被讨论。我的教学思路是把它连接到具体物理系统，在实验和数学之间来回探索，甚至模拟这个系统——我也确实试着做过模拟。
karpathy查看英文原文
I took 2 courses on Quantum Mechanics during my undergrad and I'm taking the Coursera course now as well for fun. My biggest quibble with how QM is presented is that it is, paradoxically, never tied into physics. Quantum Physics is really a Linear Algebra + Statistics but with extension to complex numbers. In my experience it is just another math course and there is no Physics anywhere. There's always talk of measuring states, applying Hadamard gates and writing down their decomposition in the eigenbasis but it's abstract and meaningless. What does a particle with some particular wave function look like? As it evolves in time it "smears" out according to the time evolution equations, but how fast is the process? Does it occur on scale of nanoseconds? Seconds? Hours? How is it suspended, or operated on? What exactly is an example of a measurement? There are so many tangible questions that would help with intuition, but are never addressed. This would be my approach to teaching quantum mechanics. Connect it to a concrete physical system, explore in detail going back and forth between experiment and math. And best, maybe even simulate the system somehow. Some time ago I made an effort to try to simulate what I learned and this was the result (as an example): http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp I'ts ugly and it's probably wrong, but it's tangible and the best I could do because finding this kind of Quantum Mechanics, as opposed to a lot of talk about measuring things is very hard.
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