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What's the main logic of path integral formulation ? (Feyman path integral formulation)
I mean what's the reason to think this way ?
Thanks
I mean what's the reason to think this way ?
Thanks
Given those facts, we can write the amplitude [itex]\Psi(x_f, t_f, x_i, t_i)[/itex] in the approximate form: (Picking [itex]N[/itex] to be a large number, and letting [itex]\delta t = \frac{t_f - t_i}{N+1}[/itex])stevendaryl said:The main goal of path integrals is to compute transition probabilities. If a particle is known at time [itex]t_i[/itex] to be at a location [itex]x_i[/itex], then you want to know what the probability is that the particle will be found at postion [itex]x_f[/itex] at a later time [itex]t_f[/itex]. Quantum mechanically, this is computed in terms of probability amplitudes, and you square the amplitude to get the probability.
The two facts about this amplitude, [itex]\Psi(x_f, t_f, x_i, t_i)[/itex] that are important for computing path integrals (I'm only doing single-particle nonrelativistic quantum mechanics here, but it should give you the general idea) are:
- [itex]\Psi(x_f, t_f, x_i, t_i) = \int d x_1 \Psi(x_f, t_f, x_1, t_1) \Psi(x_1, t_1, x_i, t_i)[/itex], where [itex]t_1[/itex] is any time between [itex]t_i[/itex] and [itex]t_f[/itex]. The idea behind this is that in order to go from [itex]x_i[/itex] at time [itex]t_i[/itex] to location [itex]x_f[/itex] at time [itex]t_f[/itex], the particle must be somewhere, call it [itex]x_1[/itex], at time [itex]t_1[/itex]. The amplitude for the entire path is the amplitude to get from [itex]x_i[/itex] to [itex]x_1[/itex] times the amplitude to get from [itex]x_1[/itex] to [itex]x_f[/itex]. To include all possibilities for the intermediate location, [itex]x_1[/itex] we integrate over all possibilities.
- In the limit as [itex]\delta t \rightarrow 0[/itex], the amplitude [itex]\Psi(x_f, t_i + \delta t, x_i, t_i)[/itex] approaches the value [itex]K(\delta t) e^{\frac{i}{\hbar}} L(x, v, t) \delta t[/itex] where [itex]x = \frac{x_i + x_f}{2}[/itex] (the "average" position), and [itex]t = t_i + \frac{\delta t}{2}[/itex] (the "average" time), and [itex]v = \frac{x_f - x_i}{\delta t}[/itex] (the "average" velocity), and where [itex]L(x,v,t)[/itex] is the classical Lagrangian, which for nonrelativistic single-particle motion is given by: [itex]L = \frac{m v^2}{2} - V(x)[/itex], where [itex]m[/itex] is the mass and [itex]V[/itex] is the potential energy.
Quarlep said:I understand that every trajectory is a wavefunction.Some of them cancels or some of them makes interferance and it makes the real trajectory.(In double slit experiment every slit)
Let's suppose I am in London I will go to Berlin but Before go there I must be somewhere between this two points.Lets call it Paris so The probability to go Berlin is London-Paris probability ( wave function squared) multiply by Paris Berlin probability. Is that true ?stevendaryl said:The main goal of path integrals is to compute transition probabilities. If a particle is known at time tit_i to be at a location xix_i, then you want to know what the probability is that the particle will be found at postion xfx_f at a later time tft_f. Quantum mechanically, this is computed in terms of probability amplitudes, and you square the amplitude to get the probability.
The two facts about this amplitude, Ψ(xf,tf,xi,ti)\Psi(x_f, t_f, x_i, t_i) that are important for computing path integrals (I'm only doing single-particle nonrelativistic quantum mechanics here, but it should give you the general idea) are:
Quarlep said:Lets suppose I am in London I will go to Berlin but Before go there I must be somewhere between this two points.Lets call it Paris so The probability to go Berlin is London-Paris probability ( wave function squared) multiply by Paris Berlin probability. Is that true ?
The path integral formulation is a mathematical framework developed by physicist Richard Feynman to describe the behavior and interactions of particles in quantum mechanics. It is based on the principle of least action, where the probability of a particle moving from one point to another is calculated by summing up all possible paths it could take.
In the path integral formulation, the probability amplitude for a particle to travel from one point to another is represented as a sum of all possible paths that the particle could take. Each path is assigned a weight based on the action of the particle along that path. The path with the highest weight is the most likely path that the particle will take.
The path integral formulation provides a more intuitive understanding of quantum mechanics compared to other mathematical approaches. It allows for the calculation of probabilities for complex systems by breaking them down into simpler interactions between particles. It also allows for the incorporation of both quantum and classical mechanics in a unified framework.
One limitation of the path integral formulation is that it can be computationally intensive, especially for systems with a large number of particles. It also does not provide a complete picture of particle interactions, as it does not take into account the effects of virtual particles. Additionally, the path integral formulation may not be applicable to all physical systems.
The path integral formulation is used in various fields of physics, including particle physics, condensed matter physics, and quantum field theory. It is used to calculate probabilities and make predictions about the behavior of particles and systems. It has also been applied in other fields such as finance and biology.