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As it turns out, analytic solutions of the Schrödinger equation are available for only a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom are the most important representatives. Even the helium atom — which contains just one more electron than does the hydrogen atom — has defied all attempts at a fully analytic treatment. (Wikipedia)

Why, for example, doesn't the helium atom have an analytic solution? What about the other noble gases? Are the other formulations of QM more successful?

If it's to hard to answer all subquestions please answer partially, maybe all answers together will draw a full(er) picture.

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    $\begingroup$ You can write down a differential equation but the chances are that you will not be able to solve it analytically. $\endgroup$ – Farcher Sep 7 '17 at 6:54
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    $\begingroup$ Exact solutions to the equations of any theory of physics are rare. For instance the three-body problem in Newtonian gravitation has no general analytic solution. $\endgroup$ – tfb Sep 7 '17 at 6:59
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    $\begingroup$ Basically every field of physics and engineering. Typically, only the most elementary and artificial applications of a model end up being analytically solvable. $\endgroup$ – J. Murray Sep 7 '17 at 7:04
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    $\begingroup$ Related: physics.stackexchange.com/q/123185/2451 $\endgroup$ – Qmechanic Sep 7 '17 at 10:44
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    $\begingroup$ @HolgerFiedler Try predator-prey problems as per en.wikipedia.org/wiki/… Analytical solutions are extremely rare in any fields. $\endgroup$ – ZeroTheHero Sep 7 '17 at 15:23
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Even a single particle Schrodinger equation is a complex second order differential equation in three dimensions. There is no general mathematical way to solve such equations in a closed form.

This is why approximations (quite accurate ones in many cases) and numerical methods are used.

Why, for example, doesn't the helium atom have an analytic solution?

The Helium atom is a three body problem. Given that the single particle equation have no general solution method, it should be easy to imagine that the three body problem is beyond the ability of mathematics to solve in a closed form.

Even the classical (non quantum) three body problem has no general solution.

What about the other noble gases?

Complex but quite effective approximation methods are used. They have no general solutions.

Are the other formulations of QM more successful?

Here you labor under a misconception.

Success of a theory is not defined by how easily you can express results in closed forms.

It is defined by how accurately the results are, regardless of how hard you have to work to calculate the result.

You should also note that when we write down a closed form like $f(x)=sin(ax)$ this only appears to be a closed form. In fact calculating a relatively simple function like $sin(x)$ turns out to be quite a complex problem which cannot be done perfectly for all numbers.

Even using $\pi$ in a formula requires we use an approximation, as $\pi$ cannot be written in finite number of digits.

If it's to hard to answer all subquestions please answer partially, maybe all answers together will draw a full(er) picture.

The fuller picture is that physicists are perfectly happy to find a method of solving a problem that's practical and accurate enough. They don't need a convenient expression (but it's nice if you can get one).

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Because that is the natural state of affairs for any theory based on partial differential equations. More generally, exact analytical solutions are much more sparse on the ground, throughout physics and mathematics, than a first undergraduate pass at physics and its maths would lead you to believe.

Other fields have it rather worse than quantum mechanics, with the go-to example being fluid dynamics: there, the foundational dynamical equations are the Navier-Stokes equations, which are easy and natural to formulate, but for which we don't even have existence or smoothness theorems about its solutions (that being one of the Clay Institute Millenium problems). For Navier-Stokes, as well as the simpler Euler equations, there is a huge number of problems that are easy to formulate but which don't have analytical solutions.

Now, quantum mechanics is simpler than fluid dynamics because the equations are linear, which makes the formalism much easier to handle, but it also suffers from the curse of dimensionality: the dimension of the space in which the Schrödinger equation plays out, as a PDE, grows linearly with the number of particles. High-dimensional PDEs are extremely hard to solve anywhere they occur (with other examples being the Black-Scholes equation when describing multiple financial assets, or the Hamilton-Jacobi equation over a large number of degrees of freedom); quantum mechanics does stick out as requiring high-dimensional PDEs more often than other parts of physics but it is not unique in that regard.

