Plz, I need the solution of the problem (14.3.9 part b) Arfken, Mathematical Methods for Physicists as soon as possible. I need the solution with details, because i tried hardly to solve it,but i could NOT. regards

what is a directional derivative/gradient?

Directional Derivative

The directional derivative  is the rate at which the function  changes at a point in the direction . It is a vector form of the usual derivative, and can be defined as

			(1)
			(2)

where  is called "nabla" or "del" and  denotes a unit vector.

The directional derivative is also often written in the notation

			(3)
			(4)

where  denotes a unit vector in any given direction and  denotes a partial derivative.

Let  be a unit vector in Cartesian coordinates, so

(5)

then

(6)

Q. The matrices representing the angular momentum components jx jy jz are all hermitian .Show that the Eigen values of j2 is equal TO J2 =JX2 +JY2+JZ2 are REAL AND NON-NEGATIVE.

 Ans. This question is taken from the "Mathematical Methods for Physicist" Problem number 3.5 from chapter no 3

Finding the eigen values for J²=Jx² + Jy² + Jz²

Then we get

(Jm |J²|Jm)= (Jm |Jx²|Jm) +(Jm |Jy²|Jm)+ (Jm |Jz²|Jm)

Which can also be written as

|Jx(jm)|² + |Jy(jm)|² + |Jz(jm)|²

The above equation shows that the eigen values for J²=Jx² + Jy² + Jz² are real and non negative

   for complete understanding download Solution Manual : Mathematical methods for physicists 5th edition Arfken and Weber  and See Chapter 3 page no. 19