Re: finite v. infinite spectrum Fair enough. But I don't really need a full harmonic oscillator; I just need something that has periodic activity -- a heartbeat, if you will. I presume that this can be done with qbits? But now you're going to tell me that this periodic activity *requires a constant application of some sort of unitary operator*; it can't simply sit there and vibrate/spin/whatever without any forces coming into play? Can't there be some oscillations between two discrete energy levels -- e.g., two different energy levels of an outer electron of an atom? For example, an atom in a 1-D echo chamber with perfect reflectors at both ends whose distance is consistent with the wavelength of the difference in energy levels. At 01:54 PM 12/7/2018, Mike Stay wrote:
On Fri, Dec 7, 2018 at 2:24 PM Henry Baker <hbaker1@pipeline.com> wrote:
Is the "state" of the system a 3-vector of complex numbers or a 3x3 (constant?) matrix of complex numbers?
It's a 2^3=8-dimensional vector of complex numbers |Ï> = âáµ¢ cáµ¢|i> where i ranges from 000 to 111.
Note that I've implicitly chosen an orthonormal basis for this vector; this amounts to picking a direction "z" in space.
By "state change", I presume that we pre- (or post-) multiply this initial "state" by a 3x3 unitary matrix; correct?
Premultiply by an 8x8 matrix U(t) = exp(iHt) where H is a Hermitian perturbation of the system.
So our computation consists of k steps -- an ordered sequence of k matrix multiplications; correct? Yep. The time-reversed computation is the conjugate transpose of the entire sequence; correct? Yep. So how can I represent a simple harmonic oscillator using one (more more) qbits? Wouldn't a harmonic oscillator "DO" something even if isolated from the rest of the universe? Wouldn't there be some time-variation of the "state" -- i.e., a periodicity?
A quantum harmonic oscillator assumes a quadratic potential V(x) = (x^2)/2.
The nth eigenvector of the system has energy proportional to (n+1/2).
In the Schrodinger picture, the states with energy E change phase by exp(-iEt/â).
In the Heisenberg picture, that phase is absorbed into the Hamiltonian operator.
To get the particles in the QHO to change energy levels, you perturb H = (p^2 + x^2)/2 for some period of time, which multiplies the state by a unitary matrix as above.