Where can I learn how to manipulate operators?

  • Thread starter k4ff3
  • Start date
  • Tags
    Operators
In summary, the author feels like they don't understand how to do the chain rule when working with operators, and is looking for help.
  • #1
k4ff3
39
0
Nice to be back here at PF and to physics after a year off in the software industry. Now it's time to catch up again :)

I feel like I never really get the grasp of manipulating operators. In QM there's a lot of trixing and mixing going on, and I really would like to learn to do the magic. Oddly, none of my books deal with these techniques, they just assume that you know them.

I can give a simple example of a wave equation:

[tex]\frac{ \partial^2 \eta }{\partial x^2} - \frac{ \partial^2 \eta }{c \cdot \partial t^2} = ( \frac{ \partial}{\partial x} - \frac{ \partial}{c \cdot \partial x} )( \frac{ \partial}{\partial x} + \frac{ \partial}{c \cdot \partial x} ) \eta = 0 [/tex] (*)

By letting [tex]u=x-ct[/tex] and [tex]v=x+ct[/tex] one can show that

[tex]\frac{ \partial}{\partial x} - \frac{ \partial}{c \cdot \partial t} = 2 \frac{ \partial}{ \partial u} [/tex] (**)

[tex]\frac{ \partial}{\partial x} + \frac{ \partial}{c \cdot \partial t} = 2 \frac{ \partial}{ \partial v} [/tex] (***)

by the difference of two squares.

( Which simplifies (*). )

But I don't get how the operator formulas (**) and (***) are obtained! It feels terrible to miss out on something this basic.

Any suggestions?
 
Last edited:
Physics news on Phys.org
  • #3
Corrected. Thanks.

Still stuck though. Despite the article. I know how to use the chain rule when I have variables. But it's when I'm dealing purely with operators I struggle. I.e

[tex]\frac{ \partial}{\partial u} [/tex]

How can you apply a chain rule on this guy?
 
  • #4
Consider u and v as functions of x and t: u(x,t) and v(x,t).
Then apply the chain rule as is done in the example in the section I've linked.
 
  • #5
So for f a function of u and v, which are themselves a function of x and t you have according to the chain rule:

[tex]\frac{\partial f}{\partial x}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}=\frac{\partial f}{\partial u}+\frac{\partial f}{\partial v}[/tex]

Also:

[tex]\frac{\partial f}{\partial t}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial t}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial t}=c\frac{\partial f}{\partial v}-c\frac{\partial f}{\partial u}[/tex]

Now you can multiply the second equation by 1/c and subtract it from the first to get **, and add to get ***.

You put the trial function f in there to see what happens, and then remove it at the end.
 
  • #6
Exactly!
 
  • #7
Be a quantum mechanic.
 
  • #8
k4ff3 said:
I know how to use the chain rule when I have variables. But it's when I'm dealing purely with operators I struggle. I.e

[tex]\frac{ \partial}{\partial u} [/tex]

How can you apply a chain rule on this guy?
The definitions u=x-ct, v=x+ct imply x=(u+v)/2, t=(v-u)/2c, so

[tex]\frac{\partial}{\partial u}g(x(u,v),t(u,v))=\frac{\partial g}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial g}{\partial t}\frac{\partial t}{\partial u}=\frac{\partial g}{\partial x}\frac{1}{2}+\frac{\partial g}{\partial t}\left(-\frac{1}{2c}\right)[/tex]

and therefore

[tex]\frac{\partial}{\partial u}=\frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)[/tex]
 

1. What is the purpose of manipulating operators in science?

Manipulating operators allows scientists to perform mathematical operations on data and variables, enabling them to make calculations and draw conclusions about their experiments.

2. Where can I find resources to learn how to manipulate operators?

There are many online resources available, such as tutorials, videos, and interactive exercises, that can help you learn how to manipulate operators. You can also consult textbooks and ask for guidance from experienced scientists.

3. Is manipulating operators difficult to learn?

Manipulating operators may seem complex at first, but with practice and patience, it can be easily understood. As with any new skill, it takes time to become proficient, but the end result is worth the effort.

4. Are there different types of operators in science?

Yes, there are different types of operators in science, such as arithmetic, comparison, logical, and assignment operators. Each type serves a specific purpose and is used in different scenarios.

5. How can I apply my knowledge of manipulating operators in my scientific research?

Manipulating operators can be applied in a variety of ways in scientific research, such as analyzing data, creating and testing hypotheses, and developing mathematical models. It is a fundamental skill that is essential for many fields of science.

Similar threads

Replies
7
Views
459
Replies
3
Views
283
Replies
3
Views
775
  • Quantum Physics
2
Replies
56
Views
3K
Replies
17
Views
1K
Replies
14
Views
1K
Replies
29
Views
3K
Replies
14
Views
1K
Replies
1
Views
811
  • Quantum Physics
Replies
11
Views
1K
Back
Top