What are the experiments that established Quantum Field theory beyond the hydrogen atom solution? I took a full undergrad curriculum in physics, got through and understood Quantum Mechanics as taught from an undergraduate level. I've always been curious about what happens "after" though. The development of Quantum Field theory was an important next step in our understanding of physics. In quantum mechanics, the hydrogen atom spectrum was key experimental evidence that confirmed the theory. Is there an experiment or set of experiments that does the same for Quantum Field theory and beyond?
 A: This is an experimentalist's view, take it as a long comment:
As doing experiments in particle physics since 1970 , I can guarantee that all data were fitted with quantum field theory models. The history is long and step wise, and there is a wiki article on it.

I took a full undergrad curriculum in physics, got through and understood Quantum Mechanics as taught from an undergraduate level. I've always been curious about what happens "after" though. 

The following is to update your quantum mechanical background with respect to quantum field theory: 
Quantum mechanics is all one framework, with given postulates and principles, which pick up the solutions of differential wave equations which describe and predict the data in the microcosm ( and not only, there is superconductivity which goes for miles). Within this framework the simple wave functions, Dirac, Klein Gordon, quantized Maxwell, (the Schrodinger is not relativistic , so not good for high energies) describe two body bound  potential interactions very well , giving energy levels that fit the data. Scatterings and decays were a problem.
Also, once one goes to many bodies, it is not possible in general to create potentials to be inserted into the differential equations and get exact solutions for the wavefunction. Instead, quantum field theory was invented, which is a way of expanding into a series the solutions needed for a specific problem, as electron positron  scattering, for example, at high energies, with creation of many particles.
Quantum field theory for the standard model of elementary particles, posits that each elementary particle in the table for all (x,y,z,t) exist as an operator  field , this consists of the plane wave  solution  for the corresponding particle ( electron: dirac with zero potential for example) and  differential creation and annihilation operators on this field generating or destroying a particle ( an electron). This is illustrated in the Feynman diagrams, which have a prescription for expanding into a series the needed solution, the first order terms dominant. 
Thus quantum field theory is a sophisticated mathematical model , utilizing   the plane wave  solutions of the basic quantum mechanical equations to extend calculational possibilities for decays, scatterings and many body problems.

An example of e- e- scattering which evaluated ,successfully fits e-e- scattering.
(Real particles, as plane waves are not localized , wave packetshave to be  used to model them.)
