Sunday 27 February 2011

Wave-particle duality

Compton scattering: the phenomenon that when the incident photon hits a free electron, the free electron will be scattered and change its direction, with photon with lower frequency bounced away. This gives the evidence that photon can exist in matter form.
Recall the wave nature of light: diffraction (single slit) and interference (Young's double slit)
Particle nature of light: photoelectric effect and Compton scattering.
Einstein states that E=pc, where E is the energy of the photon, p is its momentum and c is the speed of light. Rewrite the equation: E = pc = pfλ = hf, we have the wavelength-momentum relation: p = h/λ. This describes the particle characteristic of photon.
In 1924, de Broglie claims that λ = h/p in his doctorial thesis called de Broglie relation. The corresponding λ is called De Broglie wavelength. This describes the wave nature of a matter which is revolutionary at the time, which awards him the Nobel Prize in Physics in 1929.
Students should note that though p = h/λ and λ = h/p is mathematically equivalent but they should be proved independently since these two equation describes different things. Of course, their validity has been no doubt after some experimental effort, but this is beyond the scope of this set of note.
No matter wave behavior is observed in daily life because their wavelength is far too small to be observed. (smaller than 10-35m). However electrons are proved to have wave nature in experiment: a fast-moving electron has wavelength about 10-10m while the interatomic separation is about the same. Interference pattern was observed when electron beam hits nickel foil and reflect with different angles. The graph of frequency of electrons VS deflected angle shows several maxima and minima. Very soon, Sir George Thomson (son of J.J.Thomson) created another experiment which hits the electron beam on a metal foil and the scattered electron beams interfere. Rings are shown on the fluorescent screen, which is similar to result by X-rays. The process that "electron shown on fluorescent screen" is a particle behavior, so this experiment actually shows the duality nature of electrons.
Double-slit experiment of electrons was also done to show the wave behavior of electrons. Some small molecules like C60's wave behavior have been experimentally verified too.
More about quantum theory
We say that a matter is described by a wavefunction. We do not know the state of a matter until it's observed. The state of a matter is finite and among those "allowed state". For example the allowed state of a piece of stationery can be "on the floor" or "on the desk", but never "between the desk and the floor". Then "between the desk and the floor" is not an allowed state. We never know the exact location of a matter because an observation only allows us to observe one of the allowed states. Cases with probability > 0 can be one of the allowed states.
Heisenberg's uncertainty principle
In classical view, the location of a matter is absolute and exact. Under a probability-position graph, we see only one spike which the matter is exactly there. However in modern physics the probability actually forms a wave instead of a spike only. Take interference of light as an example: When we allow exactly one photon to pass through the double-slit, interference between the waves still occur. We don't know the exact location of the photon, but we know that there's a large probability to find the photon in those maxima. The resulting probability-position graph forms a wave.
In semi-classical view, we observe the state of a matter by receiving reflected waves like visible light for illumination. But the Compton scattering shows that the electron's momentum has been changed since the photon collide with electron. The photon only gives the momentum or position of the electron before it changes its states. Therefore we never know them exactly. The uncertainty principle states that ΔxΔp ≥ h/4π, where Δx is the position uncertainty and Δp is the momentum uncertainty. Again we don't observe uncertainty in daily life because the uncertainty is far too small and is neglected.
Assume T=0K, according to classical theory: kT = mc2/3, (c is the r.m.s. speed), its speed is zero (which is exact), but it is impossible to have a infinitely exact value for its momentum. As a result the atoms, in fact, contain zero-point energy to allow its quantum vibration.
Quantum tunneling
In classical physics energy conserved, when the potential barrier is higher than the K.E. that the matter have, it can't pass through the barrier. However in quantum physics, there's a certain probability that a electron pass through the potential barrier without obtaining enough energy. Most of the matter wave is reflected back while a small part of them pass through. The probability (or proportion) of wave passed through is proportional to a-x where x is the length of barrier and a is a constant. i.e., is exponentially related.

