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## Does Wolframalpha's Android app support solving quantum mechanics equations and generally all differential equations and support step-by-step solutions and exporting the answer to PDF without extra costs?

Familiar with this equation? E = PE + KE If you’ve been in any middle or high school physics class, you probably said yes. That thing on the left there represents the total energy of a system - all the total “stuff” that can happen. On the right side, we’ve got the potential energy - all the stuff that could happen, being added to the kinetic energy, which is all the stuff that is happening. In physics, we call this the Law of Conservation of Energy. If nothing comes in from the outside, the total energy of the system - potential plus kinetic - stays constant. However, in quantum physics, the concepts of “potential” and “kinetic” energy become… difficult. For example, we might measure the potential energy, then measure it again a bit later, and get two different numbers! Obviously, this makes the concept of just adding both energies together difficult, if what we get may or may not actually make sense for the given time. Enter “operators”. Operators, in quantum mechanics, are like functions. We tell them what a quantum state is like, and t spit back out a number that describes the system. So, in order to “quantumize” our equation, we need to replace the potential and kinetic energy terms with operators. \hat{E} = \hat{K} + \hat{P} Now, we just have to figure out what the hell our operators mean, and we’re good. We can start with thinking about what exactly we mean when we talk about energy. Energy isn’t really a physical “thing” - you can’t pick it up and hold it - so when we talk about “energy”, we usually mean “how something changes with time. Now, if you’ve taken any calculus, you’ll know that this means we should be taking the first derivative with respect to time. Since we describe things in quantum mechanics with wavefunctions (\psi), that means we can rewrite the left side like this. i\hbar \dfrac{\partial \psi}{\partial t} = \hat{K} + \hat{P} The “K” is a bit harder. Our major hint comes from Newton’s Second Law, which states that kinetic energy is proportional to the acceleration of an object. And since acceleration is the second derivative with respect to time, we can with a bit of figuring rewrite \hat{K} as -\dfrac{\hbar ^2}{2m} \triangledown ^2 \psi where \triangledown ^2 is just a fancy way of writing “the total acceleration”. We won’t even try to rewrite the potential energy. There’s just way too many things that would go into it. All we really know is it’ll depend on the position of the object, so we’ll rewrite it as a function and leave it there. i\hbar \dfrac{\partial \psi}{\partial t} = -\dfrac{\hbar ^2}{2m} \psi \triangledown ^2 + P(x)\psi And we’re done! That’s the one-dimensional time dependent Schrodinger equation - the quantum-mechanical version of the law of conservation of energy. TL;DR. The Schrodinger equation is a second-order differential equation that quantumizes the fact that an object’s state depends on the sum of its potential and kinetic energies.

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I haven’t personally got involved in automata research, but I imagine when I get around to it, I would love to check out things like Grieg and Brouwer, in particular the theory is very similar to the kind of automata theory that I know from math class. Now if I want to make a decision about what type of solution I want to have I get in to that kind of system. So now, I want to ask you whether there are any examples of automata that are not solvable at all? That is basically my main question here and my answer is no. In general, we can do this. There are cases where an automaton cannot be solved because that system is a non-solvable system. We just ignore that as if it doesn’t exist. Now, we can get into some pretty exotic things, like systems with zero volume, zero.

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