Practical Quantum Mechanics

Most recent answer: 06/14/2013

Q:
In my quantum classes, we often discuss things like the measurement problem, the Quantum harmonic oscillator, Potential wells,Gaussian wave-packets, etc. Although I am beginning to grasp the formalism of my studies quite well,there is not a complete picture. I am unaware of any practical realization of many of the things we discuss. For example: How, exactly, are quantum measurements made? How is the potential of the Quantum harmonic oscillator (1/2*m*w^2*x^2) created? A common exercise in my class was to find the time evolution of a wave function, given its initial state. The question I now ask is; how would one create an this initial state? How can any wave function be made?
- Jeff (age 22)
Pocatello,ID,USA
A:

There are lots of techniques for making particular types of quantum states. Just for illustration, it's simplest to consider a system in which all the states are combinations of just two basis states. Single spins, such as those of an electron, are familiar examples.

Let's say that you want to prepare an electron spin in the state where its magnetic moment points down. Applying a very large magnetic field and cooling the environment down to very low temperatures has a very high probability of leaving the electron spin in that state. Now let's say you want the moment to point a different direction. Changing the magnetic field causes the moment direction to change in an exactly predictable way. In practice, since it's often hard to change the main field, the spin would probably be manipulated by applying radio-frequency radiation, but that's not needed in principle. The new state, say pointing east, can be expressed as a particular combination of the up and down basis states. The same basic principles apply to more complicated systems.

Harmonic oscillator potentials show up all over the place. Systems tend to dribble down toward the lowest available energy, i.e. toward minima in some potential. In general, the potential near the bottom of some well will go as the square of the distance from the minimum, at least for small displacements, giving a harmonic oscillator. The vibrations of atoms in crystals (sound waves) are essentially harmonic oscillators. Likewise the potential for electromagnetic waves looks like a harmonic oscillator, in terms of the fields. You can make a more literal harmonic oscillator by building a little cantilever of a pure silicon crystal. (Larger ones, or ones with more defects, have more complicated states available to them.)

The less cut-and-dried part of the answer concerns "measurement". The interpretation of the measurement process remains controversial. Within all of the modern interpretations (I think) the core of the process is "decoherence". This occurs when some quantum system interacts with enough of its environment to leave what amounts to a permanent record of what state the system was in. That destroys the possibility of interference between the different quantum possibilities. For example, maybe that electron got sent through a spatially-varying magnetic field. Spin up would go one way, spin down the opposite way. (We'll ignore some complications here.) These paths can be  recombined and show interference, so they both really are there. The electron went both ways.  The spin has not yet been measured.If the electron then hits a screen, jostling a bunch of atoms and sending out some photons, the up and down possibilities can no longer be recombined with interference, because they are not really part of the same world anymore. The electron spin has been "measured". If you look at a picture of the screen you'll see one possibility or the other. Does that mean that the possibility you don't see truly ceased to exist? Or does it mean that all the possibilities, including your state of mind, decohered into different versions? We don't know.

Mike W.


(published on 06/14/2013)