Random Thoughts

Two Commutation Relations

In previous post I mentioned that the First Commutation Relation and Second Commutation Relation are equivalent. Here is a proof from Classical Perspective.

We assume one of them to be true. For example, we assume First Commutation Relation to be true. i.e.

[\hat{x}, \hat{p}] = i\hbar\rightarrow \hat{p} = -i\hbar\frac{\partial{}}{\partial{x}}

Classical Hamiltonian is given by

\hat{H} = \frac{\hat{p}^2}{2m} + \hat{V} = -\frac{\hbar^2}{2m}\frac{\partial{}^2}{\partial{x^2}} + V

We next compute the commutation between Hamiltonian and Position

[\hat{H},\hat{x}] = [\frac{\hat{p}^2}{2m} + \hat{V},\hat{x}]=\frac{1}{2m}[\hat{p}^2, x] =i\hbar\frac{\hat{p}}{m}

We see the operator in the last term is precisely the velocity operator \hat{v}

\hat{v}\psi = v\psi

Therefore from definition of velocity

\hat{v}\psi = \frac{\partial{x}}{\partial{t}}\psi = [\frac{\partial{}}{\partial{t}}, \hat{x}]\psi

So we conclude that

[\hat{H},\hat{x}]\psi = i\hbar[\frac{\partial{}}{\partial{t}}, \hat{x}]\psi

\hat{H} = i\hbar\frac{\partial{}}{\partial{t}}

It then follows

[\hat{H},\hat{t}] = [i\hbar\frac{\partial{}}{\partial{t}}, \hat{t}] = i\hbar

Equivalence proved.

(Note it’s only proved for non-relativistic hamiltonian, i still haven’t worked out the most general case at the moment)

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On Schrodinger Equation

There is a concept in Schrodinger equation that has never been taught in any of my Quantum Mechanics lectures.

We assume the validity of the following commutation relation

[x, p] = i\hbar

We call it First Commutation Relation. Equivalently we conjecture the validity of another commutation relation

[H, t] = i\hbar

We call it Second Commutation Relation. Based on these two assumptions, we build up Quantum Mechanics as follows

From First Commutation Relation:

[p, x] = -i\hbar \rightarrow p = -i\hbar\frac{\partial{}}{\partial{x}}

From Second Commutation Relation:

[H, t] = i\hbar \rightarrow H = i\hbar\frac{\partial{}}{\partial{t}}

From Classical Mechanics, the definition of Hamiltonian for a particle is

H = \frac{P^2}{2m} + V

Substitute H and P into this equation

H=\frac{P^2}{2m} + V = -\frac{\hbar^2}{2m}\frac{\partial{}^2}{\partial{x^2}} + V = i\hbar\frac{\partial{}}{\partial{t}}

This produces Schrodinger Equation. Next we consider the case for photon. The Hamiltonian for photon is given by Einstein’s Special Theory of Relativity

H^2 = p^2c^2 + m^2c^4

Since photon has no mass

H^2 = p^2c^2

Substitute our formula for Momentum and Hamiltonian

H^2 = -\hbar^2\frac{\partial{}^2}{\partial{x^2}}c^2=-\hbar^2\frac{\partial{}^2}{\partial{t^2}}

We can easily see

\frac{\partial{}^2}{\partial{x^2}} = \frac{1}{c^2}\frac{\partial{}^2}{\partial{t^2}}

It reproduces Maxwell’s Light Equation. Given a standard EM wave equation to be

\psi = Ae^{i(kx - wt)}

We can find its energy by hamiltonian operator

H\psi = i\hbar\frac{\partial{}}{\partial{t}}Ae^{i(kx-\omega t)} = \hbar\omega\psi \rightarrow E=\hbar\omega

It predicts correct energy of each photon as predicted by Einstein and Planck.

As a concluding remark, the assumption of the validity of two commutation relation turns out to be generally true even in relativistic case. Interestingly, one can also show that the two commutations are essentially equivalent. Dropping the First Commutation Relation, our Quantum Mechanics builds itself upon the incompatibility of energy and time, which it’s absolutely mysterious, utterly intriguing and extremely suggestive that an internal structure of space-time must remain undiscovered that governs this incompatibility.

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Principle of Concentration

We first define a term called Interest to represent one thing that a person may potentially doThen we collect all Interests of a person, and organize them in such a way that associated Interests are interconnected. This will give us a huge network of Interests. We then topologically fold our network of Interests into a multidimensional space while keeping connected Interests as adjacent points in space. This space formed is called Interest Space.

