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I'm studying quantum mechanics and I'm trying to understand the concept of operators. They can be represented in general by the equation:

$$ \hat{A}ψ=ψ'. $$

Here the wavefunction is changed to $ψ'$ after the operator acted on the wavefunction. Since the wavefunction specifies the complete state of a particle, it seems that the state has been changed.

I've learned about eigenvalue equations, where an operator acting on a wavefunction results in the same wavefunction multiplied by a constant.

$$ \hat{A}ψ=a ψ $$

An example is the momentum operator if I'm correct:

$$ \hat{p}ψ=pψ$$

However, I'm wondering about operators that don't result in eigenvalue equations. What kind of physical process would correspond to such operators? I understand that real measurements have to correspond to real eigenvalues; e.g. the operators have to be hermitian.

What kind of physical process would correspond to an operator that doesn’t result in a eigenvalue equation:

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    $\begingroup$ Hint: E.g. time-evolution operator. Despite that, not all operators a priori have a physical meaning. $\endgroup$ Commented Feb 11 at 10:36
  • $\begingroup$ @tobias that looks like the start of an answer 👍 $\endgroup$
    – Kyle Kanos
    Commented Feb 11 at 12:19
  • $\begingroup$ Creation and annihilation operators are an example operators which give rise to useful non-eigenvalue equations. $\endgroup$ Commented Feb 11 at 13:27
  • $\begingroup$ @AlbertusMagnus isn't coherent light (the Glauber state), and eigenstate of the annihilation operator? $\endgroup$
    – JEB
    Commented Feb 11 at 14:23
  • $\begingroup$ @TobiasFünke I don't understand, could you elaborate $\endgroup$ Commented Feb 11 at 16:09

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The wavefunction, $\psi(x)$, is the position representation of the more abstract vector (or "ket"), $| \psi \rangle$, which is an element of the Hilbert space (a complex vector space with an inner product). These vectors can be thought of as the "states" of a particular quantum system, and the Hilbert space represents the system. Operators, generally, are linear maps from the Hilbert space to itself.

Observables in quantum mechanics are represented by "Hermitian operators" on the Hilbert space: those linear maps from the Hilbert space to itself that are Hermitian (the technical definition of which guarantees that they have real eigenvalues and some other properties). The eigenvalues of a Hermitian operator are the possible measurement values for the associated observable, and its eigenvectors are the corresponding states of the system which would return that measurement value. A state, $| \psi \rangle$, is almost never exactly equal to the eigenstate of an operator, instead it is in some superposition of the eigenstates. Because the eigenvectors of an observable form a basis for the Hilbert space, you can decompose the state into a linear combination of its eigenstates. The coefficients in that decomposition tell you the probability of getting the measurement value corresponding to that eigenstate, when you make a measurement of that observable on the state $|\psi\rangle$.

More generally, other linear maps from the Hilbert space to itself accomplish different tasks. The "unitary operators" (those for which their adjoint is also their inverse), are essentially measure-preserving maps on vectors/states. Unitary maps can invoke, e.g., the time evolution of a state and its translation in space.

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