Review of Quantum Physics

1.5. PROBLEMS 17Thus if  commutes with Ĥ then we have a conserved quantity. As you know conserved quantitiesplay a crucial role in mechanics. There is also a very deep connection between conserved quantitiesand symmetries of a system (Noether’s theorem), but this is beyond the scope of this course.1.5 ProblemsIf you are short of time, then focus on the following problems: 2, 3, 4, 6, 12, 13, 16, 171A. Withoutcalculation,sketchthewavefunctionoftheexcitedstatewithenergyE 4 forthepotentialwell shown in the figure . Label the important features of your sketch.E 4(hint: the wavefunctionshould decayin the forbidden regionswhere V > E 4 , and oscillateinsidethe well).2A. (i) Observables are represented by linear operators. What is meant by linear?(ii) Two state functions are orthogonal. What is meant by orthogonal?(iii) Functions ψ n form a complete set. What is meant by a complete set?3A. Write down the operators corresponding to the following observables(i) The position along the x axis(ii) The momentum in the x direction(iii) The momentum in the y direction(iv) The total momentum(v) The total momentum squared(vi) The kinetic energy for a particle confined to the x axis(vii) The kinetic energy for motion in three dimensions.(viii) The potential energy(ix) The total energy for a particle confined to the x axis(x) The total energy for a particle in three dimensions4A. Expand the Dirac notation to prove the following (note c is a complex number).(i) 〈ψ 1 |cψ 2 〉 = c〈ψ 1 |ψ 2 〉(ii) 〈cψ 1 |ψ 2 〉 = c ∗ 〈ψ 1 |ψ 2 〉(iii) 〈ψ 3 |ψ 1 +ψ 2 〉 = 〈ψ 3 |ψ 1 〉+〈ψ 3 |ψ 2 〉(iv) If the wave function ψ can be written as the sum of orthogonal functions ψ n , show that〈ψ n |ψ〉 = c n where c n is the corresponding expansion coefficient of ψ.