Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Understand Schroedinger's equation. 薛定谔方程 Solutions
Operators
Eigenfunctions
Probability
Measurement
Gauge theory
Assumptions
Equation
Factors
Solution Overall
Motherfunction
Derivation
Combinatorial Content  Orthogonal polynomials
Superposition
Consequences
Hamiltonian
Similarities
Reformulation of the Schroedinger equations The timedependent 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 onedimension {$x$} 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 ({$pp_{max}< \epsilon_{large}$}). And momentum will be best defined {$p< \epsilon_{small}$} when the system has extreme position ({$xx_{max}< \epsilon_{large}$}). 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 onedirection, 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