Saturday 26 February 2011

Bohr's Hydrogen Model and spectrum lines

I promise I'll put all equations into MS word mode when doc version is published. At the same time, this is the 100th pieces of note is this blog. ^_^

Quantization of energy in atoms
Assumption of Bohr’s model
-          Validity of Rutherford’s partial model and classical physics, i.e., electron orbiting the nucleus, and the classical laws like circular motion and electric force is valid.
-          The orbit of electron is stationary (stationary orbit). Electron stayed on a stationary orbit is called in a stationary state.
-          Stationary orbits have definite energy levels. Each transition due to emission or absorption of photon must start and end on the energy levels of the atom. i.e., hf = ΔE = Ehigher - Elower for emitting photon and hf = ΔE = Elower - Ehigher for absorbing photon.
-          Bohr’s quantum condition: angular momentum is fixed to integral multiples for h/2π, i.e., Angular momentum = mevr = nh/2π, where n is a positive integer for energy level En and n is called the principal quantum number of that orbit.
Energy of hydrogen atom
-          In Rutherford’s model we know that mev2/r = e2/(4πε0r2)           ---(1)
-          Put mevrn = nh/2π, v = nh/2πmern into (1) we will have rn = (n2h2ε0)/(e2πme)
-          In Rutherford’s model we have Etot = - e2/(8πε0r) = -(1/n2)(e4me/8ε02h2)
-          r1 is called the Bohr’s radius, and E1 = -13.6eV. For simplicity we have rn=n2r1 and En = E1/n2 = -13.6eV/n2
-          Note that all energy levels are negative so that the electron is bounded to the atom and energy is required to excite the electron into a higher level.
-          E refers to 0eV and the state of n = ∞ in which theoretically has infinite radius and both K.E. and P.E. is zero. The electron is just to escape from the atom. Beyond E, the electron becomes free electron and its energy can be in any positive value.
-          n=1 is called the ground state while n=k>1 is called the (k-1)th excited state.
-          Excitation energy is the energy to excite an electron to a higher energy level (excitation). i.e., excitation energy = ΔE = Ehigher - Elower
-          Ionization energy is the minimal energy to remove an electron from an atom, i.e., ionization energy = E - Einitial = -Einitial. Note that -Einitial is positive, so work done is required to ionize the electron.
Emission line spectrum
-          Ideal body (black body) with T > 0K emits EM waves at all wavelengths, called continuous spectrum. The higher frequency, the more EM waves of higher frequency emitted.
-          However reality atoms only emit EM waves in specified wavelengths only (spectral lines) when gaseous atoms are heated under low pressure or gas discharge tubes.
-          Atoms must be in gaseous form and excited before it emits photon. It can be done by collisions (lost of energy in inelastic collision excites the electron, elastic collision cannot excite the electron), heating or applying high voltage across the gas.
-          hfemitted = Ehigher - Elower = (e4me/8ε02h2)(1/nlower2 – 1/nhigher2)
femitted = (e4me/8ε02h3)(1/nlower2 – 1/nhigher2)
Rewrite the formula by c=fλ we have 1/λemitted = (e4me/8ε02h3c)(1/nlower2 – 1/nhigher2)
-          For hydrogen atom we have 1/λemitted = (13.6eV/hc)(1/nlower2 – 1/nhigher2) or       R(1/nlower2 – 1/nhigher2). This is called the Rydberg formula and R = 1.097*107m-1 is the Rydberg constant.
Absorption spectrum
-          When a continuous spectrum of light is passes through the gas at low pressure, the gas will only absorb EM waves at specified wavelengths only, and produce a spectrum with discrete dark lines called absorption spectrum.
-          Light fringes of emission spectrum = dark fringes of absorption spectrum for the same atom. (Or we can say emission + absorption spectrum = continuous spectrum)
-          Those absorbed EM waves will be emitted by the electrons on the atom again since when the electrons absorbed the photon it is excited and unstable. Emitting the photon away make it back to ground state. However this emission is in random direction. Therefore we see very dark fringes.
-          To produce an absorption spectrum, continuous spectrum like white light must be used.
-          We have 1/λabsorbed = (13.6eV/hc)(1/nlower2 – 1/nhigher2) = R(1/nlower2 – 1/nhigher2).
Spectrum series
-          The emission spectrum lines from En to E1 or the absorption spectrum lines from E1 to En are called Lyman series which is UV radiation.
-          The emission spectrum lines from En to E2 or the absorption spectrum lines from E2 to En are called Balmer series which is visible light.
-          The emission spectrum lines from En to E3 or the absorption spectrum lines from E3 to En are called Paschen series which is IR radiation.
-          These series do not overlap each other.
-          Spectrum lines are packed closer as n increases. (The lines are packed closer for the higher frequency part for each series)
-          Only one photon is emitted / absorbed for each transition of an atom.
X-ray spectrum
According to classical theory, X-ray is produced when fast moving electrons are decelerated by the target. However we compose the X-ray spectrum (Intensity VS wavelength), there are several spikes where the intensity of a specific wavelength is abnormally high. It is because the electron beam knocked out an electron in the inner orbit, than the electron in the outer orbit will emit a photon and fall to the inner orbit. As a result, the spikes show the characteristic spectrum of the target atom.

