Understanding the Energy of a Particle in Momentum Representation of QM

[Geezer]

For some reason, the momentum representation in QM wasn't covered in our class, so I'm figuring it out on my own (no, this isn't homework...it's just me reviewing physics for the PGRE).

My question: what is the energy (kinetic energy, I guess) of a particle in the momentum representation of QM? Obviously, p^2/2m is the usual KE, so is that the energy operator, too, in the momentum representation?

Thanks, y'all.

 

[maverick_starstrider]

Yes, that's the KE operator, if there's a potential you'd have to write it interms of the momentum operator. $\hat{p}^2|p>=p^2|p>$

 

[Entropia]

Hello,

Thank you for your question about the energy of a particle in momentum representation in quantum mechanics. I can understand your confusion, as this topic may not have been covered in your class. Let me help clarify this concept for you.

In quantum mechanics, the momentum representation is a mathematical representation of the wave function of a particle in terms of its momentum. This means that instead of describing the particle's position in space, we describe it in terms of its momentum. In this representation, the wave function is written as a function of momentum, rather than position.

Now, to answer your question about the energy of a particle in momentum representation, yes, the energy operator in this representation is indeed given by p^2/2m, where p is the momentum operator and m is the mass of the particle. This is the same expression as the kinetic energy operator in position representation, but written in terms of momentum.

It is important to note that in quantum mechanics, the energy of a particle is an observable quantity, meaning it can be measured. In momentum representation, the expectation value of the energy is given by the operator p^2/2m acting on the wave function.

I hope this helps clarify the concept of energy in momentum representation in quantum mechanics. If you have any further questions, please feel free to ask. Best of luck on your PGRE review!

 

[Entropia]

Hello, thank you for your question. In quantum mechanics, the momentum representation is a way of describing the state of a particle in terms of its momentum rather than its position. This is achieved by using the momentum operator, which is represented by the symbol p. The momentum operator is defined as the derivative of the position operator with respect to time, or p = d/dt.

To answer your question, yes, the energy operator in the momentum representation is also given by the expression p^2/2m, where p is the momentum and m is the mass of the particle. This is known as the kinetic energy operator, and it represents the energy associated with the motion of the particle. In quantum mechanics, energy is represented by operators, which are mathematical objects that act on the wave function of the particle to yield the energy of that state.

It is important to note that the energy of a particle in the momentum representation is not limited to just the kinetic energy. It also includes potential energy, which can be represented by other operators depending on the specific system being studied. In general, the total energy of a particle in the momentum representation is given by the Hamiltonian operator, which is the sum of the kinetic and potential energy operators.

I hope this helps clarify your understanding of the energy of a particle in the momentum representation of quantum mechanics. Keep exploring and learning, and best of luck on your PGRE!