Re: [math-fun] Stupid QM question
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.
On Fri, Dec 7, 2018 at 4:42 PM Henry Baker <hbaker1@pipeline.com> wrote:
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?
Yes.
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?
Unitary matrices don't necessarily involve forces or expenditure of energy. Take a single qubit where the eigenvalues are not degenerate---for instance, the spin of a single electron in a magnetic field. The field doesn't use up any energy, and the electron doesn't move (that would take an electric field). The ground state and the excited state have different energies, so in the Schrodinger picture, the relative phase between them changes over time.
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.
In the example above, if you change the strength of the field, then the energy levels will change. If you increase the strength of the field for a fixed time and then decrease it to its original strength, the amplitudes on the original states will, in general, have changed. So from the perspective of the original basis, so long as the new, stronger field strength is in play, the amplitude is moving back and forth between the two states; but from the perspective of the new energy basis, they're remaining stationary. This is called "Rabi flopping". I would recommend Griffiths' _Introduction to Quantum Mechanics_ as a good starting text that covers all of this. If you're so inclined, http://gen.lib.rus.ec has all versions of the text, the errata, and the solutions manuals.
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 |ψ> = ∑ᵢ cᵢ|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.
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