Things become complicated when it is tried to introduce the concept of paths in the domain of Quantum Mechanics. Though the position in Quantum Mechanics is completely a measurable quantity, the position in consecutive time intervals is not a determinate one.
In the sense, even if it observed a perfect value by making a position measurement on a quantum mechanical system at time $t_1$, there is no way of predicting, what would be the result of a position measurement at time $t_2$. All one can talk about is the average value of the position of the particle [known as the expectation value of position operator].
Nevertheless the wave function in quantum mechanics is absolutely defined in terms of probability. Because of this, even for a single particle there is a non-zero probability of the particle to be found at any point in the entire three dimensional space.
The particle can be found anywhere in the universe in any two consecutive position measurements. This indeterminacy makes it extremely difficult to apply the concept of paths for a particle in the Quantum World.
To put forth the idea, it is to be started with a simple definition and expanded in terms of probability arguments.
For instance, a classical path in position space is defined as the consecutive value of the position of the particle over a time interval. If a particle is found at position $x_1$ at time $t_1$ and found at a later time in position $x_2$ at time $t_2$ then it is described as, the particle travels from the position $x_1$ at time $t_1$ to position $x_2$ at time $t_2$ in the specific path determined by the extremum principle of action. This same classical path by incorporating the concept of Probability can be restated as,
the path of the classical particle is the one where the quantum wave function reduces to Dirac delta function at every point.
Similar to the above, first it is started with the restated definition of well known classical concepts in terms of probability argument and then the concepts are extrapolated to the Quantum domain.
This way of extrapolation of the 'concept of classical paths' to the quantum particles was first done by Richard Feynman in 1948. The idea is basically described in the simplest form as,
the quantum particles can follow any path as well as every path in the three dimensional Euclidean space.
This gives rise to an infinite number of possible paths for a quantum particle even if the particle wants to go the most nearest point. And the mathematical model consists of two types of strategies for the summation procedure of these paths in the form of integrals. To demonstrate, let us consider the case of a free particle going from region one to region two, where it is restricted with an infinite wall with only two slits for the particle to cross between the regions [Figure (1)].
It is known for sure that, to reach the point $(r_2,t_2)$ from $(r_1,t_1)$, the particle can only take either of the two slits. But just from making a single observation at $(r_2,t_2)$, one cannot say anything about the initial point from where the particle has started its path. The particle could have started from anywhere within the left side region of the wall.
From the figure(1), it could be from $(r_1,t_1)$ or $(r_1^*,t_1^*)$ or from some other point. But the essential point is that, a particle found at position $r_2$ at time $t_2$ will never tell anything about the initial position $r_1$ from where it started its motion.
This lack of information leads to the first of the two strategies that needs to be addressed in the general theory.
The first one is,
In the general motion of a Quantum particle in the three dimensional space, if the particle is observed at a specific position [It is possible to take out the wall by considering infinite number of slits instead of two], then the particle is said to have come from every possible initial point in the entire three dimensional space.
And each of those paths contribute to the net probability amplitude.
So, it is introduced an integral such that it is carried over every possible initial position of the particle.
The second strategy is,
Once the initial position and the final position of the particle is fixed, the next difficulty comes from the fact that the particle can now follow any path as well as every path between those two points.
In the figure(1), once the initial position is fixed, the particle can take any one of the infinite possible paths via slit 1 or slit 2 or through any one of the infinite number of slits.
To account for this new fact, it is introduced a second integral within the first integral to take into account the contribution from every possible paths.
Technically, using the definition of wave function, $$ \psi_1(r_1, t_1) $$ is the probability amplitude of finding the particle at the position ${r_1}$ at time $t_1$ and $$ \psi_2({r_2},t_2)$$ is the probability amplitude of finding the particle at ${r_2}$ at time $t_2$. Using the first integral, one would like to obtain $\psi_2({r_2},t_2)$ from $\psi_1({r_1},t_1)$ by defining a correlating function called transition amplitude $$ K (r_2,t_2,r_1, t_1)$$ which is the probability amplitude for finding a particle at $({r_2},t_2)$ when it was initially found at $({r_1},t_1)$.
