The first two sub-questions concern a particle-in-a-box that has a length of 2 (i.e. a particle of mass m where the potential is: V(x)=0 for 0x2 and infinite elsewhere) Consider a wavefunction: (x)=Bsin(2x), where B is a normalization constant. A. [2 mark] Sketch the probability density associated with this wavefunction. Label your axes and the location of any nodes that do not occur at the edges of the 'box'. B. [1 mark] Write down an equation involving a definite integral of explicit functions that determines the value of B so that 3(x) is normalized. (There is no need to evaluate this expression) C. [2 marks] Consider the usual position operator, x^, and it's self-commutator acting on an arbitrary function (i.e. [x^,x^]f(x) ) - Expand this expression, and use this to determine the value of [x^,x^] D. [2 marks] Consider the arbitrary linear operator A^, the arbitrary functions f(x) and g(x), and the following three "equations"-exactly one of which is not true (in general.) - From what you know about operator algebra, state with brief justification which one is not valid. 1) A^(f(x)+g(x))=A^f(x)+A^g(x) 2) A^f(x)=f(x)A^ 3) f(x)(A^f(x))=(A^f(x))f(x)