Richard Feynman was one of the most famous and respected theoretical physicists of the 20th century. He was responsible for incredible advances in the field of Quantum Electrodynamics, for which he received in 1965 the Nobel Prize in Physics. He also worked in the development of the atomic bomb during WWII and became known to a wide public in the 1980s as a member of the Rogers Commission, the panel that investigated the Space Shuttle Challenger disaster.  The Euler-Lagrange equation imposes conditions on the Lagrangian $L$ so that the action $S$ is minimized. To see the derivation of the Euler-Lagrange equations [click here](http://martin-ueding.de/en/physics/euler-lagrange-equation/). Since the generalized coordinates are independent $$ \frac{\partial M}{\partial \psi^{*}}= (U-E)\psi $$ $$ \frac{\partial M}{\partial(\partial \psi^{*}/ \partial x) } = \frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial^2 x} $$ Feynman's Nobel Lecture can be read [here](http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html). This derivation shows not only Feynman's mathematical ability and intuition for what's relevant in a problem, but also his insatiable desire to see things in a different way. This new way of looking at the Shrodinger equation was fundamental for Feynman's later work on the path integral formulation. $$ -\frac{i\hbar}{\psi}\frac{\partial \psi}{\partial x}= -\frac{i\hbar}{ae^{i\frac{S}{\hbar}}}\frac{i}{\hbar}\frac{\partial S}{\partial x}ae^{i\frac{S}{\hbar}}=\frac{\partial S}{\partial x} $$ $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$ and so the first order expansion is: $$ e^{-U\frac{1}{2}(x+y)\frac{\epsilon}{\hbar}}= 1-U\frac{1}{2}(x+y)\frac{\epsilon}{\hbar} $$ $$ \frac{\partial L}{\partial x}= \frac{\partial (K(v)-U(x))}{\partial x}=-\frac{\partial U}{\partial x} \\ \frac{\partial L}{\partial (dx/dt)}=\frac{\partial L}{\partial v}=\frac{\partial (K(v)-U(x))}{\partial v}=mv $$ Now if we use the same approach for quantum mechanics as we did for wave optics, we start with a wave function of the form: $$ \psi = ae^{i\phi} $$ As we saw in the previous section, the path of a particle in classical mechanics can be determined by using the variational principle - minimize the action $S$. In Geometric optics the path of the light ray is the one that minimizes time (Fermat's Principle) , which is mathematically equivalent to minimizing the phase $\phi$. On the basis of this analogy we can say that in the classical limit the phase $\phi$ of the wave function must be proportional to the action $S$, $\phi = constant \ \times S$. The constant is called Planck's constant and is denoted by $\hbar$. As a result, the wave function of an almost classical physical system is: $$ \psi = ae^{i\frac{S}{\hbar}} $$ Planck's constant plays a fundamental role in the quantum world. The transition from Quantum Mechanics to Classical Mechanics occurs when the phase $\phi$ is very large, which can be described as $\hbar \rightarrow 0$ (remember that the transition from Wave to Geometric optics occurs when $\lambda \rightarrow 0$ which makes $\phi$ very large). Los Alamos was filled with remarkable physicists that deeply influenced Feynman. One of them was the italian physicist Enrico Fermi, who used scraps of paper to estimate the power of the atomic bomb dropped at Trinity (you can find more information about how Fermi did it [here](https://www.quora.com/How-did-Fermi-estimate-the-power-of-the-Trinity-bomb)). Herbert Jehle was a german physicist that became known for his work on the two-component field equations to allow for neutrinos with mass. Feynman, in his Nobel acceptance speech, credits Herb for helping in the initial ideas toward the path-integral approach to quantum theory.  Quantum Field Theory (QFT) is the natural extension of Quantum Mechanics (QM). While QM is the theory of one (or very few particles) like the Hydrogen atom, QFT is the theory used for the analysis of systems with many particles and their interactions with fields. One of the first field theories to be developed was Quantum Electrodynamics (QED), which described the interaction of particles with the electromagnetic force. Huygens' principle can be seen as a consequence of the homogeneity of space—the space is uniform in all locations. Any disturbance created in a sufficiently small region of homogenous space (or in a homogenous medium) propagates from that region in all geodesic directions. The waves created by this disturbance, in turn, create disturbances in other regions, and so on. The superposition of all the waves results in the observed pattern of wave propagation. Homogeneity of space is fundamental to quantum field theory (QFT) where the wave function of any object propagates along all available unobstructed paths. When integrated along all possible paths, with a phase factor proportional to the action, the interference of the wave-functions correctly predicts observable phenomena. Every point on the wave front acts as the source of secondary wavelets that spread out in the light cone with the same speed as the wave. The new wave front is found by constructing the surface tangent to the secondary wavelets.  $\hbar$ has dimensions of Joules $\times$ Seconds, while the action $S$ is an integral of the Lagrangian (energy units) with respect to time. Thus, their dimensions are the same. The idea here is that Quantum Mechanics (QM) contains Classical Mechanics (CM) in the form of a certain limiting case. The question is how to do the passage to the limit. In QM the electron is described as a wave function that determines the probability of the electron being in a certain position, while in CM the electron is a material particle moving in a path determined by the equations of motion. Schrodinger observed that there was an analogy between the transition from QM to CM for the electron and from Wave Optics to Geometrical Optics for light. In Wave Optics light is treated as an electromagnetic wave described by Electric and Magnetic fields that obey Maxwell's Equations. In Geometric Optics however light is treated as small particles that travel along well defined rays. Schrodinger thought that if he applied the same limit that gets us from Wave to Geometrical Optics to Quantum Mechanics he would get Classical Mechanics. One simple way of analyzing the transition from Wave to Geometrical optics is by using the diffraction through a slit experiment. If a wave is diffracted through a slit of width $d$, a diffraction pattern with an angular width $\theta \approx \frac{\lambda}{d}$ (in radians) is formed. When $\lambda$ is small compared to $d$, $\theta$ gets small. In the limit where $\frac{\lambda}{d}\rightarrow 0$, $\theta \rightarrow 0$, and a ray comes through the slit.  Now in wave optics a wave is described by a function of the form: $$ Z = ae^{i\phi} $$ Where $\phi = kx= 2\pi\frac{x}{\lambda}$ (assuming propagation along $x$). For Geometrical Optics: if $\lambda \rightarrow 0$, it means that $\phi$ will be large over a sufficiently small distance $x$.