Hence forth, the fundamental dynamical equation is stated as, \[\psi_2({r_2},t_2) = \int_{space} K({r_2},t_2;{r_1},t_1) \psi({r_1},t_1) d^3{r_1} \,\,\,\tag{1.14}\]
The only unknown term in this equation is $K({r_2},t_2;{r_1},t_1)$. It is also called the Feynman propagator. Once the explicit form of this propagator is known, it is then possible to determine how it controls the dynamical development of the Schr\"{o}dinger wave function.
To obtain the Propagator, one can make use of the second integral where it is summed over all possible paths between the two points A$({r_1},t_1)$ and B$({r_2},t_2)$. So, it is taken a general point C between A and B and considered the motion of the particle along this point. i.e. Path from A to B via C. The probability amplitude for this path A-C-B is denoted as $\phi_{BA}(C)$ and the propagator is obtained by integrating through all possible A-C-B paths, \[K(B,A) = \int_{all\,possible\, paths} dC \,\phi_{BA}(C) \,\,\,\tag{1.15}\]
Determining this integral for $({r_1}, t_1) \rightarrow ({r_2},t_2)$ in general consists of an infinite number of possible paths with their corresponding probability amplitudes. And obviously there is no fundamental physical principle that determines this amplitude $\phi_{BA}(C)$ directly.
This difficulty was first overcame by Dirac, who postulated that each path contributes same amount of probability amplitude to the final result with different phase factors given by, \[\phi_{BA}(C) = e^{\frac{i}{\hbar}S(C)}\,\,\,\tag{1.16}\] where S is the classical action integral.
Thus one finally obtains the formula for the Feynman propagator as, \[ K({r_2},t_2; {r_1},t_1) = \int_{{r}(t_1)={r_1}}^{{r}(t_2)={r_2}} d{r(t)}\, e^{\frac{i}{\hbar}\int_{t_1}^{t_2}dt\,L({r}(t), \dot{{r}}(t),t)}\,\tag{1.17}\]
It is easily seen that, in the classical limit when $\hbar\rightarrow\,0$ or $S/\hbar>>1$ the exponential factor oscillates rapidly for all regions except where "S" remains stationary. Thus, the corresponding amplitudes of rapidly oscillating factor will be washed out by destructive interference and the major contribution will come from stationary "S" value which occurs for the classical action. Thus consistent with the correspondence principle, classical results are obtained as a limiting case of the quantum theory.
In addition, if it is considered $K(x_2,t_2,x_1,t_1)$ (for simplicity it is considered in one dimension) , and $(x_1,t_1)$ kept fixed, one obtains \[K(x,t;x_1,t_1) = K_{(x_1,t_1)}(x,t)\,\,\,\tag{1.18}\] which is a function of $x$ and $t$ alone.
From (1.14) for $t=t_1$ as, $$ \psi(x,t)=\int_{-\infty}^{\infty} dx_1 K(x,t;x_1,t_1) \psi(x_1,t_1)$$ and
$$\psi(x,t)=\int_{-\infty}^{\infty} dx_1 K(x,t;x_1,t) \psi(x_1,t)$$
And from the definition of Dirac delta function, our new function reduces as, \[K(x,t;x_1,t) = K_{(x_1,t_1)}(x,t)\vert_{t=t_1} = \delta(x-x_1)\,\tag{1.19}\]
It is seen from the above results that the propagator function $K_{(x_1,t_1)}(x,t)$,
at the initial point $(x_1, t_1)$ reduces to Dirac delta function where the particle was found certainly without the amplitude being smeared out. And at later times, the propagator is just the probability amplitude for finding the particle at a variable point $(x,t)$.
