Particle Momentum and Position Calculation in Heisenberg Picture

[eljose]

Let,s suppose we have a particle moving under a potential V:
By Heisenberg picture we know that:

$$\frac{dx}{dt}=p/m$$

so if we knew x(t) and x(t+h) we could calculate the expresion:

$$x(t+\epsilon)-x(t)=p\epsilon/m$$

so knowing x(t) and x(t+e) we could calculate the momentum of the particle:

 

[seratend]

Let,s suppose we have a particle moving under a potential V:
By Heisenberg picture we know that:

$$\frac{dx}{dt}=p/m$$

so if we knew x(t) and x(t+h) we could calculate the expresion:

$$x(t+\epsilon)-x(t)=p\epsilon/m$$

so knowing x(t) and x(t+e) we could calculate the momentum of the particle:

Please do not mix the differential equation between operators and eigenvalues. They are completely different.


Seratend.

 

[denian]

p=m(x(t+\epsilon)-x(t))/\epsilon

This calculation in the Heisenberg picture allows us to determine the momentum of a particle without directly measuring it. By knowing the position of the particle at two different times, we can calculate the change in position over a small time interval and use it to determine the momentum. This is an important concept in quantum mechanics, as it shows that the position and momentum of a particle cannot be known simultaneously with absolute certainty. The Heisenberg uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. Therefore, the Heisenberg picture provides a way to calculate these quantities without violating this fundamental principle. This is crucial in understanding the behavior of particles at the atomic and subatomic level, where the laws of classical mechanics do not apply.