General Principles
A quantum mechanical system with arbitrary N degrees of freedom obeys a relation
(I) IPsi(t)>=
{e^[-i/hH(t-t0)]}IPsi(t0)>
which is evidently notationally similar to that obeyed by a very familiar electrical engineering device - - a linear N port filter.
I am referring to an electrical network constructed of wires and linear components
(e.g., coils, ohmic resistances, capacitances) so as to extend parallel to some spatial coordinate axis, say x. Typically, one applies to electrical contacts at some location x0, N "input" sinusoidal voltages of the same frequency w but differing phases and magnitudes and is interested in computing or else measuring the magnitude and phase of the same frequency voltage component for some N "output" contacts at location x. It is well known in EE and experimental physics that
(II) V(x)=R(x,x0)V(x0)
where R(x,x0) is an NxN matrix and the N components of vector V(x) are the port voltages at spatial cooordinate x in "phasor" form, i.e.,
jth component of V(x)=[voltage amplit.]e^[i voltage phase]of the jth port at
location x
Matrix R typically has a w dependence but no other dependence on input voltages V(x0).
So far, the mathematical similarity between (I) and (II) is provocative but certainly not identical. I intend to add a bit more detail to the discussion of filter construction; if the filters already alluded to are constructed to obey just a few more constraints it will turn out that one may indeed achieve a perfect quantum mechanical analog.
Consider just for the space of this paragraph a single sinusoidal voltage input instead of a multiplicity N. A voltage filter composed of a series connected sequence of input terminal, capacitor, inductor, output terminal, resistor, ground, - - by itself or coupled to either a voltage inverting or non inverting op amp - - is all that is needed to produce output/input voltage ratios of arbitrary magnitude and phase; this result follows from basic ac circuit calculations.
A small amount of additional consideration will thence show that one may construct filter networks solely out of coils, resistors, capacitances and op amps (together with the odd voltage follower to head off any loading effects), which are characterized by any desired NxN dimension complex number matrix. In particular, then, utilizing only the very familiar electronic components just enumerated one may construct electrical filters corresponding to arbitrary unitary matrices (which in particular preserve the norm of vector V(x)).
We are almost there. Suppose one constructs a filter by cascading many layers each comprised of a filter plate of thickness delta x, each plate being characterized by the matrix R so that
V(x=right boundary of R)=
R V(x=left boundary of R). Let us suppose that delta x is sufficiently diminutive so that to acceptable accuracy we may then write
V(x)=(R^[(x-x0)/delta x])V(x0)
as regards the filter made up of many cascaded plates. (For sticklers who object that one cannot connect filters in series without degrading their individual performances - - this potential flaw is simply avoided by putting a voltage follower immediately after each filter plate's N input
terminals.)
Suppose that we furthermore construct this filter R to be unitary; we are perfectly capable of doing so for the reasons outlined in the preceding paragraph. Since it is exactly true that any unitary matrix may be expressed as the exponential of i times a Hermitian matrix, we may recast
R = e^[iH{R}]
(read H{R} as matrix H sub R
- - the operative HTML web page generator being limited regarding scientific notation)and substitute into the immediately preceding equation to obtain
V(x)=
e^[iH{R}(x-x0)/delta x]V(x0)
Finally, if we decide to relable the coordinate x, previously denoting depth location within the filter, as t instead, we immediately arrive at a characterizing equation for our analog system of filter plus multiport voltages that exactly corresponds to the Heisenberg equation for a quantum mechanical system having Hamiltonian operator
H = -hH{R}/delta t.
Turning this around, if one constructs the filter by stacking plates each characterized by thickness delta t and matrix
R=e^[-iH delta t/h]
one obtains an analog system simulating arbitrary quantum mechanical system with Hamiltonian H. At the risk of flogging a dead horse, since the proposed simulation represents the wave vector as a vector array of fully measureable voltage phasors, all necessary information is apparently accessible in this proposed analog simulation of quantum mechanics.
(Filter simulation of quantum mechanics first announced within academia 1/18/2006 via a widely disseminated email
attachment)