I'm still studying for quals. I got stuck for awhile on problem 2.11 of Griffith's Introduction to Quantum Mechanics first edition. The problem involves manipulating raising and lowering operators to perform a normalization. The problem I was having was doing too much of the math too early. I tried to expand all the operators and got stuck. I tried to apply integration by parts as the book suggests and applied it to the wrong term the first time, and carried the operation too far on a second try. The series of conservative steps that finally did the trick is shown in the video below.
The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!
What do we actually want?
To convert the Cartesian nabla
to the nabla for another coordinate system, say… cylindrical coordinates.
What we’ll need:
1. The Cartesian Nabla:
2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:
3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:
How to do it:
Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.
The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…
What do we actually want?
To convert the Cartesian nabla
to the nabla for another coordinate system, say… cylindrical coordinates.
What we’ll need:
1. The Cartesian Nabla:
2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:
3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:
How to do it:
Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.
The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…
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