Friday, March 15, 2013

Separated at Birth? Quantum Mechanics and Electrical Engineering Systems Analysis

I've been working on a history project peripherally for months, I'm just recording a few notes here as I still haven't gotten to the bottom of it.  Because I haven't arrived at the answer to my research yet, the following will ramble on a bit, but I wanted to capture my notes so far.  You see, my old electrical engineering courses keep creeping into my quantum classes and vice versa.  It's not that it's just the same math, it's also the same notation.  The ultimate answer to all of this may be that both subjects pulled their notation from pure mathematics.

The latest inspiration for really looking into this came up as I was studying for my quantum midterm yesterday. I came across the following integral in Merzbacher that I felt certain I'd seen before (picture 1).  Merzbacher certainly felt it should be familiar since not a bit of explanation was given for its execution.


When I got home, I pulled out my EE systems engineering book, "Discrete Time and Continuous Time Linear Systems" by Robert Mayhan and sure enough, there was something fairly similar, but not exact, in the table of Laplace transforms.


I know that seems tenuous, but it was only the latest in a series of similarities.  In quantum mechanics, we denote the hamiltonian by H.  In EE, the response function of a circuit is denoted by h(t) in the time domain and H(omega) in the frequency domain.  What do we do with that function you might ask?  We operate on state vectors of course!

Not only does the systems book define  operations on state equations, (although never mentioning the word opeartor), it also uses the same notation.  Here's Mayhan


and here's Merzbacher on quantum mechanical operators


Both fields are interested in determining the energy in the system as indexed by frequencies.


Finally, (for the moment), what we call a Green's function in quantum mechanics is called a convolution in EE systems analysis.

Getting into the History
I finally started looking into the history all this last night and came up with the following rather disjointed points.

The first journal article reference I could find to the use of Laplace transforms in circuit analysis was in the Proceedings of the Institute of Radio Engineers[3] circa 1949 in the form of a reference to a 1942 paper by the same author (picture 6)
I found references to the subject of Laplace transform circuit analysis, (frequently called operational calculus), in books a few decades earlier[1] (picture 7),


If you're a Philadelphia Experiment fan like I am, yes that V. Bush.

I also found a portion of an electrical engineering thesis[2] that looks very similar to the sudden approximation in perturbative quantum mechanics (picture 8).


That's all I've got for now.  Does anyone else have any thoughts on how the two fields wound up with such similar notation?

References:

1.  http://books.google.com/books?id=BUg4AAAAIAAJ&lpg=PA1&ots=MRLPb1EID5&dq=circuit%20convolution%20laplace&lr&pg=PA3#v=onepage&q&f=false

2.  The Principle of Equivalent Areas
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6371578&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6371578
10.1109/TAI.1960.6371578

3.  Clavier article
Clavier A.G. (1949). Application of Fourier Transforms to Variable-Frequency Circuit Analysis, Proceedings of the IRE, 37 (11) 1287-1290. DOI:


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