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--- theories:quantum_mechanics:path_integral [2018/05/15 06:47]
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+++ theories:quantum_mechanics:path_integral [2018/05/15 06:47]
jakobadmin [Intuitive]
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 <cite>The Lazy Universe by Coopersmith</cite></blockquote>
+----
+**Origin of the Path Integral**
+The path integral idea emerges naturally when we consider what happens when we have many [[experiments:double_slit_experiment|double slit experiments]] in a row:
+{{ :theories:quantum_mechanics:pathintegral.png?nolink&800 |}}
+<blockquote>Recall the famous double-slit experiment
+in quantum mechanics, in which a beam of electrons is
+fired through two slits. If the electrons are classical particles like tiny balls, then we should expect the screen to
+have two bright strips corresponding to where the electrons hit, i.e. we would not expect interference pattern,
+which is a characteristic of wave. However, when the experiment is conducted, we observe interference pattern
+{ electrons do have wave properties! It is not that the
+electrons are interfering with each other and thus somehow cause the interference pattern, since by firing the
+electrons one at a time, interference pattern still build
+up gradually as more and more electrons go through the
+slits. Quantum mechanically, we often say that the wave
+function will be the sum of two possible states: one that
+passes through slit A and one that passes through slit
+B, and the wave function is in a superposition of states.
+However there is no reason why we should stop at two
+slits, we could have three, and then the wave function
+will be the sum of three possible states. We can also
+have more than one screen. Therefore we could have say first screen with 2 slits, second screen with 3 slits etc.
+and stack them all together. That is, we have to consider all the probabilities of particle passing through the
+i-th slit of the k-th screen. Now imagine that we increase
+the number of screens and the number of slits and continue to do so in the limit towards infinity. In the limit
+with infinitely many slits, the slits are not there anymore! Therefore we reached a seemingly absurd [what
+isn’t in quantum mechanics?] conclusion that even in
+empty space without physical screens, we have to consider the probabilities of the particles taking all possible
+paths from one point to another instead of just the classical path [which is the unique path determined by solving
+differential equation of the Newtonian equation of motion given some initial condition.] As Zee described it,
+this is almost Zen. <cite>{{ :quantum_theory:path-integral.pdf |Where is the Commutation Relation Hiding in the Path Integral Formulation}} by Yen Chin Ong </cite></blockquote>
+<blockquote>
+Perhaps the best way to introduce the path integral formalism is by telling a story, certainly apocryphal as many physics stories are. Long ago, in a quantum mechanics class, the professor droned on and on about the double-slit experiment, giving the standard treatment. A particle emitted from a source S (fig. I.2.1) at time t = 0 passes through one or the other of two holes, $A_1$ and $A_2$ , drilled in a screen and is detected at time t = T by a detector located at O. The amplitude for detection is given by a fundamental postulate of quantum mechanics, the superposition principle, as the sum of the amplitude for the particle to propagate from the source S through the hole $A_1$ and then onward to the point O and the amplitude for the particle to propagate from the source S through the hole $A_2$ and then onward to the point O. Suddenly, a very bright student, let us call him Feynman, asked, “Professor, what if we drill a third hole in the screen?” The professor replied, “Clearly, the amplitude for the particle to be detected at the point O is now given by the sum of three amplitudes, the amplitude for the particle to propagate from the source S through the hole $A_1$ and then onward to the point O, the amplitude for the particle to propagate from the source S through the hole $A_2$ and then onward to the point O, and the amplitude for the particle to propagate from the source S through the hole A 3 and then onward to the point O.” The professor was just about ready to continue when Feynman interjected again, “What if I drill a fourth and a fifth hole in the screen?” Now the professor is visibly losing his  patience: “All right, wise guy, I think it is obvious to the whole class that we just sum over all the holes.” To make what the professor said precise, denote the amplitude for the particle to propagate from the source S through the hole $A_i$ and then onward to the point O as A(S → A_i → O). Then the amplitude for the particle to be detected at the point O is
+$$ A(\text{detected at O})= \sum_i A(S → A_i → O) $$
+But Feynman persisted, “What if we now add another screen (fig. I.2.2) with some holes drilled in it?” The professor was really losing his patience: “Look, can’t you see that you just take the amplitude to go from the source S to the hole $A_i$ in the first screen, then to the hole $B_j$ in the second screen, then to the detector at O , and then sum over all i and j ?” Feynman continued to pester, “What if I put in a third screen, a fourth screen, eh? What if I put in a screen and drill an infinite number of holes in it so that the screen is no longer there?” The professor sighed, “Let’s move on; there is a lot of material to cover in this course.”
+But dear reader, surely you see what that wise guy Feynman was driving at. I especially enjoy his observation that if you put in a screen and drill an infinite number of holes in it, then that screen is not really there. Very Zen! What Feynman showed is that even if there were just empty space between the source and the detector, the amplitude for the particle to propagate from the source to the detector is the sum of the amplitudes for the particle to go through each one of the holes in each one of the (nonexistent) screens. In other words, we have to sum over the amplitude for the particle to propagate from the source to the detector following all possible paths between the source and the detector (fig. I.2.3).
+$$A(\text{particle to go from S to O in time T} ) = $$ $$\sum_{\text{paths}} ( \text{A particle to go from S to O in time T following a particular path }) $$
+<cite>Page 7 in QFT in a Nutshell by A. Zee</cite>
+</blockquote>

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