This problem with dimensionality is one of the things that besets, say, the helium atom. In its simplest form, assuming a clamped nucleus, the Schrödinger equation is a six-dimensional PDE: $$ \left[-\frac12\left( \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial y_1^2} + \frac{\partial^2}{\partial z_1^2} + \frac{\partial^2}{\partial x_2^2} + \frac{\partial^2}{\partial y_2^2} +\frac{\partial^2}{\partial z_2^2} \right) -\frac{Z}{\sqrt{x_1^2+y_1^2+z_1^2}} -\frac{Z}{\sqrt{x_2^2+y_2^2+z_2^2}} \right. \\ \qquad \qquad \qquad \qquad \qquad\left. +\frac{1}{\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}} -E \right]\psi(x_1,y_1,z_1,x_2,y_2,z_2)=0 . $$ Now, we do know that this has at least two useful conserved quantities (the total angular momentum, and one of its components) but that still leaves us with an effective four-dimensional space to fill up ─ as opposed to the hydrogen atom, where this was a single dimension and could solve the resulting ODE.

Now, is it all that terrible that we're unable to solve this exactly? Well, not particularly (and, in fact, if we were able to solve it analytical, that solution might turn out to be less useful than the existing methods). For atomic physics, you can get a huge amount of useful information via Hartree-Fock methods, which essentially try their best to pretend that the Schrödinger equation is actually a bunch of single-electron PDEs, and in the process you get core concepts of atomic physics, including atomic electron configurations like, say, $\rm 1s^2\ 2s^2\ 2p^6$. Of course, Hartree-Fock isn't really up to scratch for quantitative predictions, but there's plenty of numerical post-Hartree–Fock methods that build on the Hartree-Fock physical picture to get quantitative answers to the required precision.

That's, by the way, where the other noble gases fit in: you can get reasonable electronic structures via Hartree-Fock, and you can refine them (via Configuration Interaction, Coupled Cluster, or your method of choice) to calculate and understand whatever property it is you want to calculate. As for analytical solutions, though, neon's TISE is over a 30-dimensional space, argon clocks in at a 54-dimensional PDE, and it keeps climbing from there, so there's no hope of getting an analytical solution.

I also want to add some remarks about how useful analytical solutions actually are with complicated problems, and here the shining example is the Rabi model. This model describes a two-level system coupled to a single mode of radiation, with the hamiltonian $$ H = \omega a^\dagger a + \omega_0 \sigma_z + g\left[(\sigma_+a + \sigma _-a^\dagger) + (\sigma_+a^\dagger + \sigma _-a) \right], $$ where the second set of coupling terms (i.e. $\sigma_+a^\dagger + \sigma _-a$) are known as counter-rotating terms, and can often be neglected, via the rotating-wave approximation leading to the Jaynes-Cummings model. Now, the Jaynes-Cummings model is integrable and easily solvable (because it has an additional conserved quantity, the excitation number $a^\dagger a + \sigma_z$), but the Rabi model was long thought not to be solvable ─ until Daniel Braak proved otherwise in 2011.

Now, Braak's solution is within the envelope of what you'd call analytically solvable, but it's really worth having a closer look at what the solution actually looks like: it exists, but it's so complicated that it's hard to use it to actually do anything useful with it. In fact, for several uses, you're better served by going back to the model and solving the Schrödinger equation numerically. What does that tell you? well, that analytical solutions are rather over-rated.


And finally, as Stephen mentions, your other query,

Are the other formulations of the QM more successful?

is way off the bat. The success of a scientific theory is measured in its capacity to explain and predict phenomena within a coherent, explicative framework, and this is entirely independent of the quirk of whether X or Y equations have full analytical solutions or whether they require the tools of numerical analysis to solve in full.

