$$-i \hbar \frac{\partial}{\partial x} (\frac{\partial}{\partial t}) = -kx$$
Now, why wouldn't that be an equation?
Because you don't equate differential operators like that. If, for example, $$\partial_{t}\phi = \partial_{x}\phi$$ that doesn't mean you can write $$\partial_{t} = \partial_{x}$$, such an expression is meaningless. $$\partial_{t}\phi = \partial_{x}\phi$$ says that there is
some field $$\phi$$ where at every point in space and time it's partial derivative wrt time is equal to its partial derivative wrt space. $$\partial_{t} = \partial_{x}$$ is saying "These two operators are identical, they not only have the same action on
some function, they have the same action on
all functions".
Yes, there's the relation $$\hat{p} \leftrightarrow -i\hbar \partial_{x}$$ used in quantum mechanics but this is relating two operators which are identical. You arrived at this dubious conclusion because you assumed all you had to do to go from classical to quantum is just change notation, ie assume all classical p's become quantum p's etc and then just mess with things using quantum mechanical expressions.
The correct way to express the classical equation you're looking at would not to change the classical p into a quantum operator p but instead into the
expectation of that operator. This is well known to anyone who has actually done quantum mechanics and not just copied it from Google hits, in that classical quantities relate to
measurements of operators. Want to see how it's done, seeing as you obviously don't know how?
We've got some wave function $$|\psi(t)\rangle$$, which satisfies the Schrodinger equation, $$-i\partial_{t}|\psi(t) = \hat{H}|\psi(t)\rangle$$. This is trivially solved to $$|\psi(t)\rangle = e^{i\hat{H}t}|\psi(0)\rangle$$. We want to measure some quantity A, which is associated to a quantum mechanical operator $$\hat{A}$$. This operator is time independent, it has an infinite dimensional matrix representation where all entries are constant. Therefore we get $$A(t) = \langle \psi(t)|\hat{A}|\psi(t)\rangle$$. Putting in the solution for the wave function we can collect all of the time dependence into a single term, $$\langle \psi(t)|\hat{A}|\psi(t)\rangle = \langle \psi(0)| e^{-i\hat{H}t}\hat{A}e^{i\hat{H}t}|\psi(0)\rangle \equiv \langle \psi(0)| \hat{A}(t)|\psi(0)\rangle$$. This is how you get different 'pictures' in quantum mechanics. The Schrodinger picture considers the operators time independent and the states time dependent, while the Heisenberg picture is the other way around. Either way we might want to ask what the time derivative of this observable is, $$\partial_{t}A(t)$$. Well we just use the chain rule and the fact $$[f(\hat{O}),\hat{O}] = 0$$ always, $$\partial_{t}A(t) = \langle \psi(0) | -\hat{H}\hat{A}(t) + \hat{A}(t)\, \hat{H} |\psi(0)\rangle = \langle \psi(0) | [\hat{H},\hat{A}(t)] | \psi(0)\rangle \equiv [H,A(t)]$$. In the Heisenberg picture the states are time independent so we get $$0 = \langle \psi(0)| \Big( \partial_{t}\hat{A}(t) - [\hat{A}(t),\hat{H}] \Big) | \psi(0)\rangle$$. This is the
Heisenberg Equation of Motion for the time dependent operators and their observables.
So let's put in $$\hat{A} = \hat{p}$$. Well we know that $$\hat{H} = \frac{1}{2}\hat{p}^{2} + U(\hat{x})$$ and the first term goes and we're left with $$\partial_{t}\hat{p}(t) = \langle [\hat{p},U(\hat{x})] \rangle$$. It's easy to show that $$ [\hat{p},U(\hat{x})] |\psi(0)\rangle = -(\partial_{x}U)(\hat{x})|\psi(0)\rangle$$ and thus we have not an operator equation but an equation for the
observables of operators as $$\dot{p}(t) = -\nabla U(x(t))$$, exactly as the Newtonian mechanics would say if you started with the classical Hamiltonian $$\frac{1}{2}m\dot{x}^{2} + U(x)$$.
The fact you don't know this shows how you've never done any real quantum mechanics, you would have come across this in an introductory course or textbook on quantum mechanics, as it is a nice example to illustrate quantum mathematical methods while returning a result the students should be familiar with. Yet
another piece of evidence you're trying to talk about things you don't understand.
