# Derivation of momentum operator

• pcflores
In summary, the conversation discusses the derivation of the momentum operator and its physical interpretation, mentioning two methods and recommending resources for further understanding. It also touches on the symmetries of Galilean relativity and their role in both quantum mechanics and classical mechanics.
pcflores
hello,

i am trying to learn the derivation of the momentum operator and i found 2 ways of deriving it. one is using Fourier transform and the other is taking the time derivative of the expectation value of x.

i just want to know what is the physical interpretation of the time rate of change of <x>

thank you

Its related to Stones Theorem and infinitesimal generators:
http://en.wikipedia.org/wiki/Stone's_theorem_on_one-parameter_unitary_groups

If you really want to understand the momentum operator get a hold of a copy of Ballentine - Quantum Mechanics - A Modern Development and have a look at chapter 3:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

Its true basis is the symmetries of Galilean relativity - but that revelation is something you need to discover for yourself - its very profound and deep. Revelations like that are best understood by working through the detail.

Then get a hold of Landaus classic on Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

It shows classical mechanics has exactly the same basis. The symmetries of Galilean relativity is the key to both just as the symmetries of the Lorentz transformations is the key to relativity.

Thanks
Bill

Last edited by a moderator:
bhobba said:
Its related to Stones Theorem and infinitesimal generators:
http://en.wikipedia.org/wiki/Stone's_theorem_on_one-parameter_unitary_groups

If you really want to understand the momentum operator get a hold of a copy of Ballentine - Quantum Mechanics - A Modern Development and have a look at chapter 3:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

Its true basis is the symmetries of Galilean relativity - but that revelation is something you need to discover for yourself - its very profound and deep. Revelations like that are best understood by working through the detail.

Thanks
Bill

thank you

Last edited by a moderator:
pcflores said:
thank you

You are most welcome.

Also see if you can have a look at Landau as well - you may have missed that bit since I added it later.

Thanks
Bill

## What is the momentum operator?

The momentum operator is a mathematical operator that represents the momentum of a particle in quantum mechanics. It is denoted by the symbol p and is defined as the product of the mass of the particle and its velocity.

## Why is the momentum operator important?

The momentum operator is important because it allows us to calculate the momentum of a particle in quantum mechanics. This is crucial for understanding the behavior of particles at the microscopic level and for solving quantum mechanical equations.

## How is the momentum operator derived?

The momentum operator is derived from the classical definition of momentum, p = mv, where m is the mass of the particle and v is its velocity. In quantum mechanics, the momentum operator is represented by the differential operator, p = -iħ∂/∂x, where i is the imaginary unit and ħ is the reduced Planck's constant.

## What are the properties of the momentum operator?

The momentum operator has several important properties, including linearity, hermiticity, and commutativity with the position operator. These properties allow us to use the momentum operator in various mathematical operations and to make predictions about the behavior of particles in quantum mechanics.

## How is the momentum operator used in quantum mechanics?

The momentum operator is used in quantum mechanics to calculate the momentum of a particle, to solve the Schrödinger equation, and to describe the dynamics of a system. It is also used in the Heisenberg uncertainty principle, which states that the position and momentum of a particle cannot be simultaneously known with absolute certainty.

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