地獄篇 8

Posted on March 8, 2022 by rpflee

地球表面，就已經是一個地獄。

幸好，那不是一個永恆的地獄；有很多事情可以做，去改變這個地獄（，即使只是一點點）。

— Me@2011.10.08

— Me@2022-03-08

伏線驅動程式 1.1

Posted on March 8, 2022 by rpflee

這段改編自 2010 年 10 月 14 日的對話。

其實，根據剛才的討論內容，這個理論本來，可以叫作「種子靈感天線搞 gag 間書太極謀事在人成事在天可遇不可求有心栽花花不香無心插柳柳成蔭踏破鐵鞋無覓處得來全不費功夫原理」。

今年三月時，我和一位朋友，想為這個原理，起一個簡潔的名字，因為沒有的話，每次要提起它時，也十分麻煩。

途中猜想了很多可能的名稱。卒之找到一個，還未太滿意，但尚可接受的名字——種子論。

你要不斷「播種」，即是不斷地播「原因」，因為「種」，就是「原因」，因果鏈的起點。眾多已播之種中，有些會發芽，有些不人發芽，有些又會生長到中途就死了。只有極少數，會開花結果。你不能主宰，哪棵生哪棵死；你可以做的，就只有不斷播種。

然後，我上星期時，想到一個較佳的名字——伏線論。

不斷播種，就是不斷埋下伏線，即使你在事前，不知每條伏線，究竟是在伏什麼；甚至不知，究竟最終伏不伏到，任何東西。

時間的結構是這樣的，未來片面不全知，成果可預不可期。

我發覺人生，即使愛情以外的範疇，主要的演變都是，身不由己式，依靠伏線來驅動。例如，此時此境此地，我是來教你們物理，順便講道理。

我又是如何認識你們的呢？

…

— Me@2022-03-08 12:06:18 PM

2014

Posted on March 7, 2022 by rpflee

— Me@2022-03-07 01:12:19 PM

Ex 1.23 Fill in the details

Posted on March 6, 2022 by rpflee

Structure and Interpretation of Classical Mechanics

Show that the Lagrange equations for Lagrangian (1.97) are the same as the Lagrange equations for Lagrangian (1.95) with the substitution $\displaystyle{c(t) = l}$ , $\displaystyle{Dc(t) = D^2 c(t) = 0}$ .

~~~

[guess]

The Lagrange equation:

$\displaystyle{ \begin{aligned} D ( \partial_2 L \circ \Gamma[q]) - (\partial_1 L \circ \Gamma[q]) &= 0 \\ \end{aligned}}$

Eq. 1.97:

$\displaystyle{ L''(t,q,\dot q)}$

$\displaystyle{ = \sum_\alpha \frac{1}{2} m_\alpha \left( \partial_0 f_\alpha (t, q) + \partial_1 f_\alpha (t, q) \dot q \right)^2 - V(t, f(t,q,l)) }$

Eq. 1.95:

$\displaystyle{ L'(t;q,c,F; \dot q, \dot c, \dot F)}$

$\displaystyle{ = \sum_\alpha \frac{1}{2} m_\alpha \left( \partial_0 f_\alpha (t, q, c) + \partial_1 f_\alpha (t, q,c) \dot q + \partial_2 f_\alpha (t, q, c) \dot c \right)^2 }$

$\displaystyle{ - V(t, f(t,q,c)) - \sum_{\{ \alpha, \beta | \alpha < \beta, \alpha \leftrightarrow \beta \}} \frac{F_{\alpha \beta}}{2 l_{\alpha \beta}} \left[ c_{\alpha \beta}^2 - l_{\alpha \beta}^2 \right] }$

$\displaystyle{ \begin{aligned} &L'(t;q,c,F; \dot q, \dot c, \dot F) \\ &= L'(t;\{q_{\alpha i}\}, \{c_{\alpha \beta}\}, \{F_{\alpha \beta}\}; \{\dot q_{\alpha i}\}, \{\dot c_{\alpha \beta}\}, \{\dot F_{\alpha \beta}\}) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= \sum_\alpha \frac{1}{2} m_\alpha \left( \partial_0 f_\alpha (t;\{q_{\alpha i}\}, \{c_{\alpha \beta}\}) + \partial_1 f_\alpha (t;\{q_{\alpha i}\}, \{c_{\alpha \beta}\}) \dot q + \partial_2 f_\alpha (t;\{q_{\alpha i}\}, \{c_{\alpha \beta}\}) \dot c \right)^2 \\ &- V(t, f(t;\{q_{\alpha i}\}, \{c_{\alpha \beta}\})) - \sum_{\{ \alpha, \beta | \alpha < \beta, \alpha \leftrightarrow \beta \}} \frac{F_{\alpha \beta}}{2 l_{\alpha \beta}} \left[ c_{\alpha \beta}^2 - l_{\alpha \beta}^2 \right] \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= \sum_\alpha \frac{1}{2} m_\alpha \left( \frac{\partial f_\alpha}{\partial t} + \sum_i \frac{\partial f_\alpha}{\partial q_{\alpha i}} \dot q_{\alpha i} + \sum_\beta \frac{\partial f_\alpha}{\partial c_{\alpha \beta}} \dot c_{\alpha \beta} \right)^2 \\ &- V(t, f(t;\{q_{\alpha i}\}, \{c_{\alpha \beta}\})) - \sum_{\{ \alpha, \beta | \alpha < \beta, \alpha \leftrightarrow \beta \}} \frac{F_{\alpha \beta}}{2 l_{\alpha \beta}} \left[ c_{\alpha \beta}^2 - l_{\alpha \beta}^2 \right] \\ \end{aligned}}$

Consider the mass $\displaystyle{m_\alpha}$ :