Actually, I consider what I have done relatively easy.
It's generally the most ignorant who have the most inflated, unjustified view of their abilities.
All I did was swap some classical physics for the canonical quantization method.
No, you read what the classical expresions where, you looked up a bunch of formulae from quantum mechanics and you indiscriminantly applied them on the assumption such usages were valid. As I just explained, they are not.
So easy for you to say I am ''playing physics'' when I have a genuine interest in it like the next person here.
No, you don't. You have an interest but not a genuine one. A genuine one wouldn't start thread after thread throwing around expressions he doesn't understand, pretending to be doing actual physics when it is neither actual physics nor demonstrating any understanding of actual physics (quite the opposite in fact). It is very very clear you do not have a working understanding of this area of physics, both from this thread and your Planck thread, never mind all your previous accounts. There's plenty of people here who don't understand quantum mechanics and most of them, the half rational ones, don't start thread after thread spewing out algebra they can't do properly, claiming its about physics they don't understand. You don't have a genuine interest, you have an interest in
appearing to be genuine and informed when in fact you are neither. I get it, you want to put on the proverbial white coat, say something which sounds complicated and delude yourself you've achieved something. Unfortunately it results in precisely the opposite.
you're not exactly saying my post is wrong in any way.
You regularly try this line of flawed reasoning, ie if I don't explicitly say "This is wrong" then you assume I'm accepting it is right. The fact I might neither have the time nor the inclination to go through a post line by line doesn't mean I accept it correct. You should have learnt that by now, given all the times you've tried this, only to have me turn around and nail your post to the wall in extreme detail. Besides, even if there were no one here explaining your mistakes you'd still be practically innumerate and extremely bad at quantum mechanics and relativity. Even if this forum were filled with thousands of people posting "Wow, what a genius you are!" and "Surely you are the most insightful physicist ever!" and giving you offers of marriage or free beer and blow jobs, it wouldn't accomplish anything. No one with any mathematics or physics education would believe you, it wouldn't get you a research job or a PhD or a publication. It would accomplish sweet F.A..
Your behaviour of sticking your fingers in your ears and declaring I don't respond to certain posts or how you try to pick up on little things like "Is it an equation or an expression!" show you know you've backed yourself into a corner. If you had something good to retort with you'd do it. Instead you're stuck trying to magnify out of all reasonable proportions completely pathetic responses.
Your interest is diverted on me alone, which is lousy moderating.
Because if I reply to you then surely I cannot be replying to or reading any other threads
Lousy logic.
Maybe then if you agree with him so much, you can start off where he left off and explain why that specific equation isn't actually an equation, secondly, explain why this is psuedoscience! Because to be honest, this only looks like you trying to have a dig at me!
....
You have put this in psuedoscience which I think is way harsh - considering I am not attempting to ''explain any fringe sciences''
I moved this here because I do not believe you started this thread in good faith. Even if you weren't Reiku you've presented enough evidence that you don't understand the necessary physics to have a proper discussion on this stuff so starting a thread where you try to do physics, badly, is not a terribly good idea. You have shown you're happy to carry on waffling nonsense even when your mistakes are explained to you by multiple people and since you open this thread with a load of algebra it's clear you're going down the route again. Furthermore, as Tach commented and as I just explained, you've made several mistakes already which show you don't understand the notation and also that you haven't actually learnt any quantum mechanics. Someone not knowing quantum mechanics isn't a problem if they behave themselves, ask reasonable questions and enter into honest, informed discussion. You have shown you won't. You having a thread about quantum mechanics in the main forum will only serve to feed your delusions of competency, which I have no wish to prop up.
If and when you demonstrate a sound understanding of the material you post lengthy mathematics about,
then you can post such things in the maths/physics forum. You can ask qualitative questions, you can post
single equations for discussions, you can reply to threads, but threads like this, where it's line after line after line of mathematics you cannot do properly, on a topic you blatantly don't understand, you cannot do. If you'd asked the question "Is there a quantum version of classical equations from Newtonian mechanics?" along with an example equation from Newtonian mechanics then the thread would have stayed. As such you attempted to develop your own answer, a task you are clearly ill equipped mathematically to do. You want to put forth your own take on things, your own derivations, your own results, do it in pseudo or alternative theories. Consider this your last 'friendly' warning in this regard.