Druggie Druggie Addict!

There's a hidden birthday message, can you find it?

So here's the case; as is the case with the USA, Singaporean soldiers are not allowed to take illegal drugs (besides the usual smoke I guess, which is perfectly legal). The question that I pose to you now, is:

If you'd like to do a survey to find out how many Singaporean soldiers consume illegal drugs, how would you do it?

You must keep in mind that if they do confess or are caught answering "Yes", then they will face the death penalty or some other kind of huge fine. The nice guy you are, you'd want to come up with a way to prevent this, and yet come up with some sort of estimate of figures.

Knowing this blog, it's definitely a statistical trick that you'll have to employ, but what? Heh.

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Well, the thing is, you've got to let them confess in secret, but yet you have to know what they confessed somehow. But you need to ensure their anonymity and safety as well!

As such, leaving any high-technology mind reading devices out, we are left with no choice but to play Jedi mind tricks.

Yeah right. We just use what we learnt in high school - that's right, simple statistics, for a good estimation.

We come up with three types of cards, as shown (of course their backs will be all of the same colour):


Each type of card is associated with one type of question - these questions need not be written on the card, but may be posed to the soldiers verbally. Now, to ensure that the soldiers draw out a card at random, so the chance of drawing a red, blue or green card is 1/3 for all, we have say, a good number of cards (say 72 cards) laid down like this:


So now, the chance or probability that any soldier draws the green, blue or red card, is exactly 1/3, even if he or she has a propensity or tendency to draw from the corners or from the middle of the deck. Make sure that the cards are laid in the order as shown for the probability to be true.

Let's give it a go; let's say for example, that out of 12000 soldiers who were surveyed, 5600 of them answered "Yes". Assuming all soldiers are sane, and that they completely understand English and are well, disciplined enough to not want to play with the system, they'll be truthful.

Therefore, all "Yes" replies must mean that the soldiers either chose the red or blue card.

Since 12000 is a huge enough number, and this is a fair test, we should be confident enough to say that on average, we expect 4000 soldiers to say "Yes" to the question of whether "Is this card red?"

If that is the case, then we expect that on average, at least 1600 soldiers do take some kind of illegal drug.

On the other hand, we also expect 4000 soldiers to say "Yes" to the question of "Do you take drugs?"

So what does all these mean? It means that on average, we expect at best, 1600 soldiers to be taking drugs, but at worst, 4000 soldiers to be taking drugs.

It'd be good to perform another test on another group first, to obtain the standard deviation! Haha.

So... what do you think? Aren't statisticians rather useful? Heh.

Representations

Well, for those who have no knowledge of matrices, here's a challenge I have for you; given that the rules of 2x2 matrix mutliplication are summarised as such:


Can you find me a 2x2 matrix, such that when it's multiplied by itself, yields:


There's about 10000000000000000000000000000000000 such matrices by the way. So if you can find 1, and you think you're smart, think again. Haha.

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Well, even though this post has been here for the longest time, it seems that no one has ventured beyond perfunctory reading, so I guess I'll just carry on.

One possible matrix would be:


And another would be:


There's actually a pattern: the number in the upper left and lower right positions are the same! But of course, you'd need the right combination. Heh. Can you find any more? There's probably a million more of these, haha.

Stay Still!

I think I’ll go into the idea of stationary states today; what exactly is a stationary state? Well, it’s essentially a state of a system, where the energy of the state remains constant over time. Defined more rigorously, it’s a state where the expectation values of observables remain constant over time.

For instance, let us take a look at the particle-in-a-box wavefunctions (as functions of time and position):


So let’s consider the probability density function of finding the particle within the box:


Notice that the phase factors (exponential time factors) cancel out when the complex conjugation is taken and multiplied! That is, the probability density function isn’t a function of time! It’s solely a function of position and thus, doesn’t vary with time.

How about the expectation value of the position? Well let’s take a look:


Hey! It turns out that the expectation value of the position doesn’t depend on time as well! So, it turns out that for all states of the system that are eigenstates (that is, if the system exists only as an eigenstate), the state is a stationary state!

So what isn’t a stationary state? Well, let’s look at linear combinations of eigenstates; we know that any linear combination (properly normalized of course) of eigenstates will still result in an arbitrary state that is a solution of the Schrödinger equation, so let’s try a positive linear superposition of the ground state and first excited state (I’ve normalized it for all of you already):


Let us now evaluate the probability density function (click on it to enlarge it):


Notice that now the probability density function is a function of time as well! The time phase factors no longer cancel out! It turns out that for any linear combination of eigenstates, the state will no longer be a stationary state – and therefore observables like its energy, momentum and even position will not have time-constant expectation values.

If you’ve heard physics professors go, “It’s all because of the cross terms!” this is what it means. Haha.

Postulato Negativito!

So I was looking, and I was just a little bit amused:

"So... there are two types of fundamental postulates in science. The first type is the self-evident kind, like how heat flows from hot to cold regions. The second type is one that is more subtle, and needs to be unfolded in a series of arguments before you get the point of it all."

And why am I laughing?

"Unfortunately, in Quantum Mechanics, the fundamental postulates all belong to the second type."

Lol!

Time Evolution

So if you look at it this way, then the time evolution of the wave function of a system is no more than an evolution that is governed by the energy of the system:


Then only separating out the time-dependent wavefunction:


It would appear that the time evolution of a system lies only in the changing of the phase of the system!

Strange. Weird. I still don't get what this means, haha.

Double Pendulum Revisited

Remember this post: http://wulidancing.blogspot.com/2008/07/double-pendulum.html?

Well, it turns out that you don't really need any high-level mathematics for this question, and I'm just wondering how come I didn't figure it out at that time, haha. So here's the whole set up again:


It'd be useful to see that the equilibrium position should have zero gravitational potential energy:


The trick to solving this question is to lift each pendulum up one after the other; so let's give the first pendulum a push and see what happens to the potential energy:


Notice that both pendulum bobs are raised by the same length, which explains the coefficient of '2'. Now let's give the lower pendulum another push and let's see what happens:


Notice that we've just added another term to the potential energy, and this time the coefficient is '1' because only one bob is raised.

Quite easy right? Haha.

Lagrangian in Action

As a follow up to my previous post, let's see some Lagrangian Mechanics in action, with the use of a very easy example - the simple pendulum! So let us consider the following set-up:


For your convenience, I've worked out the various quantities to take note already, and I've labelled all of them on the above diagram. Now, with all of these observables in place, we can now write down the Lagrangian immediately:


Notice that I'm no longer using the x coordinate! This explains why the Lagrangian is so versatile, because it can be expressed in terms of any general coordinate, and still work! So given this, let us work out the Euler-Lagrange Equation:


And we see that if we equate the two derivatives, we obtain the equation of motion:


Notice that we didn't even have to go about resolving out the various forces, like tension or weight or whatever! It's really easy with Lagrangian Mechanics!

I believe your teacher might have told you to "only make small oscillations when setting up the pendulum", and the rationale for that being that "small oscillations result in simple harmonic motion." But how?

Easy, let us consider small angular displacements, and our equation of motion automatically reduces to:


Voila! Notice that the second derivative with respect to time of the angular displacement (i.e. angular acceleration) is now directly proportional to the angular displacement! This is the very definition of simple harmonic motion!

Haha. Easy right? Lagrangian Mechanics really simplifies a lot of things. :)