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So firstly, let's think about what it means to solve the Schrodinger equation for an atom? So the Schrodinger equation describes the evolution of something called a wave function, usually denoted $\Psi({\bf{x}},t)$. The wave function in essence encapsulates all the information about the system you want to model. So if you were, let's say, trying to model the Helium atom, you'd try to work out what the wave function for a Helium atom is. Now the cool thing about the Schrodinger equation is that it can be understood in a fairly intuitive way; let's take a look: $$i\hbar\cfrac{\partial}{\partial t}\Psi({\bf{x}},t)=\left(-\cfrac{\hbar^2}{2m}\nabla^2+V(x,t)\right)\Psi({\bf{x}},t)$$ So what does any of that mean? The L.H.S. is specifically referring to how the system in question evolves in time. The R.H.S. tells you $\bf{how}$ the evolution is driven. In fact, the R.H.S. is telling you that the dynamics of your system are governed by the total energy of the system, i.e. the Hamiltonian. Now, the great thing about this equation is that it really talks about measurements. To $\bf{measure}$ the dynamics of your system, you effectively need to measure the total energy of your system. Now for $V=0$, you get the Schrodinger equation for a $\textbf{free particle}$. Now when it was said that 'for a single particle, there exists no closed-form solution', well that's not quite the case. The fact that the Schrodinger equation is a second-order partial differential equation is really just a statement that its solutions involve equations with multiple variables, e.g. space in 3 dimensions and time $\textbf{and}$ the equation involves second derivatives. Let's consider for simplicity, the 1D case: $$i\hbar\cfrac{\partial}{\partial t}\Psi(x,t)=-\cfrac{\hbar^2}{2m}\cfrac{\partial^2}{\partial x^2}\Psi(x,t)$$ This is really just the wave equation. We know the general solution for the wave equation and it's not too tricky to get, that is, the standard plane wave: $$\Psi(x,t)=Ae^{i(kx-\omega t)}$$ Plug this into the Schrodinger equation and out pops the de Broglie relations, specifically $E=\hbar\omega$. So there you go, the plane wave is a solution to the Schrodinger equation as long as the following relation is true: $E=\hbar\omega$, which in essence tells you that plane waves really just describe particles that carry a discrete packet of energy, or $\textbf{Quanta}$. Now the problem comes when you try to physically realize this solution. The special thing about the wave function is that taking the square of the modulus gives you a probability distribution, which tells you the probability of finding a particle at some $x$ and $t$; specifically $$|{\Psi(x,t)}|^2 = Pr(x,t)$$ For this solution to be $\textbf{Physical}$, or shall I say describe reality, we would like the following condition to hold: $$\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx <\infty $$ This means that we can find some number $A$ such that $$A\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx =1 $$ This tells us that the probability of finding the particle somewhere in space is 1, e.g. the particle (or system) exists. When it comes to free particles, such a condition is not met, so in general, we can't realize the solution to a free particle as a physical solution, despite such a solution being a perfectly fine solution to the Schrodinger equation (you can do a bit of tinkering to show that the free particle can be realized physically as a discrete wave packet).

Now the potential term in the equation describes some configuration that determines the dynamics of your system. More generally, your system will be described by a Hamiltonian. The most simple example is probably the Hamiltonian which describes a particle trapped in a box/infinite potential well. The potential for this system will impose boundary conditions such that your particle can never overcome the potential energy of the system. The boundary conditions allow you to derive a set of solutions by solving the Schrodinger equation. These solutions will correspond to the discrete energy levels that the particle can occupy. There are of course many other boundary conditions you can impose to try and model different systems; this is one way the potential works. The quantum harmonic oscillator is another very important example. In fact, it has a myriad of incredible properties! For example, it can be used as a building block for more complex systems. In fact, you can show that if you have a system of coupled (joined together) simple harmonic oscillators, the Hamiltonian of such a system is exactly the same as the Hamiltonian for an un-coupled set of simple harmonic oscillators, which is actually a big motivating factor for QFT.

We can also have even more complex systems, for example, the Hydrogen atom. The Hamiltonian which models the Hydrogen atom is one that involves Coulomb's law: $$E\Psi({\bf{x}},t) = -\cfrac{\hbar^2}{2\mu}\nabla^2\Psi({\bf{x}},t)-\cfrac{q^2}{4\pi\epsilon_0r}\Psi({\bf{x}},t)$$ This is effectively a two-body problem and with some clever manipulation, it's not too hard to arrive at the solutions. We can decompose such a system into two parts: one that describes a free particle, e.g. the dynamics of the atom itself (we know how to solve that), and one that describes the internal dynamics of the atom, i.e. the electron orbitals. Solving the latter is a bit tricky, but not too bad. Once you arrive at a solution, you get some interesting properties that describe the hydrogen atom for free. So as you can see, we can actually describe quite a few systems analytically and the benefit of doing so is that we end up deriving properties that are intrinsic to the system, e.g. quantum numbers, energy levels (energy spectrum). Now there's so much more that I can go into, but that's the idea behind solving the Schrodinger equation. But why can't you do the same for the Helium atom? I mean, surely there must be some clever trick you can do! Well, the problem is that the Hamiltonian for the Helium atom is really a 3 body problem: $$E\Psi({\bf{r_1,r_2}},t) = \left(-\cfrac{\hbar^2}{2m_1}\nabla_1^2-\cfrac{\hbar^2}{2m_2}\nabla_2^2-\cfrac{q^2}{4\pi\epsilon_0}\left(\cfrac{1}{r_1}+\cfrac{1}{r_2}\right)+\cfrac{q^2}{4\pi\epsilon_0|{\bf{r_1}-\bf{r_2}}|}\right)\Psi({\bf{r_1,r_2}},t)$$ The problem with such a system is that there aren't any tricks we can do to separate it into simpler constituents which we can solve individually. It's really the last term of the Hamiltonian that messes everything up. For the Hydrogen atom, we can separate the general solution into smaller parts, each of which describes some part of the system and is solvable. Ideally, we would like to get a solution that has one part describing the atom itself (free particle) and one part that describes the electron orbitals. This is, in fact, the case for all atoms with more than 1 proton-electron pair. In such cases, we result in approximations.