$\displaystyle{ \begin{aligned} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot q_{\alpha i}} \right) - \frac{\partial L}{\partial q_{\alpha i}} &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot c_{\alpha \beta}} \right) - \frac{\partial L}{\partial c_{\alpha \beta}} &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &\frac{\partial L}{\partial q_{\alpha i}} \\ &= \frac{\partial}{\partial q_{\alpha i}} \left[ \sum_\alpha \frac{1}{2} m_\alpha \left( \frac{\partial f_\alpha}{\partial t} + \sum_j \frac{\partial f_\alpha}{\partial q_{\alpha j}} \dot q_{\alpha j} + \sum_\beta \frac{\partial f_\alpha}{\partial c_{\alpha \beta}} \dot c_{\alpha \beta} \right)^2 - V(t, f) \right] - 0 \\ &= \sum_\alpha \frac{1}{2} m_\alpha \frac{\partial}{\partial q_{\alpha i}} \left( G \right)^2 - \frac{\partial}{\partial q_{\alpha i}} V(t, f(t;\{q_{\alpha j}\}, \{c_{\alpha \beta}\})) \\ &= \sum_\alpha m_\alpha G \frac{\partial G}{\partial q_{\alpha i}} - \frac{\partial V}{\partial q_{\alpha i}} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} G &= \left( \frac{\partial f_\alpha}{\partial t} + \sum_j \frac{\partial f_\alpha}{\partial q_{\alpha j}} \dot q_{\alpha j} + \sum_\beta \frac{\partial f_\alpha}{\partial c_{\alpha \beta}} \dot c_{\alpha \beta} \right) \\ \frac{\partial G}{\partial q_{\alpha i}} &= \frac{\partial}{\partial q_{\alpha i}} \left( \frac{\partial f_\alpha}{\partial t} + \sum_j \frac{\partial f_\alpha}{\partial q_{\alpha j}} \dot q_{\alpha j} + \sum_\beta \frac{\partial f_\alpha}{\partial c_{\alpha \beta}} \dot c_{\alpha \beta} \right) \\ &= \left( \frac{\partial}{\partial q_{\alpha i}} \frac{\partial f_\alpha}{\partial t} + \sum_j \frac{\partial}{\partial q_{\alpha i}} \frac{\partial f_\alpha}{\partial q_{\alpha j}} \dot q_{\alpha j} + \sum_\beta \frac{\partial}{\partial q_{\alpha i}} \frac{\partial f_\alpha}{\partial c_{\alpha \beta}} \dot c_{\alpha \beta} \right) \\ \end{aligned}}$

…

[guess]

— Me@2022-03-06 05:24:18 PM

EPR paradox, 11.12

Posted on March 6, 2022 by rpflee

Measuring spin and polarization

…

In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup. It is possible to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin-up, while in the other setup it registers as spin-down. Thus, for the de Broglie–Bohm theory, the particle’s spin is not an intrinsic property of the particle; instead spin is, so to speak, in the wavefunction of the particle in relation to the particular device being used to measure the spin. This is an illustration of what is sometimes referred to as contextuality and is related to naive realism about operators. Interpretationally, measurement results are a deterministic property of the system and its environment, which includes information about the experimental setup including the context of co-measured observables; in no sense does the system itself possess the property being measured, as would have been the case in classical physics.

— Wikipedia on De Broglie–Bohm theory

2022.03.06 Sunday ACHK

預防離婚

Posted on March 6, 2022 by rpflee

有什麼事情，結婚後才可以做？

離婚。

不結婚，就沒有可能離婚。

— Me@2022-03-05 07:13:14 PM

單身悲，雙身慘；

兩害取其輕。

各人情況不同。

同一個人的不同時代，也未必一樣。

大方向是：

「好婚姻」勝過「無婚姻」；

而「無婚姻」卻勝過「壞婚姻」。

— Me@2022-03-06 10:33:05 AM

同一屋簷下

Posted on March 5, 2022 by rpflee

這段改編自 2021 年 12 月 8 日的對話。

準備結婚前，試一試做「同屋主」一兩個月，看看會不會「同住難」。同屋不同房，最為理想。

留意，我講的是「準備結婚前」，而不是「結婚前」，因為「結婚前」就經已太遲。

與另一半同住，比與原本家人同住，難度還要高一點。例如，你的哥哥不會擅自，改動你房中的物件，拋棄他覺得沒有用的東西。但是，你的潛在太太可能會。

如果注定「同住難」的話，「準備結婚前」就知道，會好一點。那樣，可以及早溝通，看看你倆究竟是「同住暫時難」，還是「同住永久難」。「暫時難」就立刻解決；「永久難」就馬上分手。

— Me@2022-03-05 07:13:14 PM

1990s, 16

Posted on March 5, 2022 by rpflee

— Me@2022-03-05 11:41:03 AM

2.11 Extra dimension and statistical mechanics

Posted on March 5, 2022 by rpflee

A First Course in String Theory

…

(a) Explicitly calculate $\displaystyle{Z(a, R)}$ in the very high temperature limit $\displaystyle{\left( \beta = \frac{1}{kT} \to 0\right)}$ .

…

