Exponential Operators: Inverting, Rearranging, Expanding

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    Exponential Operators
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Discussion Overview

The discussion revolves around the properties of exponential operators, specifically focusing on inverting, rearranging, and expanding them. It touches on theoretical aspects, mathematical reasoning, and conditions for the application of these properties.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions what happens when taking the exponential of an operator to the opposite side of an equation, specifically asking about the inversion of ##e^A##.
  • Another participant asserts that the inversion of the exponential operator follows the relation ##(e^{A})^{-1} = e^{-A}##.
  • A participant raises concerns about conditions for this inversion, noting that inverting an ordinary matrix requires a non-zero determinant and asks if similar conditions apply to the operator in the exponential.
  • It is mentioned that as long as ##e^A## exists, which is true for bounded operators, the inversion relation holds.
  • Participants agree that the order of operators matters when taking exponentials to the other side of an equation, particularly in cases where operators do not commute.

Areas of Agreement / Disagreement

Participants generally agree on the relation ##(e^{A})^{-1} = e^{-A}## under certain conditions, but there are unresolved questions regarding the specific conditions required for different types of operators and the implications of non-commuting operators.

Contextual Notes

Limitations include the need for clarification on the conditions under which the inversion holds, particularly regarding the nature of the operator and the implications of operator commutation.

dyn
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If I rearrange an equation invoving exponentials of operators and I take ex to the opposite side of the equation it becomes e-x. What happens if I try to take eA to the opposite side ? I know a exponential of operators can be expanded as a Taylor series which involves products of matrices but can this be inverted ?
 
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Yes, ##(e^{A})^{-1} = e^{-A}##
 
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Thanks. Are there any conditions for that to apply ? To invert an ordinary matrix requires a non-zero determinant. Are there any conditions on the operator/matrix in the exponential ? Also when taking the exponential over to the other side of the equation I presume order matters in case any operators do not commute ?
 
dyn said:
Thanks. Are there any conditions for that to apply ?

No. As long as ##e^A## exists (which it always does if ##A## is a bounded operator), then the above applies.

Also when taking the exponential over to the other side of the equation I presume order matters in case any operators do not commute ?

Yes.
 
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