reply to post by delusion
i have received some critisism of the apparent lack of sufficient explanation in this presentation. this is strange to me because i remember very
clearly what it is like to be utterly confused by the actual application of quantum mechanics, and this post was written from the perspective of what
i might like to have known at that time. i am still considering making a more explicit example, but honestly, i think it will be met with the same
critisism. my main purpose was to show with visual aids (i always think math is too short on the graphic visualizations), how the various jargon of
the science correlates to the evolution of an actual quantum system.
so, about the phase space. this is a great place to start your learning. think for a moment about the difference in characteristics of a laser light
versus an incandescent bulb. the primary difference between the two can be described in terms of PHASE. whereas the laser has a coherent phase, the
bulb has a mixed phase. conceptually, the phase of a quantum system is represented as a space called the 'Hilbert Space', also known as the phase
space.
i am sure you have heard talk about "superposition" in quantum science? the superposition can be thought of as the full spectrum of angles between
0 and 90 degrees. usually, say in a classical computer, a bit is either ON or OFF. in the hilbert space, we could say that the classical bit is at
0degrees (off) or 90degrees (on)....but only one or the other. but in a quantum system, the phase angle can be represented at any position between 0
and 90. the phase of a qubit is represented as a superposition of both 0 and 90 (a little bit of both). like the sine and cosine of trionometry.
because of superposition, a single quantum system can interfere with itself. if you imagine a wave in the ocean, and another wave opposite in phase
(one wave is at the top while the other wave is at the bottom), you can easily understand that the two waves will cancel each other out. so the
"wave-state" of the entire wave-like system depends on the exact phase of all of the components in the system acting upon each other, cancelling and
amplifying and producing all kinds of sometimes very striking dynamics (re: cymatics).
and this is the biggest difference between a classical and quantum computer. a classical computer requires a single bit to represent every single
transition state in the system....and the system is dynamically changing from moment to moment....that is a LOT OF BITS! and very difficult for a
classical computer to handle.
but a quantum computer, because of superposition, requires only one qubit to represent every single transition state in the system. this is because
the qubit IS THE TRANSITION STATE, itself. this is why, during the OP i stressed so heavily the nature of the STATE FUNCTION. if you can understand
this, you have understood quantum mechanics.
so, going back to the OP: you are looking at a single state function = a single qubit. the information content of the qubit is contained as the
state of the entire system. but representation in a classical computer would represent all of the positions of the matrix as a single bit, as a
sequence of one after the other, and the classical computer will never "comprehend" the state of the system, or that the system has a coherent state
at all.
quantum mechanics is a description of the state of the wavefunction in the phase space according to a given unit of measurement.
.....for anyone wanting to learn the math specifically, you may view the sequence of spin values down the left column as the eigenstates of the KET
and the sequence of values along the top row as the BRA, using the standard bra-ket notaion. the bra and ket represent the 0 and 90 degrees 'basis
states' of the phase space, and the superposition of the wave function is expressed in the pattern of intermediate spin values over the full matrix
of possible quantization values between the bra and ket.
and at the very heart of it all is the (totally anti-intuitive) mathematical definition known as the canonical commutator relation:
pq - qp = ih
where p is the bra, q is the ket , i is the imaginary number, and h is ??????
in standard math the following is usually true:
pq - qp = 0
but, you see, the wavefunction on the hilbert space always leaves you with some spare change. this should strike you as very, very strange. back
when QM was developed, h had not been standardized. h was just a bunch of weird numbers that kept falling out of the math when it should have gone to
0 but didnt. the value of h was a major source of anxiety among the developers. but h has been standardized as the plank's constant.
it is the plank's constant, h, which is the quantization of the quantum math function.
the phase space NEVER goes to 0 (fully nilpotent