Introduction E9F5FC

Questions FFFFC0

• View
• Edit
• History
• Print

Physics

薛定谔方程

• 含時 time dependent
• 不含时 time independent

Equation

• {$\frac{\text{d}}{\text{dt}}\Psi=-i\frac{H}{\hbar}\Psi$}

Factors

• The number {$i$} sets up the possibility of two distinct but equally valid perspectives, a dual perspective, thus allowing for slack, as in symplectic geometry.
• Time is an inner variable and thus a hidden variable. It is there to give slack so the system can change continuously. The amplitude of the wave function, the probability density need not depend on time.
• Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy).

Solution

Overall

• The solution of the Schroedinger equation consists of a mother function which is the background, asymptotic solution, but which is pregnant in a way that differentiating yield through the chain rule a multiplicative factor. And then the product rule yields additive choices constituting position and momentum. Thus the whole point of the solutions is to distinguish between position and momentum. Differentiation evokes that distinction, much as with the Laws of Form. And those distinctions are captured by configurations - all possible ways of configuring position and momentum with the n cells of energy as provided. So these are like Wolfram's cellular automata states on which simple rules can act. But the orthogonal polynomials presents them as continuous variables, powers of x.

Motherfunction

• Gaussian integral shows that integrating over an infinite square is the same as integrating over an infinite circle - the square may be inside or outside of the circle - because the contribution at infinity becomes negligible.

Derivation

• The chain rule and the product rule work together. The chain rule yields a contribution of {$-2x$} from inside, the internal structure. The product rule yields a sum that injects the momentum by filling the new cell and an old cell.

Combinatorial Content - Orthogonal polynomials

• We know physics from the material world, in which we interact with a continuum, and do measurements accordingly. But the content of physics is not analytic, but algebraic, combinatorial, ideal. These are the two branches of the house of knowledge, as in math and also physics.
• The identity morphism goes from -i to +i for a total weight of +1. Whereas momentum relates -i and -i, or +i and +i.
• Empty space cells, stationary state levels model how we go towards simplicity, towards the ground state and the mother function.
• Momentum is like the gap between two position cells, thus requires two cells.
• The momentum pair ii brings to mind a string.
• The momentum two-cell ii (the measurement of momentum) can be thought of as an entanglement. It is the entanglement of two cells in space (or two measurements in space). The eigenfunction gives a set of possible entanglements, of which the most costly energy wise is no entanglement at all. The possible entanglements are dictated by the global constraints. A configuration that satisfies one global constraint may or may not satisfy a different global constraint.

Superposition

• Apparently, a wave function can consist of a mix of states, thus a mix of possible energies. Collapsing the wave function fixes a particular energy. As time goes on, the mix of energies evolves, smears, spreads, and thus there can be the possibility of tunneling, of achieving new energies. This violates conservation of energy. But the conservation of energy is expressed through the Heisenberg uncertainty principle which pairs energy with time.

Consequences

• In nature, what is exact is not the wave function, not the mix of energy states, but the possible energy states that can be expressed and the differences between them.

Hamiltonian

• The Hamiltonian is a unitary operator.

Similarities

• In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}.

Reformulation of the Schroedinger equations

The time-dependent Schrödinger equation is formulated as:

{$i{\hbar}\frac{\text{d}}{\text{dt}}\Psi=H\Psi$}

Below are my thoughts from 2019 on why it would be more intuitive to write it:

{$\frac{\text{d}}{\text{dt}}\Psi=-2\pi i\frac{H}{h}\Psi$}

Consider a periodic system, namely, a harmonic oscillator, like an ideal spring in one-dimension {$x$} with momentum {$p$}. Suppose that the spring moves between {$x_{max}$} and {$-x_{max}$} and at {$x=0$} achieves its extreme momentums, {$p_{max}$} and {$-p_{max}$}. And let that system have a certain amount of slack. This slack may be thought of as error bounds for measurements, or as a multitude of systems that we are choosing from, or as a looseness in the laws of nature so that momentum can transform into position and vice versa.

