- #1
k4ff3
- 39
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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?
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?
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