Thursday, November 19, 2009

Just my opinion

Anyone who uses the word "quantum" in a serious discussion, and who has never solved Schroedinger's equation, should be stuck in a box with a radioisotope, a neutron-triggered poison dispenser, and an introductory textbook on quantum mechanics. This is what we call "incentive to learn".

First nominee for this treatment: Deepak Chopra. Anyone got their own preferred woo-merchant?


Dunc said...

While I fully agree on Chopra, I've never actually solved Schroedinger's equation myself despite making it to the 2nd year of a physics degree course (as far as I can recall), yet I have used to word "quantum" in a serious discussion (usually something along the lines of "No, quantum mechanics doesn't say that.") so I have to object to the proposed general principle.

Lifewish said...

If you got to 2nd year of a physics degree then you can argue that:

1) you have the technical skills to solve Schrodinger's equation (e.g. it's physically possible)

2) there were lots of periods during uni (typically after visiting the pub) where you can't remember what happened

3) clearly you were in a state of superposition during these periods

4) therefore you have both solved and not solved Schroedinger's equation, you just haven't collapsed the waveform yet to find out which.

These days the recommended method of waveform-collapsing is for friends to post embarrassing pictures on Facebook. Your mileage may vary.

Actually, solving Schroedinger's equation is fairly easy for simple cases. Let's say you have a fixed potential V(x) in 1 dimension. Then the equation is known to be:

E.psi = -hbar^2/2m.(d^2/dx^2)psi + V(x).psi

* m is the particle's mass (assume m=1/2)
* E is the particle's energy
* V is the potential mentioned above
* hbar is Planck's constant (assume hbar=1 - we can choose our units of measurement so that this is true)
* all the d's are partial derivatives
* psi(x,t) is any wavefunction matching the equation

A common example of V(x) is: V(x) = 0 on [-L,L] and infinity everywhere else. This is your basic square potential well.

In this case, the equation has to be solved over three ranges: (inf,-L), [-L,L] and [L,inf). In the outer two ranges, the answer is 0. It has to be. Anything else causes extreme screwiness due to scary infinities.

This also gives us phi(-L,t) = phi(L,t) = 0 as our boundary conditions for the middle range.

In the middle bit we have:

2.m(E - V(x)).psi = -hbar^2.(d^2/dx^2)psi

As per the list of variables above, for this example we're going to basically ignore all the constants. The "naked" equation is:

(E - V(x)).psi = -(d^2/dx^2)psi

And V(x) is 0 in the middle range, so:

(d^2/dx^2)psi = -E.psi

This is just the differential equation for an oscillator, so:

psi(x) = A.sin(sqrt(E).x) + B.cos(sqrt(E).x)

where A and B are just arbitrary constants.

But wait! Remember the boundary conditions! We must have psi(L)=0. Therefore:

A.sin(sqrt(E).L) + B.cos(sqrt(E).L) = 0

Firstly, this tells us that either A or B must equal 0. If B=0 then:

sin(sqrt(E).L) = 0

which means sqrt(E).L must be a multiple of 2.pi. So:

E = (2.n.pi/L)^2 for some integer n

Similarly, if A is 0, we get:

cos(sqrt(E).L) = 0
sqrt(E).L = pi.(2.n+1)
E = ((2.n+1).pi/L)^2

So overall, E = (n.pi/L)^2 for any integer n. This is what we call quantisation - the particle can exist in no other state.

Congratulations! If you can get all that straight in your head, you too can comment on the use and abuse of quantum mechanics in woo!

Of course this is the boring old way of writing out quantum equations, which owes a lot to fluid mechanics and is generally a pain. These days, people use a notation called the Dirac formalism, which is a) easier to use and b) a bugger to explain. I'll leave that for a later post.

Dunc said...

You're brining back some bad memories there dude... I can feel my eyes glazing over already. It's been nearly 20 years since I did any calculus - these days I can barely manage simple algebra. I always laugh my head off when people say "Oh, you're a computer programmer? You must be really god at maths!" The closest I've ever gotten to using maths at work is figuring out the correct dimensions to resize an image to given a maximum width and height.

Lifewish said...

I still love this stuff, it's like crystallised imagination. You start off with this very simple (albeit slightly counterintuitive) description of a system, you rearrange a few expressions, and out pops quantisation. Or the Pauli exclusion principle, which is a side-effect of a) quantisation of spin and b) fairly basic linear algebra.

Or the precession of Mercury from general relativity. Or RSA encryption from number theory. Or much of economics from evolutionary game theory.

And then you start wondering what simple elegant connections we'll find next, and looking round for sources of inspiration. And that's when you're a mathematician.

Dunc said...

Oh, I appreciate the elegance - I just can't hack the math any more.

I'm pretty sure I've also forgotten how to ride a bike.

Lifewish said...

Heh. If I'm honest, I had to look up most of that proof...

Far as I'm concerned, the elegance is the math. Everything else is just implementation.