All parts of this problem refer to the harmonic oscillator. A) Use the expressions for the operators a+ and a− that we saw in class and the fact that xˆ and pˆ are Hermitian operators to show that a+ is the Hermitian conjugate of a− and vice versa. B) Express the operators xˆ and pˆ as a combination of the operators a+ and a−. C) Based on the above, show that the expected value of both xˆ and pˆ is zero at any steady state of the oscillator.

All parts of this problem refer to the harmonic oscillator. A) Use the expressions for the operators a+ and a− that we saw in class and the fact that xˆ and pˆ are Hermitian operators to show that a+ is the Hermitian conjugate of a− and vice versa. B) Express the operators xˆ and pˆ as a combination of the operators a+ and a−. C) Based on the above, show that the expected value of both xˆ and pˆ is zero at any steady state of the oscillator.

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a particle is confined to move on a circle's circumference (particle on a ring) such that its position can be described by the angle ϕ in the range of 0 to 2π. This system has wavefunctions in the form Ψm(ϕ)= eimlϕ where ml is an integer. Show that the wavefunctions Ψm(ϕ) with ml= +1 and +2 are ORTHOGONAL Show full and complete procedure. Do not skip any step
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a particle is confined to move on a circle's circumference (particle on a ring) such that its position can be described by the angle ϕ in the range of 0 to 2π. This system has wavefunctions in the form Ψm(ϕ)= eimlϕ where ml is an integer. Show that the wavefunctions Ψm(ϕ) with ml= +1 and +2 are ORTHOGONAL Show full and complete procedure. Do not skip any step
Consider a wavefunction Ψ0 prepared at t = 0 in an infinite square well of width L, such that Ψ0 = N in the interval −L/8 < x < L/8 and is zero otherwise, as illustrated below. found in image Figure 3: A wavefunction Ψ0, prepared in an infinite square well at time t = 0. a) Derive an expression for the normalization constant N, in terms of L. b) The wavefunction of this system at time t can be written as Ψ(x,t) = Pn cnψn(x)e−iEnt/~, where {cn} are a set of constants and {ψn(x)} are the set of energy eigenstates of the well. Use this equation to derive an expression for the cn in terms of Ψ0 and ψn(x). c) The energy eigenstates in the infinite square well are given by found in image Use these equations to derive expressions for the first five of the cn defined in part b) (i.e. derive expressions for c1, c2, c3, c4 and c5). d) Calculate new expressions for the cn if the constant initial wavefunction Ψ0 were changed so that it was restricted to have non-zero values…