Friday 25 February 2011

Photoelectric effect

Photoelectric effect: If an EM wave with sufficiently high frequency is shone on a piece of metal, electrons will be emitted from the metal surface.
Experiment on photoelectric effect:
Put a UV lamp against the zinc plate on a gold leaf electroscope.
1)       When zinc plate is negatively charged, gold leaf fall as UV radiation strikes on it because electrons (negative charge) are emitted away.
2)       The gold leaf stopped falling if barrier exist between the lamp and the zinc plate.
3)       When zinc plate is positively charged, gold leaf will not fall because electrons can’t escape due to electrostatic attraction between zinc plate (+) and electron (-).
Photocell is composed by a metal plate (cathode) and an electrode (anode). Under exposure of EM wave with sufficiently high frequency, it emits electron. When it’s connected with voltage supply and ammeter, photoelectric current can be measured.
1)       Consider anode with a higher potential, the photoelectric effect occurs normally.
2)       When anode has a lower potential, the current direction is still the same but the photoelectric current is decreasing.
3)       At a certain potential (-Vs) the photoelectric current becomes zero. We say that Vs is the stopping potential of the cell.
4)       At stopping potential even electrons with highest K.E. can’t reach the anode. Therefore K.E.max = eVs
5)       A smaller unit of energy, electron-volt, 1eV = (e)(V) = 1.6*10-19 J. This is equal to the gain of energy when electron accelerates through a p.d. of 1V.

 Properties of photoelectric effect
1)       Electrons emitted only when f ≥ f0, the threshold frequency.
2)       Number of photoelectrons (per second) is proportional to radiation intensity.
3)       K.E.max increases with frequency.
4)       Photoelectric effect is immediate, i.e., once radiation with sufficiently high frequency is given to the metal plate, electrons are emitted at once.
Explaining photoelectric effect by classical wave theory:
-          Wave energy transmitted in a continuous manner and spreads over the wavefront.
-          Energy transfer rate is independent of frequency.
The wave theory cannot explain property 1,3 and 4. Here’s the contradictory result by wave theory on the properties of photoelectric effect:
Property 1: Energy is independent of frequency so it should happen for all frequency.
Property 3: Energy is independent of frequency so as K.E.max.
Property 4: Since energy transfer is continuous, there’s delay before electrons get enough energy to escape.
Quantum theory: quantizing light wave into discrete packets, called light quanta or photons.
The energy of each photon is related to its frequency, E=hf, where h is the Planck constant, which is 6.63 * 10-34 J s.
Note that the behavior of photon is discrete instead of continuous manner.
Since K.E. of photoelectron = energy absorbed – energy used to escape the metal, we have Einstein’s photoelectric equation: K.E.max = hf – Φ, where Φ is the work function, in terms of eV, subjective to the metal used. (usually inversely related to its reactivity.)
The quantum theory can explain most of the photoelectric effect:
Property 1: threshold frequency is given by hf0 = Φ. Therefore we also have K.E.max = h(f – f0).
Property 2: Intensity is proportional to rate of photons transmitted, so it’s also proportional to the photoelectrons emitted.
Property 3: true by K.E.max = hf – Φ.
Property 4: true since electron gain enough energy once it absorb the photon.
Experimental verification of K.E.max = hf – Φ by showing Vs = (h/e) f –Φ/e
Direct a beam of monochromatic light of frequency f, the photoelectrons complete the circuit with voltage supply and galvanometer. Vs is found when the readings of galvanometer drops to zero. A Vs-f graph has x,y-intercept f0 and –Φ/e respectively, and slope h/e. Note that the slope is a constant and applicable to all metal and frequency.
By the above equation we have:
-          Vs is independent of intensity, but intensity is proportional to photoelectric current (Ip).
-          Under the same intensity, light with higher frequency has larger Vs but lower Ip.

