# Why the velocity operator commutes with position (Dirac equation)

• I
• zhouhao
In summary, in the classical picture, the Poisson brackets of the velocity and position operators do not equal zero. In quantum mechanics, the commutator of these operators is equal to a complex expression involving Planck's constant and the Hamiltonian operator. In classical mechanics, if we view the velocity as a field, the Poisson bracket of the velocity and position is equal to zero, indicating that the two operators commute. However, it is not clear how to explain this phenomenon.
zhouhao
##\hat{v}_i=c\hat{\alpha}_i## commute with ##\hat{x}_i##,
##E^2={p_1}^2c^2+{p_2}^2c^2+{p_3}^2c^2+m^2c^4##
But in classical picture,the poisson braket ##(v_i,x_i)=\frac{\partial{(\frac{c^2p_i}{E})}}{\partial{p_i}}=\frac{c^2}{E}+p_i\frac{\partial{(\frac{c^2}{E})}}{\partial{p_i}}=\frac{c^2}{E}-\frac{p_1c^2}{E^2}\frac{\partial{E}}{\partial{p_1}}=\frac{c^2}{E}-\frac{{p_1}^2c^4}{E^3}\neq 0##
In quantum mechanic,commutor [##\hat{v}_i,\hat{x}_i##]=##i\hbar(v_i,x_i)=\frac{c^2}{{\hat{H}}}-\frac{{\hat{p}_1}^2c^4}{\hat{H}^3}##,I do not know what the result is and how to deal with the formula,but it not equal zero...

In classic mechanic,if we regard ##v_i## as a field ##v_i(t,x_1,x_2,x_3,x_4)##,
the poisson braket ##(v_i,x_i)=\frac{\partial{v_i}}{\partial{p_i}}=\frac{\partial{v_i}}{\partial{x_i}}\frac{\partial{x_i}}{\partial{p_i}}+\frac{\partial{v_i}}{\partial{t}}\frac{\partial{t}}{\partial{p_i}}=0##,

so the velocity and position commute.

I do not know how to explain this...

## 1. Why does the velocity operator commute with position in the Dirac equation?

The velocity operator commutes with position in the Dirac equation because the Dirac equation is a relativistic quantum mechanical equation that describes the behavior of spin-1/2 particles. This equation was derived by Paul Dirac in 1928 and is based on principles of quantum mechanics and special relativity. In this equation, both the position and velocity operators are represented by matrices, and these matrices commute with each other.

## 2. What is the significance of the velocity operator commuting with position in the Dirac equation?

The significance of the velocity operator commuting with position in the Dirac equation is that it allows for the prediction of the future position of a particle based on its current position and velocity. This is a fundamental aspect of quantum mechanics and allows for the precise calculation of the behavior of particles at the subatomic level.

## 3. How does the commutation of the velocity and position operators in the Dirac equation relate to the uncertainty principle?

The commutation of the velocity and position operators in the Dirac equation does not directly relate to the uncertainty principle. However, the 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. The commutation of these operators allows for the calculation of both position and velocity, but the uncertainty principle still applies.

## 4. Can the commutation of the velocity and position operators be observed experimentally?

No, the commutation of the velocity and position operators in the Dirac equation cannot be observed directly through experiments. This is because the Dirac equation describes the behavior of particles at the subatomic level, which cannot be observed directly. However, the predictions made by the Dirac equation have been confirmed through various experiments, providing evidence for the validity of the theory.

## 5. Are there any exceptions to the commutation of the velocity and position operators in the Dirac equation?

No, there are no exceptions to the commutation of the velocity and position operators in the Dirac equation. This fundamental principle of quantum mechanics holds true for all spin-1/2 particles and is a crucial aspect of the Dirac equation. However, there are other quantum mechanical equations that do not follow this rule, such as the Heisenberg equation of motion.

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