How to form an equation of an atom? I am an amateur high school student enthusiastic about quantum mechanics. To my knowledge, Schrodinger's equation is capable of explaining all details of a particle. Similarly, is it possible to describe an atom of a particular element? Is it possible to derive an equation for, say, a hydrogen atom?
If Schrodinger's equation is not enough, what is that one needs to explain an atom mathematically? Thanks in advance and sorry if my question is a bit vague...
 A: 
If Schrodinger's equation is not enough, what is that one needs to explain an atom mathematically? Thanks in advance and sorry if my question is a bit vague...

It's not vague so much as it's complicated. Read about the 3 body problem, in astronomy.  If you read the article you will see how much we depend on computers to calculate planetary orbits. There is no equation that will tell you exactly where a particular planet will be exactly at a given time, the mutual pulling and motion of the planets is not easy to work out.
I don't want to call or compare particles to little planets, the above is just an illustration using gravity, but it is something like the same problem with electrostatic forces. We don't have an equation for anything more complicated than the simplest atom, that tells us where or what electrons are doing. Added to that, the constituents of an atom follow the laws of probability, so that's even more complication thrown in.(a lot more)....

To my knowledge, Schrodinger's equation is capable of explaining all details of a particle. Similarly, is it possible to describe an atom of a particular element? Is it possible to derive an equation for, say, a hydrogen atom?

No, Schrodinger's equation is not realistic for anything like explaining all the details of an atom. It takes a lots of study to see why, but Schrodinger's equation does not allow particle interaction, it does not cover the spin of particles or the production of particles and it is is not suitable for two experimenters moving at different speeds relative to each other, or one being in the vicinity of a black hole while the other experimenter is far away, to agree on the results of the same experiment.  
Physical laws should be the same throughout the universe, but Schrodinger's equation does not conform with that, so it's not enough, we need Dirac's Equation, (and that's just for starters.)
To describe what we know about the properties of an atom would take a library, The Standard Model Wikipedia  is a short summary, (and what we don't know, well another library, its size yet to be determined).

If someone actually did derive an equation like that, what factors will he have to look into to do so, such that the other properties just follow...

A quote from Woody Allen: Confidence is what you have, until you understand the problem. 
Anything I write here should a. not be taken personally, I am simplifying it (for myself) as well as you and b. should not be taken as in any way correct until you research the link belows. 
I self study, and so although my dog hears my questions, his answers are sometimes hard to follow. In other words, don't believe a word I say until you read someone else.
Please read the links, as far as you can, as they are more informed than this post is.
If you read the Dirac Equation article, you should see that following its discovery, the Schroedinger Equation was replaced by the concept of Quantum Field Theory, not particles, as models for  "everything". 
Particles and the forces connecting them are seen as excitations/vibrations of these invisible fields. A particle and force carrier list follows (and this is the most common picture on this site): 

Read this article on the Standard Model, which explains things bettter than I can. Please remember that the SM is a (sophisticated) list of the properties, of the particles, it does not attempt to say what the particles are actually "made of". 
It only deals with their interactions with each other, so as you are looking for an equation, I have to say there may only ever be an equation(s) describing properties and interactions, because the mass, charge, energy etc of the particles and underlying fields are measured experimentally and there are no verified equations that explain the 20 plus properties of them. 
Now here is someone else: Energy of the Vacuum by AccidentalFourierTransform, explaining in other words, what is I hope is the general point I am trying to make here, in relation to fields.
If you have covered co-ordinate systems, such as the Cartesian ($x$, $y$, $z$) system and the spherical cordinate system ($r$, $\theta$, $\phi$), you will know that problems are often easier to solve in one co-ordinate system than in another, and because they are all equally valid to use, this helps derive an equation and a solution(s).
In something like the same way, a lot of modern physics uses different areas of math to attack and understand problems that may be easier to solve in, for example, Group Theory than by using another area of math.
Again, a simple example, the equations of motion of a pendulum can be derived from its Lagrangian: 
$${\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left[\left({\dot {x}}+\ell {\dot {\theta }}\cos \theta \right)^{2}+\left(\ell {\dot {\theta }}\sin \theta \right)^{2}\right]+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta \end{array}}}  $$
From this Lagrangian, which is not an equation in itself, we can easily derive the equations of motions of pendulum, how it moves, where it will be a specific time, its' velocity and so on.
To go up a (few) gears, The Standard Model Lagrangian  gives you the nearest thing we have to a mathemathical description of what we know so far, re: fields, particles and their interactions. Again, I would stress this is not an equation in the same way as $5 = 2 + 3$ is, it is rather the basis from which equations are derived.  
This answer is not really important, but the links are if you want to get an idea of how far physics has come from Schrodinger's equation, and possibly, how far it has to go.