Comparing the above properties with the postulates of Quantum Mechanics, it is understood that the propagator function is exactly what it is meant by the Schr\"{o}dinger wave function.
Thus, we have found an intrinsic way to determine the general schrodinger wave function from the propagator function.
Another interesting property of the propagator is that, from the definition (1.14), \[\psi(x_3,t_3)=\int_{-\infty}^{\infty} dx_2 K(x_3,t_3;x_2,t_2) \psi(x_2,t_2)\,\tag{1.20}\] and \[\psi(x_2,t_2)=\int_{-\infty}^{\infty} dx_1 K(x_2,t_2;x_1,t_1) \psi(x_1,t_1)\,\tag{1.21}\] and \[\psi(x_3,t_3)=\int_{-\infty}^{\infty} dx_1 K(x_3,t_3;x_1,t_1) \psi(x_1,t_1)\,\tag{1.22}\] substituting (1.21) in (1.20) and equating with (1.22) one obtains an important group property, \begin{align*}
& \int_{-\infty}^{\infty} dx_1 K(x_3,t_3;x_1,t_1) \psi(x_1,t_1) \\ & = \int_{-\infty}^{\infty} dx_2 K(x_3,t_3;x_2,t_2) \int_{-\infty}^{\infty} dx_1 K(x_2,t_2;x_1,t_1) \psi(x_1,t_1)
\end{align*}
which gives, \[K(x_3,t_3;x_1,t_1) = \int_{-\infty}^{\infty} dx_2 K(x_3,t_3;x_2,t_2)K(x_2,t_2;x_1,t_1)\,\tag{1.23}\] In general, for $f(x_f,t_f)$ and $i(x_i,t_i)$ one can write \[K(f;i) = \int_{-\infty}^{\infty} dx_{N-1}.....\int_{-\infty}^{\infty}dx_1 K(f;N-1)\,K(N-1;N-2)\,...K(2;1)K(1;i)\,\tag{1.24}\] It is to be noted that, the intermediate times are not integrated over.
Reference: Classical and Quantum Dynamics - W.Dittrich, M.Reuter
In the sense, even if it observed a perfect value by making a position measurement on a quantum mechanical system at time $t_1$, there is no way of predicting, what would be the result of a position measurement at time $t_2$. All one can talk about is the average value of the position of the particle [known as the expectation value of position operator].
Nevertheless the wave function in quantum mechanics is absolutely defined in terms of probability. Because of this, even for a single particle there is a non-zero probability of the particle to be found at any point in the entire three dimensional space.
The particle can be found anywhere in the universe in any two consecutive position measurements. This indeterminacy makes it extremely difficult to apply the concept of paths for a particle in the Quantum World.
To put forth the idea, it is to be started with a simple definition and expanded in terms of probability arguments.
For instance, a classical path in position space is defined as the consecutive value of the position of the particle over a time interval. If a particle is found at position $x_1$ at time $t_1$ and found at a later time in position $x_2$ at time $t_2$ then it is described as, the particle travels from the position $x_1$ at time $t_1$ to position $x_2$ at time $t_2$ in the specific path determined by the extremum principle of action. This same classical path by incorporating the concept of Probability can be restated as,
the path of the classical particle is the one where the quantum wave function reduces to Dirac delta function at every point.
Similar to the above, first it is started with the restated definition of well known classical concepts in terms of probability argument and then the concepts are extrapolated to the Quantum domain.
This way of extrapolation of the 'concept of classical paths' to the quantum particles was first done by Richard Feynman in 1948. The idea is basically described in the simplest form as,
the quantum particles can follow any path as well as every path in the three dimensional Euclidean space.
This gives rise to an infinite number of possible paths for a quantum particle even if the particle wants to go the most nearest point. And the mathematical model consists of two types of strategies for the summation procedure of these paths in the form of integrals. To demonstrate, let us consider the case of a free particle going from region one to region two, where it is restricted with an infinite wall with only two slits for the particle to cross between the regions [Figure (1)].
