Huygens’ principle, 1.1

Posted on March 30, 2022 by rpflee

Why are there no backward secondary wavefronts?

A secondary wave source is of different nature from a primary wave source. Consider an one dimensional transverse wave on a string:

Primary wave source is activated by the force from above, and then the wave propagates to both directions. Secondary wave source is activated by the particle on the left.

Primary wave source energy is from outside the string. Secondary wave source energy is from an adjacent string molecule.

When the secondary source S reaches its maximum, although it drags both P and Q, they are in different situations. At that moment, while the instantaneous velocity of Q is upward, that of P is downward.

Note that the red arrow at P is the force on P by S. It is not the net force on P.

— Me@2022-03-09 11:14:07 PM

惜此際 5

Posted on March 29, 2022 by rpflee

先唔講地唔地獄，但係因為有呢種人生觀，就算瓜咗，呢種思想模式都只會帶佢到呢啲境界嘅地方，到咗嗰度，如果佢都係咁睇野，結局又係揀自殺，咁樣不斷自殺，經歷一次又一次咁嘅局面，咁嘅經歷真係唔知算唔算地獄，可能佢都唔知經歷過幾多次呢款生生死死了，只有佢自己救到自己，其實心態改變一下就ok了，無野帶到黎，亦無野帶得走，咁認真做乜呢

