04 Feb 06
I found a more complicated version of this in a book today. This equation basically represents a 'quantum harmonic oscillator', and E is a unitless version of the energy (probably why the lecturer who set it as a problem used the letter E). The allowable energy levels are of the form k+1/2 where k is any integer, which explains why E must be odd. I doubt that equations requiring E to be of other forms have physical interpretations.
05 Feb 06
Originally posted by sonhouseThanks muchly! It seems the analytical solution was correct (the polynomials the first site calls 'H' are the 'Ps' I've been going on about). That's actually very interesting; the solutions to the equation correspond to probability distributions of the location of the particle for different energy levels (defined by E).
I googled in 'quantum harmonic oscillator' and found this link, you
might find it interesting.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html
Here is a Java applet, animation:
http://www.falstad.com/qm3dosc/
"In the wavefunction associated with a given value of the quantum number n, the Gaussian is multiplied by a polynomial of order n (the Hermite polynomials above) and the constants necessary to normalize the wavefunctions."
The 'Gaussian' refers to the exponential part of the solution (it's the same as the one appearing in the Gaussian distribution). It looks like the solutions which converge to zero are the only relevant ones because they are the only ones which correspond to legitimate probability density functions.
Quantum mechanics seems to have approached this question by starting with the notion of energy states corresponding to odd integers, while we deduced those properties from some random equation. I like that continuity.
05 Feb 06
Originally posted by royalchickenIt also proves that even at zero point there is still energy which means
Thanks muchly! It seems the analytical solution was correct (the polynomials the first site calls 'H' are the 'Ps' I've been going on about). That's actually very interesting; the solutions to the equation correspond to probability distributions of the location of the particle for different energy levels (defined by E).
"In the wavefunction assoc ...[text shortened]... while we deduced those properties from some random equation. I like that continuity.
at absolute zero, the oscillations don't stop, its still moving a bit.
I wonder if anyone will ever figure out a way to overcome that and
actually stop molecular motion completely, bypassing the
quantum oscillation, or overcoming the oscillation I guess would be the
more correct way to put it. I can see where you could have detectors
of sufficient sensitivity to follow the motions of a molecule and match
the energy of motion like carefully timed laser shots that would try to
keep an atom in place. A technique like that is already used to cool
atoms down to microkelvin tempuratures, it was one of the steps
used when they finally got the Einstein-Bose Condensates, remember
all the fuss about them a few years ago? The conglomerate of atoms
all join together at a certain tempurature, a few microkelvin I think and
the wave functions merge and it becomes for all intents and
purposes one giant atom of the same stuff. That is freaky.
05 Feb 06
Originally posted by sonhouseNo, the zero pont energy of a quantum oscillator is pretty fundamental to quantum theory. It would be a violation of the uncertainty priciple, and is therfore impossible. It would be akin to inventing a perpetual motion machine; fundamentally not achievable.
It also proves that even at zero point there is still energy which means
at absolute zero, the oscillations don't stop, its still moving a bit.
I wonder if anyone will ever figure out a way to overcome that and
actually stop molecular motion completely.
05 Feb 06
Originally posted by corp1131Well there are instances where the uncertainty principle has been
No, the zero pont energy of a quantum oscillator is pretty fundamental to quantum theory. It would be a violation of the uncertainty priciple, and is therfore impossible. It would be akin to inventing a perpetual motion machine; fundamentally not achievable.
at least temporarily violated. The idea of not knowing the exact
energy and positon at the same time to arbitrary limits has been
exceeded to a certain extent by quantum trickery but overall you
are correct. Guy can dream, eh. Suppose you had a million detectors
arranged around the atom being attacked (attempted violation)
and it can sense the zero energy motion. That in itself would be
a fundamental generation of random motion which could be made into
a totally random number generator but suppose this knowlege of
the attosecond by attosecond position, I guess not, the more I think
about it. Attosecond timing would not do it. That would spoil the
physical position measurement in the first place. Dang. That sucks.
Slippery devil. But there has to be a way....
