How does one develop a Hamiltonian for a free particle?

In summary, the Hamiltonian equation, H = T + V, can be used to obtain the equation for a free particle, i\hbar|\psi'> = H|\psi> = P^2/(2m)|\psi>, where V=0 and the kinetic energy operator is KE=p^2 /2m. This can be seen by looking at Schrodinger's equation and referencing classical mechanics. For further clarification, refer to page 151 in the book "Shankar Quantum" at the given link.
  • #1
ehrenfest
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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 Schrodinger'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
 
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  • #2
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)
 
  • #3
I see. Thanks.
 

1. What is a Hamiltonian?

The Hamiltonian is a mathematical function used in classical mechanics and quantum mechanics to describe the dynamics of a system. It is a sum of the kinetic and potential energies of all the particles in the system, and it can be used to determine the equations of motion for the system.

2. How does a Hamiltonian differ from other mathematical functions?

The Hamiltonian is unique in that it takes into account both the position and momentum of particles in a system. This is different from other mathematical functions, which typically only consider one of these variables.

3. How is a Hamiltonian developed for a free particle?

A Hamiltonian for a free particle is developed by considering the kinetic energy of the particle, which is given by its momentum squared divided by twice its mass. The potential energy is assumed to be zero, since a free particle does not experience any external forces. The Hamiltonian is then the sum of the kinetic and potential energies, and can be written as H = p^2/2m.

4. Can a Hamiltonian be developed for a system with multiple particles?

Yes, a Hamiltonian can be developed for a system with multiple particles. In this case, the Hamiltonian would be the sum of the kinetic and potential energies of all the particles in the system, taking into account the interactions between them.

5. What are the applications of developing a Hamiltonian for a free particle?

Developing a Hamiltonian for a free particle is important in understanding the dynamics of particles and predicting their motion. It is also a crucial concept in quantum mechanics, where the Hamiltonian is used to calculate the energy levels of particles and their corresponding wavefunctions.

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