One of the simplest approaches would be to use something called $\textbf{variational methods}$. This is where you pick a Hamiltonian whose solution we know $\textbf{and}$ is similar to the Hamiltonian we are trying to solve, e.g. for the Helium atom, we would use the Hydrogen atom. The idea here is to first show that by doing this, we can bound the ground energy state (lowest energy) of the Hamiltonian we are approximating. This, in turn, leads to the following result: $$E[\Psi] =<\Psi|\hat{H}_0+\hat{V}|\Psi> $$ where $\hat{H}_0$ gives us the lowest energy state for the Hydrogen atom. What this basically says is that we expect the energy of the Helium atom to be the ground energy of the Hydrogen atom, plus some extra contribution from the additional potential term. We then minimize the above functional, meaning we try to find the lowest value that it can take. And since this minimum value bounds the true value, we end up with a very accurate solution, in some cases the exact solution. Using the same principle, we can look for higher energy levels and eventually get the entire energy spectrum for the Helium atom. Another method we can use is called $\textbf{pertubation theory}$ which I suppose could be understood more intuitively. We'll stick to the Helium atom example. Suppose we can write the Hamiltonian for the Helium atom in the following way: $$\hat{H}=\hat{H}_0 +\lambda\hat{V}$$ where $\lambda$ is a small parameter and $\hat{H}_0$ is a Hamiltonian whose solutions we know (i.e. the Hydrogen atom). The idea here is by doing some clever tricks, we seek a power series solution that converges to the real solution provided the necessary criteria are met. We end up with something along the lines of $$E_n=E_n^{(0)}+\lambda V_{nm}+\lambda^2\sum_{k\neq n}\cfrac{|V_{kn}|^2}{E_n^{(0)}-E_{k}^{(0)}}+...$$ where $E_n^{(i)}$ are your $i$-th order correction terms. You pretty much try to add small corrections to the know Hamiltonian to get to the new one.

These are of course fairly simple examples and there are far more sophisticated methods out there. Both of the above have their own shortcomings and require special care when dealing with more complex problems. If the necessary conditions aren't met, then these methods will simply fail. Furthermore, in some instances, perturbation theory does not produce a convergent series, or if the series does converge, you cannot always guarantee a complete solution set. The great advantage to using such methods is that we have very powerful computers to do the job for us, but again that will depend on whether the problem can be feasibly solved in an ideal time period, i.e. for perturbation theory, you'd need to consider the radius of convergence to get an idea of how long the computation might take.

Now using methods like perturbation theory and variation is absolutely essential to make progress when analyzing complex Hamiltonians. In many cases, analytic solutions either don't exist or require some exotic mathematical tools which is a mountain all in itself. The theory of PDEs, especially non-linear PDEs which many Hamiltonian systems yield require theory from all branches of maths from elliptic functions to algebraic topology, and most of this theory is used to prove existence theorems rather than for computation.

On the other hand, analytic solutions give you the foundations you need to begin exploring. Although far more challenging, if successful they can give you far more depth and understanding of the system you're trying to study. You can find unexpected results lurking in the machinery which numerical approaches simply miss. Furthermore, it is essential to be $\textbf{rigorous}$ in your work, regardless if you do numerical analysis or take a more direct approach. Each step requires careful justification and when trying to understand complex systems, we shouldn't expect to always get simple results.

As for the other formulations of Quantum Mechanics, I'm not too sure what you're specifically referring to. If it's for example something like David-Bohm then the formulation seems to just add more complexity. I mean any formulation in most ways will be equivalent, since for it to be an acceptable theory, it needs to reproduce experimental observations. Modern Quantum mechanics has a very rigorous mathematical foundation, developed first by Hilbert and Von Neumann. Then further development was made by Dirac (his work is pretty much standard QM theory) and his work was similarly rigorously developed and fleshed out. We then have your standard QFT: fields, path integrals, renormalization, gauge theories, the standard model, etc. We also have algebraic QFT which is a very rigorous framework that studies Quantum Field Theories in curved space-time and is typically used when the underlying spacetime Manifold is curved. Of course, these frameworks deal with completely different things compared to QM.

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