It is known for sure that, to reach the point $(r_2,t_2)$ from $(r_1,t_1)$, the particle can only take either of the two slits. But just from making a single observation at $(r_2,t_2)$, one cannot say anything about the initial point from where the particle has started its path. The particle could have started from anywhere within the left side region of the wall.
From the figure(1), it could be from $(r_1,t_1)$ or $(r_1^*,t_1^*)$ or from some other point. But the essential point is that, a particle found at position $r_2$ at time $t_2$ will never tell anything about the initial position $r_1$ from where it started its motion.
This lack of information leads to the first of the two strategies that needs to be addressed in the general theory.
The first one is,
In the general motion of a Quantum particle in the three dimensional space, if the particle is observed at a specific position [It is possible to take out the wall by considering infinite number of slits instead of two], then the particle is said to have come from every possible initial point in the entire three dimensional space.
And each of those paths contribute to the net probability amplitude.
So, it is introduced an integral such that it is carried over every possible initial position of the particle.
The second strategy is,
Once the initial position and the final position of the particle is fixed, the next difficulty comes from the fact that the particle can now follow any path as well as every path between those two points.
In the figure(1), once the initial position is fixed, the particle can take any one of the infinite possible paths via slit 1 or slit 2 or through any one of the infinite number of slits.
To account for this new fact, it is introduced a second integral within the first integral to take into account the contribution from every possible paths.
Technically, using the definition of wave function, $$ \psi_1(r_1, t_1) $$ is the probability amplitude of finding the particle at the position ${r_1}$ at time $t_1$ and $$ \psi_2({r_2},t_2)$$ is the probability amplitude of finding the particle at ${r_2}$ at time $t_2$. Using the first integral, one would like to obtain $\psi_2({r_2},t_2)$ from $\psi_1({r_1},t_1)$ by defining a correlating function called transition amplitude $$ K (r_2,t_2,r_1, t_1)$$ which is the probability amplitude for finding a particle at $({r_2},t_2)$ when it was initially found at $({r_1},t_1)$.
Hence forth, the fundamental dynamical equation is stated as, \[\psi_2({r_2},t_2) = \int_{space} K({r_2},t_2;{r_1},t_1) \psi({r_1},t_1) d^3{r_1} \,\,\,\tag{1.14}\]
The only unknown term in this equation is $K({r_2},t_2;{r_1},t_1)$. It is also called the Feynman propagator. Once the explicit form of this propagator is known, it is then possible to determine how it controls the dynamical development of the Schr\"{o}dinger wave function.
To obtain the Propagator, one can make use of the second integral where it is summed over all possible paths between the two points A$({r_1},t_1)$ and B$({r_2},t_2)$. So, it is taken a general point C between A and B and considered the motion of the particle along this point. i.e. Path from A to B via C. The probability amplitude for this path A-C-B is denoted as $\phi_{BA}(C)$ and the propagator is obtained by integrating through all possible A-C-B paths, \[K(B,A) = \int_{all\,possible\, paths} dC \,\phi_{BA}(C) \,\,\,\tag{1.15}\]
Determining this integral for $({r_1}, t_1) \rightarrow ({r_2},t_2)$ in general consists of an infinite number of possible paths with their corresponding probability amplitudes. And obviously there is no fundamental physical principle that determines this amplitude $\phi_{BA}(C)$ directly.
This difficulty was first overcame by Dirac, who postulated that each path contributes same amount of probability amplitude to the final result with different phase factors given by, \[\phi_{BA}(C) = e^{\frac{i}{\hbar}S(C)}\,\,\,\tag{1.16}\] where S is the classical action integral.