~~~

$\displaystyle{ \begin{aligned} Z &= Z(a) \tilde{Z}(R) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} Z(a) &= \sum_{k=1}^\infty \exp\left[- \beta \left( \frac{\hbar^2}{2m} \right) \left(\frac{k \pi}{a} \right)^2 \right] \\ \tilde Z (R) &= 1 + 2 Z(R \pi) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \Delta k &= 1 \\ Z(a) &= \frac{1}{\beta} \sum_{\beta k = \beta, 2\beta, ...} \exp\left[- \frac{1}{\beta} \left( \frac{\hbar^2}{2m} \right) \left(\frac{\beta k \pi}{a} \right)^2 \right] \Delta (\beta k) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \int_{0}^{\infty } e^{-ax^{2}}\,dx &= \frac{1}{2} {\sqrt {\frac {\pi }{a}}} \\ \end{aligned}}$

When $\displaystyle{ \beta \to 0 }$ ,

$\displaystyle{ \begin{aligned} Z(a) &\approx \frac{1}{\beta} \int_{x = 0}^\infty e^{\left[- \frac{1}{\beta} \left( \frac{\hbar^2}{2m} \right) \left(\frac{x \pi}{a} \right)^2 \right]} dx = \sqrt{\frac{m a^2}{2 \beta \pi \hbar^2}} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \tilde Z (R) &\approx 1 + \sqrt{\frac{2 m R^2 \pi}{\beta \hbar^2}} \approx \sqrt{\frac{2 m R^2 \pi}{\beta \hbar^2}} = Z(2 \pi R) \\\\ \end{aligned}}$

$\displaystyle{ \begin{aligned} Z &\approx Z(a) Z(2 \pi R) \\ \end{aligned}}$

— Me@2022-03-04 07:12:49 PM

Bit software; qubit hardware

Posted on March 4, 2022 by rpflee

Quantum information is the information of the state of a quantum system.

