AlbertYang的甘苦滋味

<Derivation of the Classical Wave Equation>

Consider a vibrating string at a instant time:

Focus on the red segment:

Assume that the string is homogeneous, and the mass of the string per unit length is r (a constant).

T₁ & T₂: tension acting on P & Q, and T₁ & T₂ are tangential to the curve of string.

Only vertical motion exists, that is, the horizontal components of the tensions at all points along the string must be equal.

Consider the horizontal part:

Consider the vertical force:

Upward motion is positive, “+”; Downward motion is negative, “-”.

According to Newton’s 2^nd law, F = ma = rDx    

tanb and tana is the slope of curve at x & x+Dx.

that is,

and let

v is the speed with which a disturbance propagates along a stretched string depends on the tension in the string and the linear mass density of the string.

<How to Solve it & Discussion>

Try separation of variables

y(x,t) = X(x)T(t)

Multiply 1/y on both side

x and t are independent variables. This equation can only be valid if both sides are equal to a constant. 

Note we have total not partial derivatives: linear, 2nd-order, and ordinary differential equation.

General solutions of have the form (see the below tables)

Also for T(t) equation

Now look at Boundary Conditions

y(0,t) = 0

y(L,t) = 0

For K > 0, try to satisfy boundary conditions

so we should study the  condition of〝K < 0 〞

For K < 0

=> 〝quantization〞

In the same way, we can solve T(t) when T(0) = 0.

Now we choose the initial condition y = y₀, that is J = y₀

This is a set of solutions, one for each value of n. Each solution corresponds to a different standing wave. The differential equation, the boundary conditions, and the initial conditions have completely determined the set of wave functions.

The velocity is given by

The velocity at t = 0

Our initial condition has specified the initial velocity. Instead of specifying the maximum amplitude we could have specified that the initial velocity is given by this function.

Typically for a standing wave, its wavelength (l) equals to 2L/n (n: number of standing wave), i.e. l is quantized.

The period t of our standing wave is the time required for the argument of the time factor sin(npvt/L) to change by 2p, so that

The frequency is quantized. It is directly proportional to the integer n, inversely proportional to the length of the string, directly proportional to the square root of the tension force, and inversely proportional to the square root of the mass per unit length.

A string does not usually move as described by a single harmonic. A linear combination of harmonics can satisfy the wave equation:

The fact that a linear combination of solutions can be a solution to the wave equation is called the principle of superposition.

Reference: 

1. Physical Chemistry 3rd Ed Robert G Mortimer

2. http://ocw.mit.edu/courses/chemistry/5-61-physical-chemistry-fall-2013/

Someone, one of my friend, has written that moon doesn't exist unless you "perceive" and "observe" it from philosopher's point of view. This is kind of perverting the fundamental idea from the great masters who are famous for their pioneering work on quantum mechanics. In order to elucidate the spirit of quantum mechanics, I'll try to explain "absolute size" and "superposition principle" in the following context as possible as I can.

First of all, it alsway been hold, either classical mechanics or quantum mechanics, that we must interact with our target if we wants to make an observation on it. Observation causes disturbances. Classical mechanics assume that we can always make object being big and neglect disturbance when makeing observation on it. Size is "relative". For example, when we throw a billiard ball at that wall, the wall is "big" and the ball is "small". That is, size depends on the object and your experimental technique, so the classical system obey the causality, i. e. an observation doesn't scatter the system we interested in. Quantum mechanics is fundamentally different from classical mechanics in the way it treats size. Dirac assumes that there is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance, a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer. That is, for a small object, an unavoidable limiting disturbance is not negligible even if we improve our experimental technique. Size is absolute. On the contrary, classical mechanics doesn't set up to describe objects that are small in an absolute sense. Due to unavoidable limiting disturbance, observation on a microscopic system breaks down the causality, and uncertainty comes in calculation of observables. The experimental result accompanies with probability.

What happened before making observation on a system? A famous example is "Schrödinger's cat." The whole story can be found on Wiki. The state of such system can be described by the following equation

To get statistical result, we perpare many similar boxes. By opening these boxes, we get the probability finding the cat is dead or alive. But we still don't know the real situation before we open these black boxes. It seems strange, and alittle bit distracting from my another main topic: "Superposition principle." All I do is to desribe our interested system by using linear combination (superposition) of a set base kets (vectors). In quantum mechanics, superposition of states is central theoretical description of nature. Observation on a superposition of A (ex: "alive") and B (ex: "dead"), which are basis, gives either a or b. Never gives anything else. Probability of getting result a or b depends on relative weights of A and B in the superposition. A great conclusion comes from Dirac: "The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being 'intermediate' between the corresponding probabilities for the original state, not through the result itself being intermediate between the corresponding results for the original states." "Absolute size" and "Superposition principle" are intimately linked. Maybe the example about the cat is untouchable, I strongly recommendated the book, Elementary Quantum Mechanics, which is written by Michael D. Fayer. He illustrate the two ideas in an attainable way. All quantum mechanics do is to calculate probabilities about our interested system.

Reference Link:
http://www.stanford.edu/group/fayer/

Furthermore video on quantum mechanics

古典的量子論學習中  最經典的模型就是  粒子於一維盒子中的運動描述(波函數)推導  及其能量量化的導式

這邊藉由簡單的一維駐波模型方式推得粒子於一維盒子中

我們知道粒子於一維盒子中的假設(如下圖)

(圖片引用自維基百科)

在盒子當中位能為零且盒子以外(含盒子的邊)其位能視為無窮大

那如何以古典的一維駐波來近似這個問題？

我的想法是藉由將粒子視為物質波的形式且為一兩端固定之一維弦(將位能無窮大視為不動之端點)

由古典的駐波模知道固定兩端點的弦長度L為駐波其半波長(l)的整數倍  亦即   

經過重排可以得到

(圖片引用自維基百科)

且由物質波公式

 式中p係指粒子動量  E係指粒子動能

結合兩式對波長l的描述可以得到

由假設知道盒中質點運動時位能為零  亦即質點能量為其動能