Degenerate energy levels
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy  alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy. Degeneracy plays a fundamental role in quantum statistical mechanics. For an -particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level all have an equal probability of being filled. The number of such states gives the degeneracy of a  particular energy level. Mathematics. The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If is a matrix, a non-zero vector, and is a scalar, such that formula_1, then the scalar is said to be an eigenvalue of and the vector is said to be the eigenvector corresponding to . Together with the  zero vector, the set of all eigenvectors corresponding to a given eigenvalue form a subspace of , which is called the eigenspace of . An eigenvalue which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e., formula_2 and formula_3, where formula_4 and formula_5 are linearly independent eigenvectors. The dimension of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional. The eigenvalues of the matrices representing physical observables in quantum mechanics give the  measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy  eigenvalue. This clearly follows from the fact that the eigenspace of the energy value eigenvalue is a subspace (being the kernel of the Hamiltonian minus times the identity), hence is closed under linear combinations. Effect of degeneracy on the measurement of energy. In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. However, if the Hamiltonian formula_6 has a degenerate eigenvalue formula_7 of degree g, the eigenstates associated with it form a vector  subspace of dimension g. In such a case, several final states can be possibly associated with the same result formula_7, all of which are linear combinations of the g orthonormal eigenvectors formula_9. In this case, the probability that the energy value measured for a system in the state formula_10 will yield the value formula_7 is given by the sum of the probabilities of finding the system in each of the states in this basis, i.e. Degeneracy in different dimensions. This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of  one and two-dimensional systems aids the conceptual understanding of more complex systems. Degeneracy in one dimension. In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function formula_10 moving in a one-dimensional potential formula_14, the time-independent Schrödinger equation can be written as Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy formula_16 at most, so that the degree of degeneracy never exceeds two. It can be proven that in one dimension, there are no degenerate bound  states for normalizable wave functions. A sufficient condition on a piecewise continuous potential formula_17 and the energy formula_16 is the existence of two real numbers formula_19 with formula_20 such that formula_21 we have formula_22. In particular, formula_17 is bounded below in this criterion. Degeneracy in two-dimensional quantum systems. Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made of monoatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include  MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems. Particle in a rectangular plane. Consider a free particle in a plane of dimensions formula_24 and formula_25 in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with wave function formula_10 can be written as