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In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum , the system's energy spectrum or its set of energy eigenvalues , is the set of possible outcomes obtainable from a measurement of the system's total energy.
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
The Hamiltonian of the particle is: ^ = ^ + ^ = ^ + ^, where m is the particle's mass, k is the force constant, = / is the angular frequency of the oscillator, ^ is the position operator (given by x in the coordinate basis), and ^ is the momentum operator (given by ^ = / in the coordinate basis).
This proof of the Hellmann–Feynman theorem requires that the wave function be an eigenfunction of the Hamiltonian under consideration; however, it is also possible to prove more generally that the theorem holds for non-eigenfunction wave functions which are stationary (partial derivative is zero) for all relevant variables (such as orbital rotations).
This shows that and † are also eigenstates of the Hamiltonian, with eigenvalues and + respectively. This identifies the operators a {\displaystyle a} and a † {\displaystyle a^{\dagger }} as "lowering" and "raising" operators between adjacent eigenstates.
The most general form of a 2×2 Hermitian matrix such as the Hamiltonian of a two-state system is given by = (+), where ,, and γ are real numbers with units of energy. The allowed energy levels of the system, namely the eigenvalues of the Hamiltonian matrix, can be found in the usual way.
In the general time-independent case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form: ^ = ^ + Here, is the mass of the particle, ^ is the momentum operator, and the potential () depends only on the vector magnitude of the position vector, that is, the radial distance from the ...
The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.