ive the definition of the term "Operator" ? (b) Name the Momentum Operator (c) Name the Total Energy Operator (d) Name the Hamilton

# quantum mechanical particde in a harmonic osci lator potential has the initial wave function y,)+4,(x), where Y. and Y, are the real wavefunctions in the ground and fist exci ted state of the harmonic osciclator Hamiltonian- for Convenience we take mzhzw= 1 for the oscillator- What ở the probabilpty den sity of finding the par ticke at x at time tza?

Using the eigenvectors of the quantum harmonic oscillator Hamiltonian, i.e., n), find the matrix element (6|X² P|7).

A particle is described by the wavefunction Ψ(t, x), and the momentum operator is denoted by pˆ. a)  Write down an expression for the differential operator pˆ. b)  Write down an expression for the expectation value of the momentum, ⟨p⟩. c)  Write down an expression for the probability density, ρ. d)  Write down an expression for the probability of finding the particle between x = a and x = b.