at and where must be zero. Thus, at , and . At , in which cannot be zero as this would conflict with the postulate that has norm 1. Therefore, since , must be an integer multiple of , This constraint on implies a constraint on the energy levels, yielding A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy. Harmonic oscillator As in the classical case, the potential for the quantum harmonic oscillator is given by This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by where Hn are the Hermite polynomials and the corresponding energy levels are This is another example illustrating the discretization of energy for bound states. Mach–Zehnder interferometer The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum eraser, the Elitzur–Vaidman bomb tester, and in studies of quantum entanglement. We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector that is a superposition of the "lower" path and the "upper" path , that is, for complex . In order to respect the postulate that we require that . Both beam splitters are