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

In summary, the conversation discusses the derivation of the path integral and the use of the identity exp(-iHΔt/ħ) = exp(-i(T+V)Δt/ħ) = exp(-iTΔt/ħ)exp(-iVΔt/ħ), which is only correct to first order. The question is asked about the general identity for factoring the exponential of a sum of operators and a request for help in understanding the derivation of the path integral.
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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.
 

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

1. What is the path integral method?

The path integral method is a mathematical tool used in quantum mechanics to calculate the probability amplitude of a particle moving from one point to another in space and time. It involves summing over all possible paths that the particle could take, taking into account the wave-like behavior of the particle.

2. How is the path integral derived?

The path integral is derived from the principles of quantum mechanics and the concept of superposition. It involves breaking down the motion of a particle into infinitesimal steps, calculating the probability amplitude for each step, and then summing over all possible paths to get the total probability amplitude.

3. What is the significance of the path integral in physics?

The path integral is a fundamental concept in quantum mechanics and is used in various areas of physics, including particle physics, condensed matter physics, and quantum field theory. It allows for the calculation of complex systems and has applications in understanding the behavior of particles and fields at the quantum level.

4. How does the path integral relate to classical mechanics?

The path integral method is a generalization of the classical action principle, which states that the path a particle takes in space and time is determined by the principle of least action. The path integral method takes into account the wave-like behavior of particles and is a more comprehensive approach to calculating the behavior of systems at the quantum level.

5. What are the limitations of the path integral method?

The path integral method is a powerful tool in quantum mechanics, but it has its limitations. It is difficult to apply to systems with a large number of particles, and the calculations can become increasingly complex. It also does not take into account the effects of gravity, which is still an area of active research in quantum mechanics.

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