- #1
gremezd
- 18
- 0
Does anyone know how to differentiate an exponential, which has an operator in its power? I found it quite a trouble in Peskin's QFT (page 84, formulas (4.17), (4.18)).
Here we have these two formulas of Peskin:
[tex]U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }e^{-iH\left( t-t_{0}\right) } [/tex];
[tex]i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }\left( H-H_{0}\right) e^{-iH\left( t-t_{0}\right) } [/tex].
I agree with this. However, if we write [tex]U\left( t,t_{0}\right)[/tex] as [tex]U\left( t,t_{0}\right)=e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }[/tex], then
[tex]i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=\left( H-H_{0}\right)e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }[/tex]
and we cannot transport [tex]e^{iH_{0}\left( t-t_{0}\right) }[/tex] to the left of [tex]\left( H-H_{0}\right)[/tex] so easily to obtain Peskin's result, since, according to my calculations, [tex]\left[ H,H_{0}\right]\neq0[/tex]. Do we have a rule, which explains where to put the operators from the exponential after differentiation, when we have several noncummuting operators in the power of exponential?
Here we have these two formulas of Peskin:
[tex]U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }e^{-iH\left( t-t_{0}\right) } [/tex];
[tex]i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }\left( H-H_{0}\right) e^{-iH\left( t-t_{0}\right) } [/tex].
I agree with this. However, if we write [tex]U\left( t,t_{0}\right)[/tex] as [tex]U\left( t,t_{0}\right)=e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }[/tex], then
[tex]i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=\left( H-H_{0}\right)e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }[/tex]
and we cannot transport [tex]e^{iH_{0}\left( t-t_{0}\right) }[/tex] to the left of [tex]\left( H-H_{0}\right)[/tex] so easily to obtain Peskin's result, since, according to my calculations, [tex]\left[ H,H_{0}\right]\neq0[/tex]. Do we have a rule, which explains where to put the operators from the exponential after differentiation, when we have several noncummuting operators in the power of exponential?