— Wikipedia on Quantum information

$\displaystyle{\begin{aligned} |a|^2 + |b|^2 &= 1 \\ \end{aligned}}$

For a system with the superposition state

$\displaystyle{\begin{aligned} | \psi \rangle &= a~| \psi_L \rangle + b~| \psi_R \rangle \\ \end{aligned}}$ ,

quantum information is the information of the superposition coefficients $\displaystyle{a}$ and $\displaystyle{b}$ .

— Me@2022-03-03 07:11:10 PM

A quantum state is not a state. Instead, it is a property of a physical system. It is a statistical property of a variable of an experimental setup.

— Me@2022-02-20 06:44:32 AM

— Me@2022-03-03 07:53:09 PM

Quantum information does not exist in physical spacetime. Instead, it exists in the experiment-setup designer’s time. In this sense, it exists in meta-physical time.

It is not information that exists in a physical system. Instead, it is the information about the physical system.

— Me@2022-02-20 11:27:31 PM

Quantum computers are implemented by using qubits, encoding information in the system property (aka quantum state) itself. A qubit is stored in the property of the system, not just arrangements of particles of the system as in a classical media.

Classical information is stored in microscopic setups, such as the arrangements of microscopic particles, of a system.

Quantum information is stored in macroscopic setups, such as the magnetic field direction for maintaining an electron spin’s superposition state, of a system.

— Me@2022-02-20 11:30:37 PM

— Me@2022-03-03 10:29:53 PM

quantum information ~ a system property

conservation of quantum information ~ a property of those system properties

— Me@2022-02-20 8:13 AM

You can locate a piece of classical information in a physical system/information media.

You cannot locate a piece of quantum information in a physical system, because quantum information is stored in the statistical properties of that physical system, which includes objects and experimental processes.

You have to do a large number of identical experiments in order to extract those statistical patterns.

For example, for a fair dice, the numbers on its faces, its weight, etc. are classical information. However, the probability value of getting (such as) 2, which is $\displaystyle{\frac{1}{6}}$ , is quantum information; it does not exist in physical spacetime.

— Me@2022-02-20 11:46:40 PM

“State” and “property” have identical meanings except that:

State is physical. It exists in physical time. In other words, a system's state changes with time.
Property is mathematical. It is timeless. In other words, a system's property does not change. (If you insist on changing a system's property, that system will become, actually, another system.)

For example, “having two wheels” is a bicycle’s property; but the speed is a state, not a property of that bicycle.

— Me@2022-02-20 06:44:32 AM

.

state ~ the easiest-to-change property of a system

— Me@2022-02-21 08:52:34 PM

Classical information is stored in the states of a system (information media).

Quantum information is stored in the properties of a system.

— Me@2022-02-21 08:52:34 PM

A qubit might for instance physically be a photon in a linear optical quantum computer, an ion in a trapped ion quantum computer, or it might be a large collection of atoms as in a superconducting quantum computer. Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers.

— Wikipedia on Quantum information

Note that while a bit is a software object, a qubit is a physical object.

— Me@2022-02-23 05:19:02 PM

Constructicons, 2

Posted on March 3, 2022 by rpflee

For a combiner transformer, whose mind (of the component robots) is it?

~ Whose mind is a group of people, such as a company?

Not a fixed one, depending on what the consensus of that group (at that time) is, who the speaker (at that time) is, etc.

In reverse, one person is also a multi-mind.

— Me@2016-06-16 09:07:46 PM

天地創造

Posted on March 3, 2022 by rpflee

心海 4 | 心韻

始於

俊男美女千篇一律

成於

心靈戀情一篇千律

— Me@2022-03-03 01:09:58 PM

Turbo Pascal for Windows

Posted on March 2, 2022 by rpflee

A day in year 1995, somehow, I needed to take a ferry to Hong Kong Island. Before that, I went to the Star Computer Centre in Tsim Sha Tsui. I found a box of Turbo Pascal for Windows. More likely, I went there when I came back from Hong Kong Island.

WinWorld screenshot

More more likely, I went there and saw the box before going to Hong Kong Island; and then went there again to buy it when I was back.

I needed that for doing my programming homework. It cost me 299 HKD.

Throughout that school year, I did almost every exercise in my programming textbook, at the cost of my Additional Mathematics and Biology time. However, for my programming projects, I still needed to test them under Microsoft Pascal, because it was what my school computers ran.

Even during moving house in 2019, I was still not willing to throw this box away.

— Me@2022-03-02 05:20:54 PM

The particle-trajectory model

Posted on March 2, 2022 by rpflee

The 4 bugs, 1.13

The common quantum mechanics “paradoxes” are induced by 4 main misunderstandings.