Wednesday 23 February 2011

Atomic model

The neoclassical view on atomic model
In classical view, matters are continuous and can be cut infinitely many times.
In modern science we have the concept that they are composed by atoms, which is made up of nucleus (containing proton and neutron) and electron.
-          J. J. Thomson suggested that atom is made up of a positively charged sphere with electrons distributed on the sphere. It’s known as plum pudding model. He also contributed to give evidence of the existence of subatomic particles.
-          Rutherford’s atomic model: he suggested that:
1)       Most volume occupied of an atom is empty.
2)       All proton and neutron are concentrated in a small nucleus at the center. They occupied most of the mass. (nucleus is 105 times smaller than the atom)
3)       Negatively charged electrons orbit the nucleus.
Evidence: α particle scattering experiment

α source is emitted and stroke into gold foil. Some of them are deflected and is detected.
The deflected angle is defined by the angle between deflected route and the original route (i.e. route that the particles were not deflected.)
The probability of finding particles at a certain angle is inversely related to the size of angle (most of them are deflected slightly only)
1)       When α particles are able to pass through the gold foil, that implies that the gold atoms has a large number of empty space.
2)       α particles are deflected due to the electric repulsive force between α particle which has charge +2e, and the nucleus. If the α particle goes nearer, it will deflect more.
3)       Large deflection is impossible for the plum pudding model. Thus Rutherford’s model is better to describe an atom.
Limitations of Rutherford’s model
1)       In classical EM theory, accelerating charged particles (electron) will emit radiation and loss energy, which cause the reduction in orbiting radius which is contradictory to the reality.
2)       Consider electron orbiting the proton, mv2/r = Qq/(4πε0r), v2 = e2/(4πε0mr), U = K.E. + P.E. = mv2/2 + Qq/(4πε0r) = e2/(8πε0r) – e2/(4πε0r) = - e2/(8πε0r) < 0 which is impossible.
3)       Atoms only emit radiation at specified frequencies; this can’t be explained by Rutherford’s model.

Sunday 20 February 2011

偽.SIMC回憶錄5

12:00 midnight, Day 5
不管有沒有慶功的意味,那晚的確玩得痛快。
一副撲克、一場足總盃還有一堆零食,我們足足玩了一整晚(11PM ~ 9AM),從鋤Dee、German、傷心小棧跟Contract Bridge都是取樂的部分之一……事實證明了數學人的確很喜歡這類思考性強的遊戲。
翌日早晨,我們坐那載我們來的旅遊巴回去機場。跟那些當地學生道別實在不容易,就正如這五日已足夠讓我們跟別隊留下深刻友誼一樣。他們道了我們一點小手信(仍掛在床頭)和道別卡,希望這些回憶不是僅僅留在紙上而真是長存心中吧。

飛機起飛,道別了星洲,也宣告著這活動的結束。

Day After I
回學校做演說,沒有甚麼好提的。我只想令自己記得用話筒的正確用法。

Day After II
今天再次拿起合照,腦海中先補上了一句see you in SIMC 2012,再想起各種有趣的回憶,百般滋味非筆墨所能形容。我們隊伍間仍然以fb, msn互相聯繫著,但願這些已能長久地保持下去吧。
回到正題,關於我一開始問自己的問題,我給出的答案是否定的。廁所的話我回程的時候自己親眼看到了被大陸人弄壞的公廁,但市內的衛生一直都不錯,至少街邊的伯伯才不會做出很多不衛生的行為。那邊在家長式鐵腕政策下這方面的確做得不錯。
至於自己的解答方面,我肯定我方的解答是唯一用直觀法做的。但不得不承認一點,Greedy Alogarithm的確是更有效的解答。在嚴謹性方面,我認為我們可以用一些比較容易的方法證明"選取非greedy alogarithm的地段使答案非最大化",不過我還沒有作出任何嘗試。 orz

Extending question
1) Prove / disprove "greedy alogarithm maximizes answer. If not, please give a map that greedy alogarithm does NOT work (and give the optimal answer at the same time).
2) Account the existance for general answer for a map that all street lies on grid square while all intersection points are in integal points.
3) Account the same thing but the map is not necessraily on the grid. Try to solve this in the light of topology concept?
4) Solve these things too, for i) constrains of Q2, ii) constrains where population density is considered.


fin.

