soft question - Are quantum mechanics calculations useful for engineering?

I heard it's is pretty tough to get results for more than a few quantum particles. Are quantum mechanical calculations useful at all for any technology that is being sold? Or do they use quasi-classical results at most?

Is there hope that advances in QM calculations would make any change to the technological world? I'm basically considering everything that could make a change to our life disregarding pure theoretical research results.

Which path of QM advances could potentially be the most yielding in terms of practical results?

EDIT: From what I experienced, experiments are already producing results, when theorists are still trying to fit their theories to the data. So why do you need the theoretical calculations then? Do they have predictive power that would be found easier and more precise with experiments?

Answer

There are several different levels of advanced in quantum mechanics. I will try to answer using these levels of quantum mechanics:

1. Basic: single particle or single particles interacting with a single atom/nucleus, or classical field picture--- anything Einstein would have been comfortable with.


2. Advanced: highly entangled many-body quantum mechanics, involving many-body effects that cannot be understood from single-body or single-field picture.


3. Inscrutable: quantum computing--- actual exponential computation as compared to classical behavior.

I will give an off-the-cuff list of things that were predicted theoretically at each of the three levels, that was hard to understand from just seat-of-the-pants non-quantum intuition.

At level 1, there are essentially as many examples as you care to list:

• Electron diffraction: The diffraction of electrons from crystals was one of the early predictions of quantum mechanics which was confirmed experimentally. Knowing that electrons diffract is important for the construction of electron microscopes, and you need to know the relation between wavelength and momentum.


• lasers come from spontaneous emission theory, and Einstein's prediction that a coherent collection of bosons will make other bosons be created with the same momentum preferentially was very surprising. This is the basic idea behind lasers, and BEC's, and these were only found because the theoretical principles were known in advance.



• Slow neutrons are dangerous: If you classically estimate the neutron scattering cross section off a nucleus, at low momenta, you are completely wrong, because of resonance effects. These effects can blow up the nucleus to look (to the neutron) as it it were the size of a barn, when it is normally the size of a kernel of corn. You can look up the unit "barn" for a more accurate etymology.


• qualitative chemistry: If you use simple quantum mechanical orbitals, and the notion of superposition (which is called resonance in chemistry), you can get an idea of which molecules will make dyes, what shapes will be preferred, and so on. These types of things were worked out by Linus Pauling, and led to the discovery of the alpha-helix, and later, to the structure of DNA.

There are too many class-1 examples to list, so consider class 2. Here, one is looking for a theoretical insight in a many-body system, with a highly entangled wavefunction, which leads to practical predictions. The easiest example that comes to mind is BCS theory.

• BCS theory: this predicts that any very cold Fermi system with the weakest of attractive interactions will produce a strange vacuum state, where it is like a Bose Einstein condensate of paired-fermions, even when the force is too weak to bind two individual Fermions into actual pairs. The presence of other Fermions in the sea is essential, it makes a condensate of particles which do not exist really.

One of the most striking prediction of BCS theory was the prediction that He3 should become superfluid at ultra-cold temperatures. There is no reason you would suspect this from experiments on superconductors, without the detailed theory of Cooper pairing. This was spectacularly confirmed by difficult experimental work of Lee, Osheroff, and Richardson, work which was awarded the 1996 Nobel prize.

The theory of renormalization is quantum, class 2--- many body. But it is equally applicable to statistical systems, where simple models allow one to predict all sorts of phenomena that were not suspected experimentally. Here is an example:

• Anderson localization in 1d and 2d: Any sufficiently long wire is insulating. Any sufficiently large sheet of conductor is also insulating. You would never guess even the 1d business from experiment, but it is true, and needs to be considered when you make very thin wires. Anderson localization itself is in class 1, but the renormalization analysis which allows you to say things like this is class 2.

Quantum field theory has made contact with experiment, most elegantly through 2d conformal field theory:

• Rational 2d critical exponents: This was predicted from sophisticated quantum field theory considerations, relying on the conformal algebra from string theory, relying on the 2d conformal field theory of Belavin, Polyakov Zamolodchikov. That in itself was an extension of 1960s work on operator product expansion, by Zimmermann, Wilson, Kadanoff and Polyakov, which defined the correct algebra for renormalized fields. The rational critical exponents are confirmed experimentally using systems as divers as polymers, 2d fluids, but also using exact solutions, and conputer simulations. You would have a hard time guessing an exponent is rational from an experiment.

But by far the most spectacular type 2 quantum theoretical application is:

• Semiconductor physics: The qualitative ideas of semiconductor physics, including the existence of "P-type" charge carriers, were understood theoretically in tandem with the experimental production of these materials. The theory of doping is not so sophisticated--- you need to know which are donors and which are acceptors, but the theory of p-type semiconductors relies crucially on many-body effects, so that you have particle hole symmetry. This is the central technological advance of the late 20th century, and made possible the computer revolution.

In class 3, there are several potential applications:

• Simulating quantum systems: As Feynman noted, a quantum computer will be able to simulate other quantum systems efficiently. This is impossible on a classical computer.


• Factoring: Given a quantum computer, Peter Shor showed how to factor numbers, which will make current cryptographic systems insecure.


• Grover's database search: This allows you to search a database with N items in $\sqrt{N}$ steps.


• Guaranteed secure communications: You can make a channel in which you can ensure that you and your communication partner are not eavesdropped on.

For the other applications in this class, I defer to Nielsen and Chuang. The problem with class 3 applications (at least the full blown computational ones) is that we aren't going to be 100% sure they will work until we build them. The other option is that quantum mechanics will fail for these.