Hermitian operators

Could you please show me how to do the problem called QM12 that I have attached in MS Word format? If you prefer pdf, it's question 1 in section 4 in the file QM-MT that is also attached. The question is called Hermitian Operators. Thank you.

Change of state following a measurement

Could you please explain the MS Word problem attached (QM10)? If you prefer pdf, it's also attached in pdf format. However, the pdf format is the whole problem sheet and I need problem 1 of section 5 in the pdf problem set. Equation 16 is shown in problem 3 in section 3. In the MS Word format, all the relevant information is on one page. Thank you.

For a particle in a 1-dimensional box confined between 0

The central potential and the 3-dimensional time independent Schrodinger equation

The Central Potential The 3-D time independent Schrodinger equation for a particle of mass m in a spherically symmetric potential is -- Equation---- a) By writing grad^ in terms of r, theta and phi show that this can be written as -- Equations--- (b) Writing shi = R Y show that Y must satisfy L^2 Y = c Y where c is a constant, and where -- Equation---- (c) Writing R = f(r)/r, show that f(r) satisfies the equation -- Equation---- (d) This is of exactly the same form as a 1-D time independent Schrodinger equation with V replaced by V + c/2mr^2. So can we conclude that the spectrum in the original 3D problem is the same as for this 1D problem?

The angular momentum operators

Could you please solve the problem attached on ang. momentum operators? Thank you.