完成了。留意上面的圖,3"E"是SIMC的標語,代表Expeior, Expono, Excedo, 也就是Experiment, Explore和Excel,這的確成了我的座右銘,因為它很簡單明確地道出了科學的精神……

*重申,此文純屬虛構,如有雷同實屬巧合。

Friday 18 February 2011

Physics DSE Electives

I'm here to say that I'll make all 4 topics of the Physics elective part.
Originally our school have chosen "medical physics" and "applications", but I'll try to study the two electives "atomic physics" and "astronomy" as in exam.
Atomic physics which include atomic model (Rutherford's, Ch.1), photoelectric effect (Ch.2), Bohr's atom model (Ch.3), Partical-wave duality (Ch.4) and nanotechnology (Ch.5), gives an pretty good introduction to the modern physics. Despite the harshness of the contents (as it even exceed A-Level), these topic is more interesting to be studied.
Astronomy physics include the GPE, the universe, laws of star motion, phase of a star, doppler effect (ya doppler, but it's just funny that they don't even talk about the application on theoratical waves) and the existance of dark matter.
These two electives are quite hard and somehow like a bridging course between secondary education and the tertiary one. No past papers can be found while the university level is far too hard (as those statistical physics were introduced), I'm now choosing some reference book for my notes.
I hope I can finish them before the end of this yearly exam.
---------------------
(Update 2014)
I found the page being so popular in the search engine...so here's the short-cut for my physics notes

Click here

Tuesday 15 February 2011

15-12-11

很久沒寫過一篇像樣的日記了--又或者從來沒像樣過。
現在的我正在聽著FFx Vivaldi的音樂,眼看著旁邊的算草紙,手不停地打著網誌。
這代表了兩件事。
一、我仍然繼續堅持長時間multi-tasking,但生病的時候專注力總會差一點,因此較為輕鬆的古典音樂能在這方面稍作彌補。
二、快要忙死了。CW剛過、清完農曆假功課、繼續雙邊同人文……此時有個令人心痛的消息傳出,那就是某篇「如有雷同」的「跨世代日誌」(嗯,不是HP那本)被「破壞」了。
不是那種化為灰燼的「殲滅」,而是那種宣洩的慾望被阻止造成對心靈的傷害。
但既然沒法將日誌公開,那我分次將文章張貼出來也沒關係了。

從前,在通往大學的道路上有一座關卡,叫會考。
有一天,他看見這批精英輕易通過會考,就加了另一個關卡叫高考。
從此,道路再不平坦。
從來沒有人--從了一個--能在高考全身而退。拿著幾支火箭離開的人足以令自己在歷史上留名。
歲月不饒人,高考一眨眼就屹立了三十年。
……忽然他改變了主意,拿掉了這兩個關卡,放了一座新的上去,叫文憑試。
第一個人以為這關卡很容易就衝了過去,於是他死了。
因為文憑試的英文不比UE淺很多。
第二個人是英文高手,衝了過去還是掛掉。
因為中文科會莫名其妙地出現從未見過的文化題。
第三個人是個老將,恃著自己三十年前會考帶走了整整八支火箭就衝了過去,結果差點要砍掉重練。

因為,守城的那一科名叫通識。
===========================================
弍寺   能嚼的喉糖

十月某天。


課室外面一片狂風暴雨,換了不久的壁報版接下了狂風扔過來的雨箭,各學會釘上去的心血就要白廢了──

可課室裏的人沒空理會這些。課室的白版堆滿了一條條的數式,那是測驗的解答。

「你們的分數都不錯,可是其他班也做得很好。要知道你們這班數學本來就不差,你不進步,別人進步了,就是不進則退……」

坐在窗邊的我,聽着老師的訓話打了個呵欠,這已經是她第N次說這番話了。眼前的試卷也確實沒有好挑剔的,只是偶爾跳了個步驟被扣了分。原本用紅筆改卷的那位給的是滿分,可是一支彩色筆重重地、硬生生地插了個箭嘴示意那該死的步驟。順帶一提,我們的測驗先會互改,然後再給老師看一次:分數不高的話確認一下分數就能通過,可是分數高的話免不了就要接受高兩個檔次的審核,最後的分數當然是有驚無喜。