…

4.1 Each particle always has a definite identity. Wrong.

identical particles

~ some particles are identical, except having different positions

~ some particle trajectories are indistinguishable

.

trajectory indistinguishability

~ particle identity is an approximate concept

— Me@2022-02-11 12:47:14 AM

physical definition

~ define microscopic events in terms of observable physical phenomena such as the change of readings of the measuring device

~ define unobservable events in terms of observable events

— Me@2022-01-31 08:33:01 AM

4.2 Each particle always has a trajectory. Wrong.

“Which trajectory has a particle travelled along” is a hindsight story.

“The electron at location x_1 at time t_1 and the electron at location x_2 at a later time t_2 are actually the same particle” is also a hindsight story.

These kinds of post hoc stories do not exist when some definitions of trajectories are missing in the overall definition of the experiment setup. In such a case, we can only use a superposition state to describe the state of the physical system.

Since such a situation has never happened in a classical (or macroscopic) system, it gives a probability distribution that has never existed before.

…

— Me@2022-01-31 08:33:01 AM

a trajectory story

~ a hindsight story

~ a post hoc story

reality

~ experimental data

~ observable events

story

~ an optional description of an unobservable event

Even though the nature of a trajectory story is a hindsight story or a post hoc story, it must be based on observable events, compatible with experiment results.

— Me@2022-03-01 07:33:31 PM

— Me@2022-03-02 10:30:41 AM

The double slit experiment (and any other experiments that have quantum effects) puts the particle-trajectory model into a stress test and breaks it. The experiment exposes the bug of the particle-trajectory model. For example, the superposition case (aka the no-detector case) cannot be explained by this model.

Another example, even if the two slits on double-slit-plate are all closed, some particles, although not many, will still “go through” the plate.

— Me@2022-02-27 09:17:23 AM

— Me@2022-03-01 11:09:24 PM

quantum effect

~ an effect that cannot use the particle-trajectory model

~ an effect that does not have a trajectory story

— Me@2022-03-02 12:23:54 PM

The 4 bugs, 1.12

Posted on March 1, 2022 by rpflee

EPR paradox, 11.11

3.2 (2.3) In some cases, the wave function of a physical variable of the system is in a superposition state at the beginning of the experiment. And then when measuring the variable during the experiment, that wave function collapses. Wrong.

A wave function (for a particular variable) is an intrinsic property of a physical system.

“Physical system” means the experimental-setup design, which includes not just objects and devices, but also operations.

…

…

The common misunderstanding comes from representing $\displaystyle{| \psi \rangle }$ as a sum of $\displaystyle{| \psi_L \rangle }$ and $\displaystyle{| \psi_R \rangle}$ . But this is not a physical superposition, but a mathematical superposition only.

This mathematical superposition has 3 meanings (applications):

…

3.

…

Only the longcut version can avoid such meaningless questions.

If you insist on answering those questions:

How to collapse a wave function?

Replace system $\displaystyle{A}$ with system $\displaystyle{B}$ .

It is not that the wave function $\displaystyle{\psi}$ evolves into $\displaystyle{\phi}$ . Instead, they are just two different wave functions for two different systems.

How long does it take? How long is the decoherence time?

The time needed for the system replacement.

How to uncollapse a wave function?

Replace system $\displaystyle{B}$ with system $\displaystyle{A}$ .

— Me@2022-02-27 12:41:31 AM

.

The wave function “collapse” is actually a wave function replacement. It “happens” not during the experiment time, but during the meta-time, where the designer has replaced the experiment-setup design (that without activated device) with another one (that with activated device).

That’s how to resolve the paradoxes, such as EPR.

Anything you are going to measure is always classical, in the sense that it is the experiment designer that decides which variable is classical, by adding the measuring devices and measuring actions to the experiment design.

It is not that the wave collapses during the experiment when you turn on the detector to measure.

The detector and the planned action of activating it have already formed a “physical definition” that makes your experiment design to have a system being in a mixed state, instead of a superposition state, since the beginning of the experiment.

Put it more accurately, since a wave function is a mathematical function, not a physical field, it does not exist in physical spacetime.

In a sense, instead of existing at the time level of the experiment and the observer, a wave function exists at the meta-time level, the time level of the experiment-setup designer.

