Canonical Transformations

Step 1: The canonical transformation is given by the relations:

Q%20%3D%20%5Cfrac%7B1%7D%7Bp%7D%2C%20%5Cquad%20P%20%3D%20q%20p%5E2.

The generating function

F

satisfies the following relations:

%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20q%7D%20%3D%20P%2C%20%5Cquad%20%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20p%7D%20%3D%20-Q.

Step 2: For

F_1(q%2C%20Q)

, we choose the generating function such that:

%5Cfrac%7B%5Cpartial%20F_1%7D%7B%5Cpartial%20q%7D%20%3D%20P%20%3D%20%5Cfrac%7Bq%7D%7BQ%7D.

This satisfies the relation, and thus:

F_1(q%2C%20Q)%20%3D%20%5Cfrac%7Bq%7D%7BQ%7D.

Step 3: For

F_4(p%2C%20P)

, we need to satisfy:

%5Cfrac%7B%5Cpartial%20F_4%7D%7B%5Cpartial%20p%7D%20%3D%20-Q%20%5Cquad%20%5CRightarrow%20%5Cquad%20F_4(p%2C%20P)%20%3D%20%5Cfrac%7BP%7D%7Bp%7D.

Chapter Questions
1 MCQs