How can I derive the path integral using the Baker-Campbell-Hausdorff formula?

Click For Summary
SUMMARY

The discussion focuses on deriving the path integral using the Baker-Campbell-Hausdorff (BCH) formula. The user references the identity exp(-iHΔt/ħ) = exp(-i(T+V)Δt/ħ) and notes that it is only valid to first order, indicating the presence of an O(Δt²) term. The inquiry seeks clarification on the general identity for factoring the exponential of a sum of operators, essential for progressing in the derivation of the path integral.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly Hamiltonians.
  • Familiarity with the Baker-Campbell-Hausdorff formula.
  • Knowledge of operator algebra in quantum mechanics.
  • Basic grasp of path integrals and their significance in quantum field theory.
NEXT STEPS
  • Study the Baker-Campbell-Hausdorff formula in detail.
  • Research the derivation of path integrals in quantum mechanics.
  • Explore the implications of operator ordering in quantum mechanics.
  • Examine higher-order corrections in the expansion of exponential operators.
USEFUL FOR

Quantum physicists, students of quantum mechanics, and researchers interested in the mathematical foundations of path integrals and operator theory.

aaaa202
Messages
1,144
Reaction score
2
On the attached picture I have started trying to derive the path integral but I don't know how I get further. Can anyone help me? Also I have used that:
exp(-iHΔt/ħ) = exp(-i(T+V)Δt/ħ) = exp(-iTΔt/ħ)exp(-iVΔt/ħ)
But my book says that this identity is only correct to first order, i.e. there is an O(Δt2). What is the general identity for factoring the exponential of a sum of operators?
But please help me to make progress in my derivation of the path integral, since I would like to understand how it comes about.
 

Attachments

  • path integral1.png
    path integral1.png
    17.8 KB · Views: 678
Physics news on Phys.org

Similar threads

Graduate Derive the Principle of Least Action from the Path Integral?
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Orthogonality of variations in Faddev-Popov method for path integral
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Can a Path Integral Formulation for Photons Start from a Massless Premise?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Understanding how to derive the Feynman rules out of the path integral
  • · Replies 33 ·
2
Replies
33
Views
7K
Graduate Where did this term in the path integral come from?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Proving Commutator Identity for Baker-Campbell-Hausdorff Formula
  • · Replies 4 ·
Replies
4
Views
4K
Graduate Help needed with derivation: solving a complex double integral
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Path Integral Approach To Derive The KG Propagator
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Flux-flux correlation function under Feynman's path integral
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Comparing Path Integral Conventions in QED
  • · Replies 3 ·
Replies
3
Views
2K