The concept of exceptional points was first defined by Kato (1995) in 1966 in a mathematical context as part of the perturbation theory of a non-Hermitian operator with complex eigenvalues (The physically more relevant definition of exceptional point is made by Heiss, for a reference see Heiss (2012)). An exceptional point can appear in parameter-dependent systems. They describe points in an at least two-dimensional parameter space at which two (or more) eigenvalues and their corresponding eigenstates become identical. In physics operators appear in quantum theory in the form of a Hamiltonian. Usually this Hamiltonian is Hermitian and has purely real eigenvalues, which are associated with a measurable energy. This is a sufficient description of a closed quantum system. A very effective description of open quantum systems interacting with an environment is often possible in terms of non-Hermitian Hamiltonians. These non-Hermitian operators possess in general complex eigenvalues. Due to their non-Hermiticity they may exhibit exceptional points. The imaginary part of an eigenvalue is interpreted as a decay rate of the corresponding state. An example is the hydrogen atom in crossed external electric and magnetic fields. Various exceptional points were identified in that system by Cartarius (2008) which were also published by Cartarius et al.(2007; 2009). This thesis deals with the temporal evolution of states at an exceptional point in the hydrogen atom as an extension to the works mentioned before. Uzdin et al.(2011) and Berry and Uzdin (2011) found that the temporal evolution of resonances, when transported around an exceptional point, has to be considered very carefully. For a closed loop around an exceptional point it is known that the two resonances connected with the exceptional point interchange. However, if a resonance is populated and then transported around the exceptional point, this exchange is not always visible. In particular, it could be shown that for sufficiently slow traversals of the parameter space loop the final population always ends up in the same state. There are suggestions to exploit this fact for technical applications, eg purification schemes, cf. Gilary, Mailybaev, et al.(2013), Atabek et al.(2011), and Gilary and Moiseyev (2012). However, Leclerc et al.(2013) showed that also the non-adiabatic exchange is only visible for isolated resonances. In real physical systems transitions to other resonances not connected to the exceptional point are always possible and can influence the dynamics. It is the purpose of this thesis to study this influence. Furthermore we investigate for the first time the transport of a populated resonance around a third-order exceptional point. Before the hydrogen atom is studied a matrix model proposed by Uzdin et al.(2011) is investigated. An eigenstate is transported along a closed loop in parameter space in its instantaneous basis. It turns out that the initial population does not need to end up in the same state. This phenomenon is called adiabatic flip. The time evolution is solved numerically exact. The result is compared to the adiabatic approximation. The main topic of this thesis is the application of the insights gained from Uzdin et al.(2011) on a physical system, viz. the hydrogen atom in crossed external electric and magnetic fields. A numerical method for calculating the eigensystem of the Hamiltonian was implemented by Cartarius (2008) and is extended in this thesis by means of the