Shankar exercise 2.5.1 - Hamiltonian

[ehrenfest]

Homework Statement

Could someone get me started with Exercise 2.5.1 in Shankar's Principles of Quantum Mechanics?
Does this forum support TeX or LaTeX?

Homework Equations

The Attempt at a Solution

 

[malawi_glenn]

https://www.physicsforums.com/showthread.php?t=8997

 

[dextercioby]

Post the text of the problem for those of us who don't have the book, but might be willing to help you.

 

[CompuChip]

Ad 2: Yes, use the [ tex] tag (without the space):
$$\left[-\frac{\hbar^2}{2 m} \nabla^2 + U(\mathbf{r}) \right] \psi (\mathbf{r}) = E \psi (\mathbf{r}).$$

 

[ehrenfest]

Show that if $$T = \sum_i\sum_jT_ij(q)q_i' q_j'$$, where $$q_i'$$'s are generalized velocities, then $$\sum p_i q_i' = 2T$$.

 

[malawi_glenn]

And was does the rest stand for?

Work done so far? etc.

 

[ehrenfest]

T is kinetic energy and pi is the canonical momentum conjugate. Also, the apostrophes are derivatives. Sorry.

There is not much work done so far. I wanted someone to give me a hint or just get me started.

 

[ehrenfest]

By the way, does anyone have Shankar's book? For a lot of his exercises you really need the context, so I want to know if I should keep posting questions from his book.

 

[dextercioby]

Show that if $$T = \sum_i\sum_jT_ij(q)q_i' q_j'$$, where $$q_i'$$'s are generalized velocities, then $$\sum p_i q_i' = 2T$$.

Since $p_{i}=\frac{\partial L}{\partial q^{i}}$, i gues the result is pretty obvious, don't you think ?