Q3:3 A particle is initially represented by a state which is one of the eigenfunctions {o, (x)} of a hamiltonian operator H, namely ø1, whose normalised form is given by 01 (1) = /2exp The eigenvalue for this eigenfunction is 3. (a) What is the mean or average value of r for this state? (b) If the wavefunction of a particle for the above system was instead represented by the normalised state 1 v (x) = (i - Vz) esp (-). exp 271/2 find the probability that a measurement of the energy of the particle yields the eigenvalue 3. ( You may assume that r exp(-x²)dr = Vñ/2.)

An observable, q, is represented by an operator,    . Assuming a system is in a state Ψ(x,t). a) Write an expression for the expectation value of the observable,<q>. b) Write an expression for the uncertainty in measuring the observable, ∆q. c) Write expressions for the operators relating to position and momentum (     and      ) and for the squares of these operators. d) If Ψ(x,t) is an eigenstate of     , write an equation relating the eigenvalue and itsoperator.

Determine the normalization constant for the following wavefunction. Write an expression for the normalized wavefunction. (8) y=(r/ao)et/2a,

The Hamiltonian for the one dimensional quantum oscillator is 1 p² 1 Ĥ = 1² + ½ k²² = 12 + √ mw² ಠ2m 2m 2 where k = mw². 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as a function of 1 and p₁ such that Ĥ = hwĤ₂. - 2) Let us define the new operators â (1 + i₁) and ↠= ½(î₁ — ip₁). Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and [1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at. 3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have the same eigenstates, what are their corresponding eigenvalues?