Path Integral From Heisenberg Uncertainty?

In summary, two questions about the path integral:a) Intuitively, how does one (as Landau does) start quantum mechanics from Heisenberg's uncertainty principle, which states there is no concept of the path of a particle, derive Schrodinger equation i \hbar \tfrac{\partial \psi}{\partial t} = H \psi the operator solution \psi = Ae^{\tfrac{-i}{\hbar}Ht}\psi_0 and from this basis express Ae^{\tfrac{-i}{\hbar}Ht} as an integral over all possible paths. In other words, starting from the assumption that no path
  • #1
bolbteppa
309
41
Two questions about the path integral:

a) Intuitively, how does one (as Landau does) start quantum mechanics from Heisenberg's uncertainty principle, which states there is no concept of the path of a particle, derive Schrodinger equation [itex] i \hbar \tfrac{\partial \psi}{\partial t} = H \psi[/itex] the operator solution [itex] \psi = Ae^{\tfrac{-i}{\hbar}Ht}\psi_0[/itex] and from this basis express [itex] Ae^{\tfrac{-i}{\hbar}Ht}[/itex] as an integral over all possible paths. In other words, starting from the assumption that no path exists we construct the wave function in terms of an integral over all paths (even though no true path exists) and get the right answer, mathematically how does that make sense?

b) Is there a way to derive the path integral from the quasi-classical approximation [itex] \psi = e^{\tfrac{i}{\hbar}S}[/itex]? I mean by using [itex] \psi = e^{\tfrac{i}{\hbar}S} \cdot 1 = e^{\tfrac{i}{\hbar}S} \cdot e^{\tfrac{i}{\hbar}S_1} \cdot e^{-\tfrac{i}{\hbar}S_1} [/itex] where [itex]S_1[/itex] is the action for a nearby path, then extending over all actions over all paths, or something?
 
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  • #2
We are talking about probability not about determinisn. The equations only involve probability amplitudes . And then what feynman did was reform the hole equation to a simple exponential.
 
  • #3
?
 
  • #4
I see how my post could confuse, it was badly expressed(sorry). I was simply noting that the equations have the form of <xl U(t) l x. for a particle traveling from x1 to x2 and at t1 to t2. What I am saying is that it is not an integral over all the paths but a integral over the probabilitys the paths.
 
  • #5
Have you read Feynman's book Quantum Mechanics and Path Integrals? I would recommend it.
 
  • #6
In quantum field theory, it can be shown that the path integral is equivalent to a Hamiltonian formulation is the Osterwalder-Schrader conditions are obeyed. Then I think (am not sure) we can get the quantum mechanics case by treating it as 0+1 dimensional field theory.

http://www.einstein-online.info/spotlights/path_integrals
 
  • #7
Yeah I have looked in Feynman but it does not answer my question, I have read the derivations in Shankar, Feynman, Galitski etc... many times and forget them because I cannot see how they sync with Landau's Heisenberg QM, and also this quasi-classical approximation idea is interesting, so I'm looking for just enough of a reason to get to those derivations from what I've said in a consistent fashion :D
 
  • #9
I'm just talking about quantum mechanics, starting from Heisenberg, Schrodinger, Quasi-classical and maybe a propagator, no field theory or axioms. Do you guys have any ideas?
 
  • #10
Paths in path integral are continuous. Take the discrete case:
We have a set of N sites. the source is on one of those sites (equiprobability) at time 0. it propagates to the other ones. At time 1 you have for each couple of sites (i,j) an amplitude ##M_{i j}## At time 2 You have ##M_{j k}## and so on.
IF you want to compute the result 0 -> t you have to multiply the matrices
##\Sigma (i,j) (j,k) (k,l) ...##
This is matrix mechanics. Heisenberg discovered how it works. Paths integrals look like a generalization in the continuous limit.
 
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1. What is the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle, also known as the Uncertainty Principle, is a fundamental principle in quantum mechanics that states that it is impossible to know both the precise position and momentum of a particle at the same time. In other words, the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

2. How does the Heisenberg Uncertainty Principle relate to the Path Integral?

The Path Integral is a mathematical tool used to calculate the probability of a particle moving from one position to another. It takes into account all possible paths that the particle could take, and sums them up to determine the overall probability. The Heisenberg Uncertainty Principle is related to the Path Integral because it shows that there is inherent uncertainty in the position and momentum of a particle, and therefore in the paths it could take.

3. What is the significance of the Path Integral in quantum mechanics?

The Path Integral is a key mathematical tool in quantum mechanics, as it allows us to calculate the probability of a particle's behavior without needing to know the exact details of its motion. It also helps to bridge the gap between classical mechanics and quantum mechanics, as it incorporates the uncertainty principle into its calculations.

4. What are the limitations of the Path Integral in quantum mechanics?

The Path Integral is a powerful tool, but it does have limitations. It can only be applied to systems in which the position and momentum of a particle are uncertain, and it is not as accurate for larger and more complex systems. Additionally, it relies on certain simplifying assumptions and may not always accurately predict the behavior of particles in more complex situations.

5. How is the Path Integral used in practical applications?

The Path Integral has many practical applications in fields such as quantum mechanics, statistical mechanics, and quantum field theory. It is used to calculate the probability of particle interactions and decays, and to model complex systems such as molecules and fluids. It has also been used in the development of quantum computers and other advanced technologies.

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