「好了,高分的人出來拿獎品吧!那是瑞士空運過來的巧克力哦……」

瑞士──空運──巧克力,聽起來不錯呢。幾位高分的同學依次走出來,看見那金光閃閃、透着香氣的顆粒也不禁垂涎三尺,有位同學甚至抓了一把回去。

到我了,我本來就對甜食沒有很大的興趣,可是不拿點回去也是說不過的。我隨便抽了一粒上來,那是一塊黑巧克力。

回到座位,後面的同學嚷着要巧克力,我就做了個人情送了給他……一切都很平淡,直至下課前的十五分鐘。

老師剛好解完一個例子,正找人出來做例題,如鷹般的目光掃過一個個或聽課、或發呆、或聊天的同學,最後落在我身後。

「你,口中的是甚麼?」

「沒、沒甚麼,我喉嚨痛在吃喉糖。」

「哈哈,世上有能嚼的喉糖嗎?」那同學無言了,旁邊的都在笑。

老師話鋒一轉,落到了我身上:「還有你啊,為甚麼將我送你的巧克力當成垃圾扔給那同學?」

「我喉嚨今天不太舒服呢,巧克力還是儘快吃會比較好。」

「那你要不要一片能嚼的喉糖?明明不要巧克力又不說出來,要不然不拿也可以呀,剩下的還不是給其他同學?」老師冷笑着道。

我也不太清楚當時為甚麼要這樣說,可是一想到就不能自制地說了出來:「正所謂『已所不欲,勿施於人』,我不喜歡巧克力的話又怎可以給他呢?」很快我就後悔說了這話。

全班渡過了尷尬的半分鐘,老師卻臉不改容地道:「所以呢,你面試時,就是你心裏不同意考官的說話,也不要反駁。這是我朋友的真人真事……」

我抓緊了下台階,便不再吭聲,低頭看書本。眼前的書本清晰地列着一個個計算要點、步驟;但當我想要看看白板上的數式,希望找出哪怕是一點點的技巧,卻又發現白板上模糊的數式與那躍然紙上的書本示範跟本是兩碼子的事。到底是不是我太笨,看不懂?我不知道。
 
(1025字)

Thursday 10 February 2011

Numerical approximation II

I found that Q8 in the last passage was quite interesting so I'm going to discuss the question a bit.
1) Modolus approximation
Recall the four equation:
Find the smallest possible integer n>2 s.t.
i) {n^0.5-2^0.5}< 0.01
ii) {|n^0.5-2^0.5|}<0.01
iii) {n^0.5-2^0.5}<0.0001
iv) {|n^0.5-2^0.5|}<0.0001
Now let's modify the question to:
i) {n^0.5-2^0.5}<0.01
ii) {2^0.5-n^0.5}<0.01
iii) {n^0.5-2^0.5}<0.0001
iv) {2^0.5-n^0.5}<0.0001
Now consider the easier one {2^0.5-n^0.5}<0.01.
2^0.5\ < n^0.5 + 0.01 +a
1.4042  < n^0.5 + a
Square:
1.97 < (n+a^2) + 2an^0.5
This may help us to estimate the level of error (like the big O function)
Now the error level is 0.01 so after root squaring it becomes 0.1.
{n^0.5- 2^0.5} < 0.1 ~ 0
n = (a^2+2) + 2a2^0.5 where a is an integer
{0.8284a} < 0.1
Magifying the system, 8284a < 1000 mod 10000
Though they are all multiple of 4, it's still troublesome to get all solutions of a.
Here's the result with (=a mod 10000, a), readers can observe why only these numbers were selected.
(4,2331)
(8,2162)
(56,134)
(60,-35=2465)
(116,99)
(236,29)
(298,-6=2494)
(352,128)
(532,23)
(828,17)
The smallest a is 17, so we will have n ~ (17+2^0.5)^2 ~ 339
{2^0.5 - 339^0.5} = 0.00226... which is true.
Similarly, {2^0.5 - n^0.5} < 0.0001 is equivalent to 8284a < 100 mod 10000, which a = 134.
(134+2^0.5)^2 ~ 18337
{2^0.5-18337^0.5} = 0.00003 which is also true.

2) Differentiatial approximation
The formula is given by f(x+h) ~ f(x) + hf'(x)
Here's an simple example:
Estimate sin (pi/90).
we have sin (pi/90) ~ sin 0 + (pi/90)(sin 0)' = pi/90 ~ 0.3490
while sin (pi/90) ~ 0.03489, quite accurate!

