(a)The time-independent, normalized wave function for the one-dimensional quantum harmonic oscillator in its ground state is given by:ψ0(x) = (a/pi)1/4 e−ax2/2where a = (x)0=momega/h, m is the mass of the oscillator, ω is the angular frequency of the oscillator, and ℏ is the reduced Planck constant, :h/2pi Let the symbol (q)s denote the expectation value of the quantity q for the state ψs(x). What is the expectation value (x)0? (Use the following as necessary: a.)(b) The time-independent, normalized wave function for the one-dimensional quantum harmonic oscillator's first excited state is given byψ1(x) = (4a^3/pi) 1/4xe−ax2/2What is the expectation value (x)1 ? (Use the following as necessary: a.)(x)1 =(c) In quantum mechanics, a particle does not have to be in a "definite" state. It can be in a "mixed" state, or superposition of states. Consider a quantum harmonic oscillator with the wave function:ψ01(x) = 1/sqrt2[ψ0(x) + ψ1(x)](x)01 = where ψ0(x) and ψ1(x) are given above. This represents a particle with a non-zero (and in this case, equal) probability of being found either in the ground state or first excited state, before a measurement is taken. What is the expectation value (x)01 for this wave function? (Use the following as necessary: a.)ψ01(x) =