—

2022.03.29 Tuesday ACHK

伏線驅動程式 1.2

Posted on March 29, 2022 by rpflee

這段改編自 2010 年 10 月 14 日的對話。

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？

因為 HYC 問我有沒有時間，教他們「數學與統計」科；她順便問我，有沒有時間去，教其他同學物理科。

為何 HYC 會問我數學呢？

因為之前臨近中五會考，我在教你們中五師妹 CKY 的「附加數學」時，HYC 見到我。

為何 CKY 會找我教「附加數學」呢？

因為再之前，我在教你一位，中七師姐 CPM 物理時，CKY 見到我。

為何我會教 CPM 物理呢？

因為我在這裡教書時，她是我的數學科學生。臨近中七高考時，我問她有沒有物理問題要問我。

為何我在幾年前，會在這裡教書呢？

因為之前我找到工作的那間補習社，言而無信。我應徵面試完時，面試職員說會聘請我，叫我等消息。一段時間後，還未有消息。

— Me@2022-03-08 12:06:18 PM

Politician

Posted on March 27, 2022 by rpflee

— Me@2022-03-26 07:51:29 PM

2022.03.26 Saturday ACHK

Ex 1.22 Driven pendulum, 3.1

Posted on March 26, 2022 by rpflee

Ex 1.24 Constraint forces, 1.1

Structure and Interpretation of Classical Mechanics

~~~

[guess]

$\displaystyle{ \begin{aligned} m \ddot{y} &= F \cos \theta - mg \\ m \ddot{x} &= - F \sin \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} m \ddot{y} &= F \frac{y_s - y}{l} - mg \\ m \ddot{x} &= - F \frac{x - x_s}{l} \\ \sqrt{(x_s - x)^2 + (y_s - y)^2} &= l \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_s &= l \\ x_s &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} m \ddot{y} &= F \frac{l - y}{l} - mg \\ m \ddot{x} &= - F \frac{x}{l} \\ \sqrt{x^2 + (l - y)^2} &= l \\ \end{aligned}}$


(define (KE-particle m v)
  (* 1/2 m (square v)))
 
(define ((extract-particle pieces) local i)
  (let* ((indices (apply up (iota pieces (* i pieces))))
         (extract (lambda (tuple)
                    (vector-map (lambda (i)
                                  (ref tuple i))
                                indices))))
    (up (time local)
        (extract (coordinate local))
        (extract (velocity local)))))
 
(define (U-constraint q0 q1 F l)
  (* (/ F (* 2 l))
     (- (square (- q1 q0))
        (square l))))
 
(define ((U-gravity g m) q)
  (let* ((y (ref q 1)))
    (* m g y))) 

(define ((L-driven-free m l x_s y_s U) local)
  (let* ((extract (extract-particle 2))
      
     (p (extract local 0))
     (q (coordinate p))
     (qdot (velocity p))
      
     (F (ref (coordinate local) 2)))
   
    (- (KE-particle m qdot)
       (U q)
       (U-constraint (up (x_s (time local)) (y_s (time local)))
             q
             F
             l))))

(let* ((U (U-gravity 'g 'm))
       (x_s (literal-function 'x_s))
       (y_s (literal-function 'y_s))
       (L (L-driven-free 'm 'l x_s y_s U))
       (q-rect (up (literal-function 'x)
                   (literal-function 'y)
                   (literal-function 'F))))
  (show-expression
   ((compose L (Gamma q-rect)) 't)))

$\displaystyle{ L = \frac{1}{2} m \left[(Dx)^2 + (Dy)^2 \right] - mgy - \frac{F}{2l} \left[ (x-x_s)^2 + (y-y_s)^2 - l^2 \right] }$


(let* ((U (U-gravity 'g 'm))
       (x_s (literal-function 'x_s))
       (y_s (literal-function 'y_s))
       (L (L-driven-free 'm 'l x_s y_s U))
       (q-rect (up (literal-function 'x)
                   (literal-function 'y)
                   (literal-function 'F))))
  (show-expression
   (((Lagrange-equations L) q-rect) 't)))

$\displaystyle{ \begin{aligned} mD^2x(t) + \frac{F(t)}{l} \left[x(t) - x_s(t)\right] &= 0 \\ mg + m D^2y(t) + \frac{F(t)}{l} [y(t) - y_s(t)] &= 0 \\ -l^2 + [y(t)-y_s(t)]^2 + [x(t)-x_s(t)]^2 &= 0 \\ \end{aligned}}$


(define ((F->C F) local)
  (->local (time local)
           (F local)
           (+ (((partial 0) F) local)
              (* (((partial 1) F) local) 
                 (velocity local)))))

(define ((q->r x_s y_s l) local)
  (let* ((q (coordinate local))
         (t (time local))
         (theta (ref q 0))
         (F (ref q 1)))
    (up (+ (x_s t) (* l (sin theta)))
        (- (y_s t) (* l (cos theta)))
        F)))

(let ((q (up (literal-function 'theta)             
             (literal-function 'F))))
  (show-expression (q 't)))


(let* ((x_s (literal-function 'x_s))
       (y_s (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression ((Gamma q) 't)))


(let* ((xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression ((compose (q->r xs ys 'l) (Gamma q)) 't)))


(let* ((xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression ((F->C (q->r xs ys 'l)) ((Gamma q) 't))))



(define (L-theta m l x_s y_s U)
  (compose
   (L-driven-free m l x_s y_s U) (F->C (q->r x_s y_s l))))

(let* ((U (U-gravity 'g 'm))
       (xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression ((Gamma
                    (compose (q->r xs ys 'l) (Gamma q)))
                    't)))



(let* ((U (U-gravity 'g 'm))
       (xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression (U ((compose (q->r xs ys 'l) (Gamma q)) 't))))


(let* ((U (U-gravity 'g 'm))
       (xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression ((L-driven-free 'm 'l xs ys U)
		    ((Gamma (compose (q->r xs ys 'l) (Gamma q))) 't))))



(let* ((U (U-gravity 'g 'm))
       (xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)             
              (literal-function 'F))))
  (show-expression ((L-theta 'm 'l xs ys U) ((Gamma q) 't))))

$\displaystyle{ \begin{aligned} L_\theta &= \frac{1}{2} m (D x_s(t))^2 + \frac{1}{2} m (D y_s(t))^2 - m g \left[ y_s(t) - l \cos \theta(t) \right] \\ &+ \frac{1}{2} m l^2 (D \theta(t))^2 + lm D \theta(t) \left[ D x_s(t) \cos \theta(t) + \sin \theta(t) D y_s(t) \right] \\ \end{aligned}}$

[guess]

— Me@2022-03-24 04:38:10 PM

EPR paradox, 12.2

Posted on March 24, 2022 by rpflee

…

Experiment C:

What if during Experiment A, the observer changes his mind to turn on the detector before the particle’s arrival?

Explanation X:

We should regard this whole process as an experiment-setup.

The probability patterns encoded in the quantum state is of this experiment-setup, which in this case is equivalent to Experiment B. With respect to this experiment-process/experiment-setup, the variable-to-be-measured is always in a mixed state.

The word “always” here means that a quantum state is not a physical state of a particle in physical spacetime during the experiment. Instead, it is a property (of a physical variable) of the experiment-setup (physical system) itself.

