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


12:00 midnight, Day 5
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。


Day After I

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。
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.


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……


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 here's the short-cut for my physics notes

Click here

Tuesday 15 February 2011


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。


弍寺   能嚼的喉糖




















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
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.
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!

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
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。
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。