Momentum Operator: Comparing p_x and p

[hyddro]

This is not really a homework problem but rather a homework-related question.. When I came across my homework (and my textbook: Atkin's physical chemistry 9th Ed.), they defined the momentum operator as:

p_x = - ( $\hbar$ / i ) * d/dx...

but i have seen in other sources that they define it as p = ( $\hbar$ * i ) * d/dx...

they multiply i rather than divide and it is also positive.. i don't known if they represent the same or not, if so.. how can you convert one to the other? Thanks and sorry if this is a very silly question..

 

[micromass]

We have

$$\frac{1}{i} = -i$$

 

[hyddro]

Thanks! I know it was something easy and silly. Thanks again.

 

[vanhees71]

The usual sign convention in physics is, however,
$$\hat{p}=-\mathrm{i} \hbar \partial_x.$$
For more details, see
https://en.wikipedia.org/wiki/Momentum_operator

 

[paranormal]

The momentum operator, denoted as p, is a mathematical representation of the physical quantity momentum in quantum mechanics. It is a vector operator, meaning it has both magnitude and direction, and it acts on a wavefunction to give the momentum of the particle described by the wavefunction.

The two expressions you have encountered, p_x = - ( \hbar / i ) * d/dx and p = ( \hbar * i ) * d/dx, are both valid representations of the momentum operator. The difference lies in the choice of convention and notation used by different sources.

In the first expression, p_x, the subscript x indicates that the operator is acting in the x-direction. The negative sign is a result of the convention used in quantum mechanics, where the momentum operator is defined as the negative of the derivative with respect to position. This convention is chosen to ensure that the momentum operator is a Hermitian operator, meaning it has real eigenvalues.

In the second expression, p, the subscript is omitted and the positive sign is used. This is a more general representation of the momentum operator, where the direction of momentum is not specified. This notation is commonly used in introductory quantum mechanics texts.

To convert one representation to the other, you can simply multiply or divide by the appropriate constants. For example, to convert p_x to p, you would multiply by -i/\hbar, and vice versa to convert p to p_x.

In summary, both p_x and p are valid representations of the momentum operator, with the difference lying in the choice of convention and notation used. It is important to understand the meaning and significance of these representations, but ultimately they both represent the same physical quantity - momentum.