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

However, for an experiment like Experiment C, doesn't the violation of Bell's inequality have proved that the spin of the particle is still in a superposition before the activation of the detector?

Let’s translate this question to a double-slit experiment version. (The consistency of the left detector and the right detector is as strange as the EPR consistency.)

Experiment C:

What if in the double-slit experiment, the detector is off when the particle is emitted, but the observer changes his mind to turn on the detector before the particle’s passing through of the double-slit plate?

Is the particle in a superposition or not when the detector is still off?

Instead of a superposition pure state, we should regard the particle path state as a mixed state, even before the detector is activated.

The reason is given in Explanation X.

However, for an experiment like Experiment C, doesn't the violation of Bell's inequality have proved that the path of the particle is still in a superposition before the activation of the detector?

The violation of Bell’s inequality should be interpreted in this way:

Experiment A:

If there is no detector activated throughout the experiment until the particle reaches the final screen, the resulting statistical pattern (aka the interference pattern) is possible only if the particle has no definite path. “An unknown but definite path exists” would not be able to create such statistics.

.

Experiment B:

If there is a detector activated since the beginning of the experiment, the particle path is in a mixed state since that beginning, meaning that the path is definite although unknown before measurement.

.

Experiment C:

What if in the double-slit experiment, the detector is off when the particle is emitted, but the observer changes his mind to turn on the detector before the particle’s passing through of the double-slit plate?

Is the particle in a superposition or not when the detector is still off?

After the detector is activated, the path is in a mixed state.

Before the detector is activated, regarding the path state as in a superposition or as in a mixed state makes no physical difference, because by definition, there is no measurement before the detector is activated.

Also, a superposition’s coefficients are always about the potential-activation of detectors.

A superposition state has a corresponding mixed state. The superposition coefficients can be modulus-squared to give the mixed state coefficients. That is the exact original meaning of the superposition coefficients. (This is the statistical interpretation given by Born rule.)

In other words, each coefficient in a superposition state by default encodes the probability of each potential measurement result.

However, in Experiment C, regarding the state as a superposition state before the activation of detector creates conceptual paradoxes, such as:

If the particle path variable has no definite state before measurement, how come the left detector and right detector always give consistent results?

This is the double-slit experiment version of the EPR paradox.

When the experimenter follows the plan of Experiment A, the particle path variable is always in a superposition, since the beginning of the experiment.

However, when he changes his mind to turn on the detector before the particle’s passing through of the double-slit plate, the particle path variable is always in a mixed state, since the beginning of the experiment.

Wouldn’t that create retro-causality?

It seems to be retro-causality, but it is not, because this description is a language shortcut only. The apparent violation of causality does not exist in the language longcut version, which is provided in Explanation X.

Also, due to the indistinguishability of identical particles, particle identities and thus particle trajectories are post hoc stories only.

In short, the violation of Bell’s inequality means that:

If there is no path detector activated throughout the experiment until the particle reaches the final screen, the path variable never has a definite value, known or unknown, throughout that experiment.

The violation of Bell’s inequality should not be applied to Experiment C, because it is not the case of “having no detector throughout the experiment”.

Also, note that we do not need the violation of Bell’s inequality to prove that the path variable has no definite state, known or unknown.

Just the interference pattern’s existence is already enough for us to do so.

— Me@2022-03-08 11:56:00 PM

What is a leader?

Posted on March 21, 2022 by rpflee

Someone who causes people to do that which they ought to do but which they would not do in his absence.

— John T. Reed

2022.03.21 Monday ACHK

十萬七千里 3

Posted on March 20, 2022 by rpflee

數學教育 7.3

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。）

…

再例如，在微積分之初，我們學到了「極限」這概念。有時一些數式，永遠不會達到某個數値，但是卻可以，無限迫近。

例如， $\displaystyle{\forall x,~~\frac{1}{x} \ne 0}$ ；意思是，無論你把 $\displaystyle{x}$ 設成任何數値， $\displaystyle{\frac{1}{x}}$ 都不會等於零。但是，你卻可以要求 $\displaystyle{\frac{1}{x}}$ ，任意接近零。

$\displaystyle{\lim_{x \to \infty} \frac{1}{x} = 0}$

例如，如果想 $\displaystyle{\frac{1}{x}}$ 與 $\displaystyle{0}$ 相差，少於 $\displaystyle{0.01}$ 的話，你就設 $\displaystyle{x = 101}$ ；如果想相差，少於 $\displaystyle{0.0001}$ 的話，你就設 $\displaystyle{x = 10001}$ ；如此類推。

這個原理用作，人生哲學的比喻的話，就是指，即使「目標不可能」，也不代表「要完全放棄」。亦即是話，即使你的一些人生目標，不可能實現，那也不代表你，完全沒有東西可以做。那「不可能的目標」或許可以，轉化為一個「可能的近似版本」。

例如，你的目標是「不勞而獲」，不用工作就有金錢收入。一般情形下，那是不可能。但是，「不勞而獲」有近似的版本嗎？

有，有一些。例如，「小勞大獲」可以嗎？又例如，找你喜歡的工作可以嗎？

快樂工作的話，工作就不再是「工作」了。你被迫做一件事的話，那件事就是「工作」。你即使沒有需要，也會因為喜好，而做一件事的話，那件事就是「娛樂」。

— Me@2022-03-14 05:11:29 PM

Game Boy

Posted on March 12, 2022 by rpflee

1990s, 17

2.11 Extra dimension and statistical mechanics, 3

Posted on March 10, 2022 by rpflee

A First Course in String Theory

(b) Assume that $\displaystyle{R \ll a}$ in such a way that …

…

