A quantum particle is subjected to the potential V(x) = ax + bx2, where a and b are constants (b > 0). What is the mean position ⟨x⟩ of the particle in the first excited state?

In quantum mechanics, the eigenvalue of a Hamiltonian operator for a particle is used to determine which of the following physical quantities?

Which of the following is a mandatory mathematical characteristic of an acceptable wave function in quantum mechanics?