Thus one finally obtains the formula for the Feynman propagator as, \[ K({r_2},t_2; {r_1},t_1) = \int_{{r}(t_1)={r_1}}^{{r}(t_2)={r_2}} d{r(t)}\, e^{\frac{i}{\hbar}\int_{t_1}^{t_2}dt\,L({r}(t), \dot{{r}}(t),t)}\,\tag{1.17}\]
It is easily seen that, in the classical limit when $\hbar\rightarrow\,0$ or $S/\hbar>>1$ the exponential factor oscillates rapidly for all regions except where "S" remains stationary. Thus, the corresponding amplitudes of rapidly oscillating factor will be washed out by destructive interference and the major contribution will come from stationary "S" value which occurs for the classical action. Thus consistent with the correspondence principle, classical results are obtained as a limiting case of the quantum theory.
In addition, if it is considered $K(x_2,t_2,x_1,t_1)$ (for simplicity it is considered in one dimension) , and $(x_1,t_1)$ kept fixed, one obtains \[K(x,t;x_1,t_1) = K_{(x_1,t_1)}(x,t)\,\,\,\tag{1.18}\] which is a function of $x$ and $t$ alone.
From (1.14) for $t=t_1$ as, $$ \psi(x,t)=\int_{-\infty}^{\infty} dx_1 K(x,t;x_1,t_1) \psi(x_1,t_1)$$ and
$$\psi(x,t)=\int_{-\infty}^{\infty} dx_1 K(x,t;x_1,t) \psi(x_1,t)$$
And from the definition of Dirac delta function, our new function reduces as, \[K(x,t;x_1,t) = K_{(x_1,t_1)}(x,t)\vert_{t=t_1} = \delta(x-x_1)\,\tag{1.19}\]
It is seen from the above results that the propagator function $K_{(x_1,t_1)}(x,t)$,
at the initial point $(x_1, t_1)$ reduces to Dirac delta function where the particle was found certainly without the amplitude being smeared out. And at later times, the propagator is just the probability amplitude for finding the particle at a variable point $(x,t)$.
Comparing the above properties with the postulates of Quantum Mechanics, it is understood that the propagator function is exactly what it is meant by the Schr\"{o}dinger wave function.
Thus, we have found an intrinsic way to determine the general schrodinger wave function from the propagator function.
Another interesting property of the propagator is that, from the definition (1.14), \[\psi(x_3,t_3)=\int_{-\infty}^{\infty} dx_2 K(x_3,t_3;x_2,t_2) \psi(x_2,t_2)\,\tag{1.20}\] and \[\psi(x_2,t_2)=\int_{-\infty}^{\infty} dx_1 K(x_2,t_2;x_1,t_1) \psi(x_1,t_1)\,\tag{1.21}\] and \[\psi(x_3,t_3)=\int_{-\infty}^{\infty} dx_1 K(x_3,t_3;x_1,t_1) \psi(x_1,t_1)\,\tag{1.22}\] substituting (1.21) in (1.20) and equating with (1.22) one obtains an important group property, \begin{align*}
& \int_{-\infty}^{\infty} dx_1 K(x_3,t_3;x_1,t_1) \psi(x_1,t_1) \\ & = \int_{-\infty}^{\infty} dx_2 K(x_3,t_3;x_2,t_2) \int_{-\infty}^{\infty} dx_1 K(x_2,t_2;x_1,t_1) \psi(x_1,t_1)
\end{align*}
which gives, \[K(x_3,t_3;x_1,t_1) = \int_{-\infty}^{\infty} dx_2 K(x_3,t_3;x_2,t_2)K(x_2,t_2;x_1,t_1)\,\tag{1.23}\] In general, for $f(x_f,t_f)$ and $i(x_i,t_i)$ one can write \[K(f;i) = \int_{-\infty}^{\infty} dx_{N-1}.....\int_{-\infty}^{\infty}dx_1 K(f;N-1)\,K(N-1;N-2)\,...K(2;1)K(1;i)\,\tag{1.24}\] It is to be noted that, the intermediate times are not integrated over.
Reference: Classical and Quantum Dynamics - W.Dittrich, M.Reuter
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