# What are the Eigenstates and Eigenvalues? [closed]

In quantum mechanics I keep hearing about them. Kindly tell about them...not at a very very high level but simple enough to understand completely

## closed as too broad by Thomas Fritsch, G. Smith, John Rennie, Nat, eranrechesJun 11 at 0:21

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• Um... This was 16 months of university for me. – puppetsock Jun 10 at 13:36
• Anyone please tell about them in a answer.... I am new here – Tanmay Siddharth Jun 10 at 15:46
• Welcome to SE.Physics! It's probably a bit too broad to ask about 4 different terms in the same question. Instead, you might be better off asking about 1 term at a time. – Nat Jun 10 at 20:24
• Eigenstate is quantum state which has definite value for some physical property (for example "particle is definitely at x=0"), eigenvalue (so in that example, x=0 is a position eigenvalue). This concept is meaningful in quantum mechanics because the quantum state does not always give a single definitive value for a physical property... Perturbation theory is a method of calculation in which a weak influence is approximated by making smaller and smaller corrections to the situation with no influence... Path integral is integral (as in calculus) which sums over all possible paths or histories. – Mitchell Porter Jun 12 at 11:48
• @MitchellPorter Thank you very much – Tanmay Siddharth Jun 12 at 11:50

Consider a matrix acting on vectors by matrix-vector multiplication. For most vectors, when the matrix acts on them, the resulting vector points in a different direction from the original vector. But for some vectors the matrix leaves the direction unchanged, while perhaps stretching or shrinking the vector. These special vectors are the eigenvectors of the matrix, and the stretch/shrink factors are the eigenvalues.

This concept generalizes to any kind of linear operator in a vector space of any dimension, even infinite dimensions.

For example, the linear operator might be a linear differential operator operating on functions of some number of coordinates. Different functions represent different “vectors” in Hilbert space. If the operator, acting on a function, gives the same function back again, but multiplied by a constant, then that function is an eigenfunction and the multiplicative constant is the eigenvalue.

In quantum mechanics, a quantum state is an abstract vector in a Hilbert space, often an infinite-dimensional one. In a particular basis for the Hilbert space, the state can be represented by a wave-function, say in coordinate space.

The time-independent Schrodinger equation is just an eigenequation,

$$\hat{H}\Psi_n=E_n\Psi_n$$

which says that the energy operator leaves the “direction” of each energy eigenstate $$\Psi_n$$ unchanged. The energy eigenvalue (the stretch/shrink factor) is the energy of that state.

The point of the eigenstates of an operator corresponding to an observable physical quantity is they they are the special states in which the measured value of the observable will be one definite value: the eigenvalue for that eigenstate. (For example, the energy eigenvalue of the least-energetic energy eigenstate of hydrogen is -13.6 eV.) When the system is in a superposition of eigenstates, you will instead observe various eigenvalues with various probabilities.