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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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