How does one develop a Hamiltonian for a free particle?

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SUMMARY

The Hamiltonian for a free particle is defined as H = T + V, where T represents the kinetic energy and V is the potential energy. In the case of a free particle, the potential energy V equals zero, leading to the Hamiltonian being expressed as H = P^2/(2m). This formulation is derived from the classical kinetic energy equation KE = p^2/2m, with the momentum operator in quantum mechanics defined as p = -iħ(d/dx). This connection between classical and quantum mechanics is crucial for understanding the behavior of free particles.

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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:

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

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
 
Last edited:
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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)
 
I see. Thanks.
 

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