When I was taking math classes in college I enjoyed topology, differential equations, calculus and combinatorial math, but somehow could not bring myself to drum up enthusiasm for linear algebra.

If I had picked physics as my major (which I almost did), I would not probably have escaped from the clutches of linear algebra while learning quantum mechanics. As it happened I picked chemistry, and most of the quantum chemistry that was served to me after that was sans linear algebra.

On his blog luysii has an excellent set of notes on linear algebra from a QM class that he audited. As I mentioned on his blog, it's interesting how much one can get away with in QC without linear algebra. Thus, take a look at some classic textbooks- Levine, McQuarrie and the classic Pauling and Wilson- and one can go a long way with very little LA. The only things that you are really required to know are eigenvalues and eigenfunctions, but even then the Dirac notation is usually skipped in elementary QC. About the only QC book I know which utilizes large doses of LA is the sophisticated book by Szabo and Ostlund.

Yet LA matters and as luysii demonstrates, there is a generality and elegance to it. There is at least one key LA theorem which is mandatory knowledge in QC. When you are learning about the variational principle (which is used to find approximations to the ground state energy of a system), you derive the so-called secular equation by utilizing a very important LA theorem; that a set of linear homogeneous equations has a non-trivial (non-zero) solution if and only if the determinant of the coefficients is zero. Further on, matrices also come into play in important ways when you are learning about the calculation of transition states, normal modes, and energy minima in molecular mechanics. In the latter exercise you have to calculate the Hessian matrix and then diagonalize this monstrosity (thank god for computer programs).

Perhaps it's not surprising that QC can go a long way without much linear algebra. QC is an application of QM to problems of chemical interest, and the whole reason why the Schrodinger formulation of quantum theory became hugely more popular than the equivalent Heisenberg matrix formulation was that it was more tractable to applications (essentially plug in the correct expression for the potential energy) and couched in the more familiar 19th century language of differential equations. If you wish to know about matrix mechanics take a look at Max Born's excellent book "Atomic Physics"; I had to give up on that particular section.

But even the great Erwin's celebrated paper introducing his equation was titled "Quantization as an Eigenvalue Problem". Maybe it is worth even for a "quantum engineer" (as the late Wolfgang Pauli once somewhat derisively called Enrico Fermi) to learn some linear algebra.

Posted: December 14, 2017 at 05:54PM

2 hours ago in Field Notes

Ashutosh --thanks for the mention, but I'm writing up these notes because linear algebra was NOT discussed or developed very much in the QM course (for one thing there wasn't time, for another I don't think the students had ever been exposed to it).

ReplyDeleteI'd studied LA on my own years ago, but didn't really appreciate what was going on (e.g. what Hermitian matrices are good for etc. etc.) until I audited QM. So it works both ways -- QM illuminates LA and vice versa.

Luysii

Ah, that explains it. I was indeed surprised that a college course on QM would include so much LA. In fact most of my physics friends also did not really learn much LA-inspired QM. Thus this is an eye-opener. By the way I vaguely remember some of my smart math olympiad friends in college recommending a classic; "Finite dimensional vector spaces" by Paul Halmos.

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