How does one develop a Hamiltonian for a free particle?

[ehrenfest]

The equation for the Hamiltonian is H = T + V. Can someone explain how you can use this to get this equation for a free particle:

$$i\hbar|\psi'> = H|\psi> = P^2/(2m)|\psi>$$

The first part is obviously Schrödinger's equation but how do you get H = P^2/2m?

Go to page 151 at the site below if you do not understand this question.

http://books.google.com/books?id=2z...sig=nQ9UgEufWOeqXgJdtGEylDqK7ok#PRA1-PA151,M1

 

[christianjb]

For a free particle V=0.

You're left with the KE operator.

Classically, KE=p^2 /2m.

QM, KE operator = p^2 / 2m, with p=-ihbar d/dx (in 1D)

 

[ehrenfest]

I see. Thanks.