Each person is represented as a collection of N Interest points living freely in the Interest Space. We assume each Interest point has equal probability of moving in any direction in the Interest Space. Given N>>1 we can approximate the distribution of  Interest points by a continuous function which we call Concentration Function, represented as C(x,t),

C(\mathbf{x}, t) = C(x_1, x_2, ... , x_d, t)

\int_V C(\mathbf{x},t)d\tau = N

where d is the dimension of Interest Space, and {x_1, x_2, ... , x_d} represents a coordinate and d\tau represents a unit volume in Interest Space.

Random walk of Interest Point

As the name suggests, the magnitude of Concentration Function will represent the level of concentration at the specific interest point. Our assumption is that each interest point has equal probability of moving in any direction. Anyone with a basic physics training will immediately realize that it reassembles a random walk. For a simplified case where d = 2, it implies a person’s concentration, which is initially at the origin, will move around like shown in Fig. 1. It shows how our concentration will be digressed graphically.

Fig 1: “Random Walk” of Interest Point. Image adapted from Wikipedia http://en.wikipedia.org/wiki/Brownian_motion

Diffusion of Concentration

When we have a collection of N interest points diffusing simultaneously from the same interest point, from statistics, they follow a Gaussian distribution around the initial interest point, and get flattened through time.

Fig. 2 Qualitative feature of concentration diffusion. (Note the axis labels and time scale don’t correspond to our discussion) Image adapted from Wikipedia: http://en.wikipedia.org/wiki/Brownian_motion

Diffusion Equation of Concentration

An important feature of our construct is the Finiteness of a person’s Interest Points. In other words, we enforce a Conservation of Interest Points. Together with our assumption that each interest point has equal probability to diffuse into its adjacent points, it’s easy to show that our concentration function must follow a continuity equation.

\nabla^2 C(\mathbf{x},t) = D\frac{\partial{}}{\partial{t}}C(\mathbf{x},t)

To simplify the problem, we first investigate a one dimensional model. Assuming there is no Boundary Conditionswe can solve the equation with standard separation of variables technique. Given a constraint that C(x,t) must remain real, we get the solution to be like

C(x,t) = const. cos(kx - \frac{k}{\sqrt{D}}t)

I call this equation Freeworking Equation as we impose no constraints on interest space and time. The implication is that during freeworking, our concentration on any specific interest point will fluctuate sinusoidally with time, and our concentration will fluctuate through related interest points sinusoidally. All these make sense to me.

Fig. 3. Cosine function. Illustration that concentration fluctuates through interest points. Image adapted from: http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphtrigfunctions.html

Importance of boundary conditions

From previous discussion, when there is no constraints, concentration merely fluctuates indefinitely throughout space and time. So here comes the question, how do we raise our concentration on a specific interest point? This is when boundary condition starts to play an important role.

Suppose we have such a constraint that anything outside interest points A and B will be completely ignored. The solutions start to look like in Fig. 4. Compared with our freeworking equation where our concentration merely flows around indefinitely, now they become localized in a certain region. A remarkable fact is that now the solution form a complete set of fourior basics that are able to compose any functions within A and B. Similar effect happens if we constrain our time, we could arbitrarily concentrate on any interest point within A and B.

Fig. 4 Constrain our concentration that any interest points outside A and B will be ignored completely. Our concentration function will look like this. Image adapted from http://webs.morningside.edu/slaven/Physics/atom/atom5.html

What does it mean after all?

We have proved that to raise concentration on a specific interest point, we constrain our interest space and time. This is exactly the rationale why we often have to constrain ourselves by setting rigid deadlines and ignore irrelevant topics to be arbitrarily concentrated and therefore achieve great things.

One may ask if we have enough time, why should we care about productivity at all? The ultimate reason is the intrinsic constraint of our lifetime. We simply don’t have enough time. That’s how nature made it to be, neither too short that we can’t achieve anything, nor too long that our concentration starts to diffuse. Nature simply assigns each species an appropriate time scale in preference of productivity. It’s absolutely fascinating!

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Some entertainment

It’s a rainy day. In California it’s rare. I felt reluctant to go to my martial art class because I am unwilling to rush in the rain and potentially get a cold. I decided to stay in the bed and entertain myself a little bit by writing this post.

The question that I want to attempt is

Why can’t we live forever?

The motivation comes about when I wonder why life has to be bound by such a restricted time frame, and wonder what would happen if we could live forever. This post serves as an introduction to the following discussions on my theory of life, in which I  conjecture how life works in its most fundamental form.

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On Fairness

Assume there exists a fair game where no one is given any privileges over the rest. The rule of the game is such that we are forced to select one of them as the winner. So, who should we choose as the winner? Since no one is given any privileges, they share no privileges to be chosen. It seems that fairness and game are terms that never coexist. If fair, it’s unfair.

Fair game must never exist. 

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