~~~

[guess]

$\displaystyle{\frac{\hbar^2}{ma^2} \ll k T \ll \frac{\hbar^2}{mR^2}}$

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

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

$\displaystyle{\beta = \frac{1}{kT} \ll \frac{ma^2}{\hbar^2}}$

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

$\displaystyle{\beta \gg \frac{mR^2}{\hbar^2}}$

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

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

[guess]

— Me@2022-03-10 10:56:10 AM

EPR paradox, 12.1

Posted on March 9, 2022 by rpflee

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.

In other words, “where and when an observer should do what during the experiment” is actually part of your experimental-setup design, defining what probability distribution (for any particular variable) you (the observer) will get.

If the experimenter does not follow the original experiment design, such as not turning on the detector at the pre-defined time, then he is actually doing another experiment, which will have a completely different probability distribution (for any particular variable).

— Me@2022-02-18 07:40:14 AM

— Me@2022-02-22 07:01:40 PM

Note that different observers see different “physical systems”. They see different quantum states, because a quantum state is actually not a “state”, but a (statistical) property of a system, encoded in the superposition coefficients.

Experiment A:

In an EPR experiment, if it is by design that an observer will not turn on the detector, then the story should be that this experiment-setup is always in a superposition state (with respect to the variable-not-to-be-measured).

.

Experiment B:

In an EPR experiment, if it is by design that an observer will turn on the detector before a particle’s arrival, then the story should be that this experiment-setup is always in a mixed state (with respect to the variable-to-be-measured).

.

No concept of “wave function collapse” is needed.

What if during Experiment A, the observer changes his mind to turn on the detector before the particle's arrival?

He has actually replaced experiment-setup A with experiment-setup B.

Then is the experiment always in a superposition state? Or always in a mixed state?

As a language shortcut, you can say that the superposition wave function collapses at the moment of system replacement.

However, it is a language shortcut, a story only.

A story is not reality.

A story is post hoc.

physical definition

~ define unobservable events in terms of observable events

Any story would be fine as long as it is compatible with reality, aka measurement results.

But some stories are better than others.

Here, only the longcut version can avoid common meaningless questions.

Experiment C:

What if during Experiment A, the observer changes his mind to turn on the detector before the particle’s arrival?

We should regard this whole process as an experiment-setup.

The probability patterns encoded in the quantum state is of this experiment-setup, which in this case is equivalent to Experiment B. With respect to this experiment-process/experiment-setup, the variable-to-be-measured is always in a mixed state.

The word “always” here means that a quantum state is not a physical state of a particle in physical spacetime during the experiment. Instead, it is a property (of a physical variable) of the experiment-setup (physical system) itself.

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

However, for an experiment like Experiment C, doesn't the violation of Bell's inequality have proved that the spin of the particle is still in a superposition before the activation of the detector?

Let’s translate this question to a double-slit experiment version.

…

— Me@2022-03-08 11:56:00 PM

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 $\displaystyle{\left( \beta = \frac{1}{kT} \to 0\right)}$ .

…

~~~

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

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

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

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

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

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

— Me@2022-03-04 07:12:49 PM

Physics Town

continues

Monthly Archives: March 2022

Huygens’ principle, 1.1

惜此際 5

伏線驅動程式 1.2

Politician

Ex 1.22 Driven pendulum, 3.1

EPR paradox, 12.2

What is a leader?

十萬七千里 3

Game Boy

2.11 Extra dimension and statistical mechanics, 3

EPR paradox, 12.1

地獄篇 8

伏線驅動程式 1.1

2014

Ex 1.23 Fill in the details

EPR paradox, 11.12

預防離婚

同一屋簷下

1990s, 16

2.11 Extra dimension and statistical mechanics