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How are eigenvalues connected to energy values in quantum mechanics?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, an eigenvalue is a scalar $k$ associated with a linear operator $P$ such that $P ψ = k ψ$ , where $ψ$ is an eigenfunction.

In quantum mechanics, physical observables—such as position, angular momentum, and energy—are represented by linear operators. When the operator is the Hamiltonian (the energy operator), the eigenvalues of that operator correspond directly to the possible energy values of the system.1

According to the axiomatic approach to quantum mechanics, any experimental measurement of a dynamical variable will yield one of its eigenvalues. While the state of a system is generally represented as a linear combination of these eigenfunctions, a measurement forces the system into a specific state corresponding to the observed eigenvalue. Consequently, the set of all possible energy outcomes for a quantum system is defined by the spectrum of eigenvalues of its Hamiltonian operator.12