Exercise
1) Complete {n^0.5-2^0.5}<0.01 and {n^0.5-2^0.5}<0.0001.
2) How many digits of 2^0.5 you should approximate to find n that {n^0.5-2^0.5}<10^-6?
3) Show that the answer obtained (339 and 18337) is really the smallest solution.
4) Show that there exist integer solution(s) in every interval [x^2,(x+1)^2] (x is real >2) which {n^0.5-2^0.5}<0.1.
5) Given tan (57pi/180) = 1.539864964..., find the smallest integer n such that {|tan (17n pi/180) - tan 57pi/180|} < 0.01 (four cases, same as {n^0.5-2^0.5})
6) Estimate 2^ 0.0101.
7) Estimate sin 79pi/180, suppose all value of sin x is NOT WELL KNOWN except sin 0 = 0 and sin pi/2 = 1. (Mean value theorem's approximation?)
8) Select integer p,q which is not bigger than 1000, in which (p/q)^2 is closet to 2. Now the n d.p. correction of (p/q) and 2^0.5 takes different value. Find the smallest n. (modified IMO prelim mock, 3M)

Wednesday 2 February 2011

偽.SIMC回憶錄 4.2

12:00 noon, Day 4, NUS High School
口頭報告的完成代表著比賽的時間已完結,剩下來的時間就可以等賽果和輕鬆一下。
今天的午飯不知到是否因為壓力頓消而胃口大開,今餐似乎異常豐富(意粉/蛋餅/布丁?),飲品還是那被再度稀釋的雜果賓治,還有超過飽和份量而好像漿糊的熱朱古力(這令我想起明將XD),兩者之下我還是選擇了飲自己的清水……
由於結果在傍晚才會公佈,中間的時間我們同樣會出外觀光,今次我們去了星洲的Science Park (有點似香港的科學園,但那邊的Science Park真的是一個Park)和濾水廠(這裡有兩個小插曲,第一個:某房間的牆壁有一個圈圈九的圖樣,其內容大概是濾水的第九個步驟,但……相信東方迷會知道另一個意思。
另一個插曲是一某個模仿水壩的模型,其中有一部分是關於下雨儲水,模型由電腦控制,所以會自己來;但當地導游似乎算準了時機(大概他也練過很多次?),大呼一聲Rain!,雨就下來了。
最後一個地方是可持展續發展的展館,最深刻的要算上用數萬個膠樽砌成的球體,以及貌似jubeat,會感應的地磚……
一眨眼時間又到了傍晚時分,我們回到了演講廳等候結果公佈(簡單來說就是看看有沒有獎)。看來星洲政府頗為重視這比賽,頒獎嘉賓為當地教育局局長(雖然這樣說有點不敬,但我隊一致認為其裝扮酷似推點心車的阿姐……XD)。
Raymond Chan的致詞中提到解答技巧的多樣性,例如loop programming, linear programming, linear algebra, matrix, algorithms,,,當然沒有提我們這種直觀解法,於是當時我似乎對自己沒啥信心可以拿到獎,始於放不下自己只是到這裡吸收經驗的事實。
當然,我是在場其中一個笑到最後的人,連同友隊的distinction,我校應該是除了主場的NUS High School以外成績最好的學校(因為他們拿到冠軍),而北京某校(上屆冠軍)啥都拿不到。
頒獎完成後來到活動的終曲:Farewell Party。
五天,不太短亦不太長的時間,能令一個完全不想講英文的人愉快地和各國好手交流,能撮合一個澳洲佬和一個日本妹(?!?),能令各人的數學技巧更上一層樓……最重要的是,SIME為我們留下了不可磨滅的美好回憶。
這一點,從我們不停向四周的對手在合照上簽名可以得證。每個人互相交換MSN,留下對對手的Comment(可惜當時未流行"Like"……),一個個簽名,代表著五日以來建立的友情……如果十年後、二十年後,當我們拿出這張照片時,腦海中還能閃過一張張當年的回憶,那我就很滿足了。
晚會基本上以簽名開始,也以簽名結束。一些隊伍因為趕夜機,已經不見人影。天下無不散之筵席,當大家各自離去,就是活動完滿結束之時。
……