About momentum operator in other coordinate system

[KFC]

Hi there, does anyone know where can I found a material or note about how to deduce momentum operator in coordinate system other than linear coordinate (especially in spherical coordinate system)?

Thanks in advanced.

 

[George Jones]

Hi there, does anyone know where can I found a material or note about how to deduce momentum operator in coordinate system other than linear coordinate (especially in spherical coordinate system)?

Thanks in advanced.

Express the other coordinates in terms of $x$, and $y$, and $z$, and then use the chain rule. For example,

$$\frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}.$$

 

[Entropia]

I can understand your curiosity about the momentum operator in other coordinate systems. The momentum operator, denoted by p, is a fundamental concept in quantum mechanics that represents the observable quantity of momentum of a particle. It is defined as the derivative of the wave function with respect to position.

In linear coordinate systems, such as Cartesian coordinates, the momentum operator is simply given by the gradient operator, which takes the form of a vector. However, in non-linear coordinate systems, such as spherical coordinates, the momentum operator may take a more complex form.

To deduce the momentum operator in a specific coordinate system, one must first understand the mathematical transformation between the coordinate systems. This involves using transformation matrices and differential operators to convert from one coordinate system to another.

There are many resources available that discuss the momentum operator in different coordinate systems. Some textbooks on quantum mechanics may have a section dedicated to this topic, or you can also find online notes and lectures on the subject. It is important to note that the form of the momentum operator may vary depending on the specific coordinate system being used.

In summary, to deduce the momentum operator in a coordinate system other than linear, one must have a good understanding of mathematical transformations and differential operators. I recommend exploring different resources and consulting with experts in the field for a deeper understanding of this topic. I hope this helps.