Degenerate orbital

In quantum mechanics, an energy level is said to be 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. 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 $N$ -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 are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.

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 $A$ is a $N \times N$ matrix, $X$ a non-zero vector, and $λ$ is a scalar, such that $AX=\lambda X$ $AX=\lambda X$ , then the scalar $λ$ is said to be an eigenvalue of $A$ and the vector $X$ is said to be the eigenvector corresponding to $λ$ . Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue $λ$ form a subspace of $C n$ , which is called the eigenspace of $λ$ . An eigenvalue $λ$ which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e., $AX_{1}=\lambda X_{1}$ $AX_{1}=\lambda X_{1}$ and $AX_{2}=\lambda X_{2}$ $AX_{2}=\lambda X_{2}$ , where $X_{1}$ $X_{1}$ and $X_{2}$ $X_{2}$ are linearly independent eigenvectors.The dimensionality 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.

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