So it is meaningless to say “the experimental setup is in a superposition state (or not) in the beginning of the experiment”. Instead, we should say:

The detector and the planned action of activating it have already formed a “physical definition” that makes your experiment design to have a system being in a mixed state, instead of a superposition state~~, since the beginning of the experiment~~.

— Me@2022-02-14 10:35:27 AM

— Me@2022-02-21 07:17:28 PM

— Me@2022-02-22 07:01:40 PM

Craftsmanship, 2

Posted on March 1, 2022 by rpflee

Shipping, 4

Real Artists Ship.

— Steve Jobs

真正的創造神，會有成果和產品，付運和上市。

— Me@2022-03-01 04:25:27 PM

數學教育 7.2

Posted on March 1, 2022 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

（安：另外，他提的另一個，有關學習數學的要點是，即使假設你在大學中，學到的數學，在日常生活中沒有用，單單是為獲取，那些嶄新的元素概念本身，就已經能夠令你有超能力；令你有一些，常人沒有的思考工具、比喻語言。）

…

又例如，之前我把向量幾何中的「完備集合」概念，應用到學習知識上，引申成「知識完備集合」。

任何一門學問，雖然在起初時，看似有無盡的細節要駕馭，但是，努力收集零碎資料，到一個程度後，你會發現，細節雖然多，原理卻只有幾個，萬變不離其中。

那就有如，雖然在三度空間中，有無數點，而每一點也可以用，一支向量箭尖去代表；但是，要表達所有點的任何之一，你並毋須在事先，就收集無數支向量箭；因為，在三度空間中，你只要收集齊，三支互相獨立的原始元素向量 $\displaystyle{\{ \mathbf i, \mathbf j, \mathbf k \}}$ ，那些任何一點，就可以透過它們的線性組合去代表。

$\displaystyle{\mathbf a = a_x \mathbf i + a_y \mathbf j + a_z \mathbf k = (a_x, a_y, a_z)}$

Wikipedia image licensed under
the Creative Commons Attribution-Share Alike 2.5 Generic license

年青時閱讀，以為將會有，無數本書要閱讀，時間不會夠用。大約六年多後，發現沒大興趣再閱新書，因為，再不覺那些新書有新知，只覺那些新書抄舊書。原因很簡單，沒有新元素。

$\displaystyle{\mathbf a = a_x \mathbf i + a_y \mathbf j + a_z \mathbf k = (a_x, a_y, a_z)}$

你在三度空間中，如果要升格，進入四度空間，必須收集到一支，全新的原始基因向量；它必須是完全獨立於，原本的那三支。

$\displaystyle{\mathbf a = a_x \mathbf i + a_y \mathbf j + a_z \mathbf k + a_t \mathbf l = (a_x, a_y, a_z, a_t)}$

如果讀者未學過「向量」那一課數學的話，那就不易明白。

再幾年後，不再只是沒有興趣閱讀，更要建立防火牆，主動抗拒大部分，只歸平庸的書籍，因為，沒有新元素的資料，會搞亂我當時已大致建立好的，自己知識體系。那是人生必經階段。

自始以後，新知識的原材料，主要只會來自，專題研究和自身實證考驗。

— Me@2022-03-01 10:37:07 AM

1930s, 2

Posted on February 28, 2022 by rpflee

— Me@2022-02-28 04:43:53 PM

The 4 bugs, 1.11

Posted on February 28, 2022 by rpflee

EPR paradox, 11.10

A wave function (for a particular variable) is an intrinsic property of a physical system.

“Physical system” means the experimental-setup design, which includes not just objects and devices, but also operations.

…

…

The common misunderstanding comes from representing $\displaystyle{| \psi \rangle }$ as a sum of $\displaystyle{| \psi_L \rangle }$ and $\displaystyle{| \psi_R \rangle}$ . But this is not a physical superposition, but a mathematical superposition only.

This mathematical superposition has 3 meanings (applications):

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3.

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If not for daily-life quantum mechanics, but for lifelong quantum mechanics understanding, you have to learn the longcut version.

For a double-slit experiment without which-way detector activated (system $\displaystyle{A}$ ), it is in a superposition state

$\displaystyle{| \psi \rangle = a~| \psi_L \rangle + b~| \psi_R \rangle}$ ,

where $\displaystyle{|\psi_L \rangle}$ and $\displaystyle{| \psi_R \rangle}$ are eigenstates of going-left and that of going-right respectively.