Position will be best defined ({$|x|< \epsilon_{small}$}) near {$x=0$}, at which time momentum may vary the most ({$||p|-|p_{max}||< \epsilon_{large}$}). And momentum will be best defined {$|p|< \epsilon_{small}$} when the system has extreme position ({$||x|-|x_{max}||< \epsilon_{large}$}). Overall, the total slack can be thought of as the product {$\epsilon_{small}\epsilon_{large}$}, a rectangle in phase space. I suppose that rectangle is an oriented area in that it circles around phase space {$(x,p)$} in one-direction, counterclockwise, and not clockwise. And I think this oriented area is conserved, although I can't prove that. And I guess that this is perhaps the simplest example of symplectic geometry, which I imagine is the study of the geometry of such slack in the form of oriented areas.

Suppose the wave function {$\Psi=0$} outside the rectangle and nonzero within the rectangle. (A problem to think about here is that {$\Psi$} needs to be continuous and even have continuous first and second derivatives.) Then the wave function {$\Psi$} is describing the location of that rectangle in phase space. And {$\frac{\text{d}}{\text{dt}}\Psi$} is describing the rate of change in that location over time.

Then my formulation explains all that is involved. The factor {$i$} means that the slack exists only because position and momentum are kept track of separately, where, say, momentum is multiplied by {$i$}. Without it, the the system would be rigid and there could be no conversion between kinetic energy (which is maximum at the extreme of momentum) and potential energy (which is maximum at the extreme of position). Multiplying by {$i$} means that slack in position arises from the value of the momentum, and slack in momentum arises from the value of the position. Note that if the rectangle was not moving, then {$\frac{\text{d}}{\text{dt}}\Psi=0$} and the probability {$\Psi$} is constant, which is to say that our rectangle is unbounded.

The factor {$-1$} simply arises from our choice of axes for position and momentum. When position becomes positive, momentum shrinks and grows negative. I think if we swapped the axes, then the factor would go away.

The factor {$2\pi$} perhaps arises in our unit of time, in that we are thinking in terms of an entire cycle of periodic activity in phase space. I imagine that the crucial thing here is that the exponential {$e^{2\pi i x}$} is the limit of {$(1 + ix\frac{2\pi}{n})^n$} whose multiplicative effect is simply to linearly add the rotations that take us around the circle an amount {$x$}.

Finally, in my formulation, we come to the crucial factor, which is the fraction {$\frac{H}{h}$}. Note that this has units {$\frac{1}{t}$} and thus is a frequency. Suppose {$h$} is fixed. Then the higher the energy {$H$}, then the greater the change in slack, which here means, the more rapid oscillation between extremes in momentum and position. I suppose for that to hold in a well defined periodic system, as above, then the fraction {$\frac{H}{h}$} is forced to be an integer.

If {$\Psi$} is a real valued probability function on the phase space, then the amount of slack never changes, but is simply transfered from position to momentum and vice versa. However, if {$\Psi$} has a complex component of probability, then I suppose that as this component is multiplied by {$i$}, {$\Psi$} smears out by a factor of {$e^\frac{H}{\hbar}$}. I suppose that is related to the problem in my (mis)understanding. Also, I am treating the Hamiltonian as a constant, when it is an operator, although in the simplest case, couldn't it act as a multiplicative constant?

Gigliola Staffilani. The Schrodinger equations as inspiration of beautiful mathematics.

Categorifying Schroedinger's equation

• John Baez. Categorifying the quantum harmonic oscillator.
• Stefano Gogioso. Categorical Semantics for Schrödinger's Equation.
• John Baez. Quantization and Cohomology (Week 10)
• John Baez. Geometric quantization.
• John Baez. Quantization and Cohomology. A diagram of the big picture in quantum mechanics.
• John Baez. Instructions on geometric quantization.
• Categorification in quantum topology and beyond