If we replace the system $\displaystyle{A}$ with another system $\displaystyle{B}$ which is identical to $\displaystyle{A}$ but with a detector activated, system $\displaystyle{B}$ will have a quantum state (schematically)

$\displaystyle{| \phi \rangle = | \psi_L \rangle~\text{or}~| \psi_R \rangle}$ ,

where $\displaystyle{| \phi \rangle}$ is either $\displaystyle{| \psi_L \rangle}$ , with probability $\displaystyle{|a|^2}$ , or $\displaystyle{| \psi_R \rangle}$ , with probability $\displaystyle{|b|^2}$ .

Note that:

1. Quantum state $\displaystyle{\phi}$ of system $\displaystyle{B}$ is not a superposition. Instead, it is a statistical mixture. So it is called a “mixed state”, which can be represented by a density matrix.

2. Although system $\displaystyle{A}$ and system $\displaystyle{B}$ are almost identical, they are not identical.

Although the superposition state coefficients, $\displaystyle{a}$ and $\displaystyle{b}$ , of system $\displaystyle{A}$ will be re-used to calculate the mixed state coefficients, $\displaystyle{|a|^2}$ and $\displaystyle{|b|^2}$ , of system $\displaystyle{B}$ , they are 2 different systems.

The coefficients $\displaystyle{a}$ and $\displaystyle{b}$ can be found by theoretical deduction or by experiment. (Theoretical deduction might not be feasible for a complicated system.) For experiment, you can use either system $\displaystyle{A}$ or system $\displaystyle{B}$ .

For a system $\displaystyle{A}$ experiment, use the resulting interference pattern to match system $\displaystyle{A}$ interference formula. However, a system $\displaystyle{B}$ experiment would be much easier, because it requires only simple counting of cases; no extra formula is needed.

— Me@2022-02-23 08:40:32 AM

Different systems will have different probabilities patterns, encoded in different quantum states‘ wave functions.

System $\displaystyle{A}$ ‘s quantum state $\displaystyle{\psi}$ and system $\displaystyle{B}$ ‘s quantum state $\displaystyle{\phi}$ are not “the same wave function at different times”. Instead, they are two different wave functions, referring to two different physical systems.

Since the shortcut presentation and the longcut one make no difference in calculations of probabilities, we should use the shortcut version whenever interpretation of quantum mechanics is not needed, except for the fact that a wave function’s squared modulus is probability density.

However, if you put the shortcut version into a stress test; if you try to use the shortcut version to interpret quantum mechanics’ foundation, you will run into different paradoxes. For example,

1. Why does a wave function collapse?

2. When does a wave function collapse?

3.1 How can a wave function ever collapse when quantum mechanics requires the evolution of any wave function to be unitary?

3.2 Wouldn't that violate the conservation of quantum information?

Only the longcut version can avoid such meaningless questions.

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— Me@2022-02-14 10:35:27 AM

— Me@2022-02-21 07:17:28 PM

— Me@2022-02-22 07:01:40 PM

The 4 bugs, 1.10

Posted on February 27, 2022 by rpflee

EPR paradox, 11.9

The common quantum mechanics paradoxes are induced by 4 main misunderstandings.

1. A wave function is of a particle. Wrong.

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2.1 A system's wave function exists in physical spacetime. Wrong.

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2.2 A superposition state is a physical superposition of physical states. Wrong.

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3.1 Probability value is totally objective. Wrong.

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A wave function (for a particular variable) is an intrinsic property of a physical system.

“Physical system” means the experimental-setup design, which includes not just objects and devices, but also operations.

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The common misunderstanding comes from representing $\displaystyle{| \psi \rangle }$ as a sum of $\displaystyle{| \psi_L \rangle }$ and $\displaystyle{| \psi_R \rangle}$ . But this is not a physical superposition, but a mathematical superposition only.

This mathematical superposition has 3 meanings (applications):

1.

…

2.

…

3. Besides calculating interference patterns in our system ( $\displaystyle{A}$ ), the coefficients in the superposition are also useful for another system ( $\displaystyle{B}$ ), which is identical to $\displaystyle{A}$ but with a detector activated.

In our double slit experiment (system $\displaystyle{A}$ ), no detector is activated. So the particle’s position variable is in a superposition state

$\displaystyle{| \psi \rangle = \sqrt{0.5}~| \psi_L \rangle + \sqrt{0.5}~| \psi_R \rangle}$ ,

where $\displaystyle{|\psi_L \rangle}$ and $\displaystyle{| \psi_R \rangle}$ are eigenstates of going-left and that of going-right respectively. The wave function $\displaystyle{| \psi \rangle}$ is for calculating the probabilities of passing through the double-slit-plate, without specifying which slit a particle has gone through, since the possible answers are still physically-undefined.

Since system $\displaystyle{B}$ has a detector to provide the physical definitions of “going-left” and “going-right”, the wave function for $\displaystyle{B}$ is not a superposition. Instead, it is schematically

$\displaystyle{| \phi \rangle = | \psi_L \rangle~\text{or}~| \psi_R \rangle}$ ,

where the corresponding probabilities are given by the squares of each superposition coefficient in $\displaystyle{| \psi \rangle}$ . In other words, $\displaystyle{| \phi \rangle}$ has 0.5 probability being $\displaystyle{| \psi_L \rangle}$ and 0.5 probability being $\displaystyle{| \psi_R \rangle}$ . Instead of being a superposition state, $\displaystyle{| \phi \rangle}$ is a statistical mixture, which is called a “mixed state”.

1. pure state

1.1 eigenstate

1.2 superposition (of eigenstates)

2. mixed state

Formally, to represent any kind of states, we need to use the mathematics formalism density matrix.

For the system $\displaystyle{A}$ (with superposition state $\displaystyle{\psi}$ ), the density matrix is

$\displaystyle{ \begin{aligned} \rho_A &= | \psi \rangle \langle \psi | \\ &= \left( \frac{1}{\sqrt{2}} | \psi_L \rangle + \frac{1}{\sqrt{2}} | \psi_R \rangle \right) \left( \frac{1}{\sqrt{2}} \langle \psi_L | + \frac{1}{\sqrt{2}} \langle \psi_R | \right) \\ \end{aligned}}$

For simplicity, assume that the eigenstates $\displaystyle{ \{ |\psi_L\rangle, |\psi_R\rangle \}}$ form a complete orthonormal set. If we use $\displaystyle{\{ | \psi_L \rangle, |\psi_R \rangle \}}$ as basis,

$\displaystyle{ \begin{aligned} \left[ \rho_A \right] &= \begin{bmatrix} \frac{1}{\sqrt 2} \\ \frac{1}{\sqrt 2} \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\ \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \end{bmatrix} \end{aligned}}$

For the system $\displaystyle{B}$ (with state $\displaystyle{| \phi \rangle}$ ), the density matrix is

$\displaystyle{ \begin{aligned} \rho_B &= \frac{1}{2} | \psi_L \rangle \langle \psi_L | + \frac{1}{2} | \psi_R \rangle \langle \psi_R | \\ \left[ \rho_B \right] &= \begin{bmatrix} \frac{1}{\sqrt 2} & 0 \\ 0 & \frac{1}{\sqrt 2} \end{bmatrix} \end{aligned}}$

Since the mixed state coefficients of system $\displaystyle{B}$ are provided by the superposition coefficients of system $\displaystyle{A}$ , we have a language shortcut in quantum mechanics:

For a system $\displaystyle{A}$ in a superposition state

$\displaystyle{| \psi \rangle = a~| \psi_L \rangle + b~| \psi_R \rangle}$ ,

if we install and activate a detector to measure which slit the particle goes through, there are two possible results. One possible result is “left”, with probability $\displaystyle{|a|^2}$ ; another is “right”, with probability $\displaystyle{|b|^2}$ .

In other words, the wave function $\displaystyle{| \psi \rangle}$ has a chance of $\displaystyle{|a|^2}$ to collapse to $\displaystyle{| \psi_L \rangle}$ and a chance of $\displaystyle{|b|^2}$ to collapse to $\displaystyle{| \psi_R \rangle}$ .

Note that this kind of language shortcut should be used as a shortcut (for the calculations in daily-life quantum mechanics applications) only. Do not take those words, especially the word “collapse”, literally. If you regard the shortcut presentation as more than shortcut, your understanding of quantum mechanics fundamental concepts will be fundamentally wrong.

If not for daily-life quantum mechanics, but for lifelong quantum mechanics understanding, you have to learn the longcut version.

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