Now we're going to rely on wave particle 
duality to deduce the form of the main 
equation of quantum physics and the 
Schrodinger equation. 
But before going to this equation I first 
would like to ask the following question. 
Which picture is more fundamental, the 
particle-based picture or the wave based 
picture? 
So here I have a simple example of what 
we usually mean by a particle. 
In this example, here is the baseball and 
so this is an object which has a 
well-defined velocity and position at any 
given time. 
We can easy to identify position just by 
looking at it and measure the velocity. 
On contrast, when we're talking about 
waves, it doesn't really make sense to us 
the question where it located. 
So here for instance, we have an example 
of wave which is generated using a rope 
and well, we cannot define the position 
of the wave, okay? 
Probably what we can do, we can define 
its phase velocity and the wavelength and 
the corresponding wave vectors, so which 
is so wavelength lambda is equal to 2 pi 
divided by k. 
k is the wavelength, right? 
Omega is the frequency. 
So the velocity of the wave is the 
coefficient between the frequency and 
the, coefficient of proportionality 
between the frequency and the wavelength. 
Now there are two common expressions 
which are two types of common expressions 
that I use to describe a simple 
sinusoidal wave that we will see pretty 
often. 
So one is a just using the sine or cosine 
function such as here, so in this example 
u is the vertical displacement, which is 
a function of x position and time, and x 
in this case, in his horizontal 
direction. 
So another way to describe a wave is 
using this exponential function of an 
imaginary constant i. 
So i carries the square root of minus 1. 
and, they're equivalent to one another. 
And most commonly we're actually going to 
use the latter expression. 
Now going back to the main question what, 
what is more fundamental, particles or 
waves? 
I would argue, is that waves are in fact 
much more fundamental in that we can't 
really represent the wave in terms of a 
particle. 
But we can surely decompose a localized 
particle into waves. 
And, this decomposition goes by the name 
of Fourier transform. 
So, here I have an example of a generic 
Fourier transform, which is a wave, which 
is represented in an arbitrary reasonable 
function F of X in terms of this 
exponentials E to the power of ikx. 
We are sure transform to this wave 
describing wave at any given time. 
And, the corresponding efficiency here, 
which multiply these exponential are both 
the upper mornings of the function f of 
x. 
In particular we can consider a function 
f of x which shows a sharp peak, see, 
with a narrow, spread out, let's say 
delta x. 
And we can consider this function as a 
mathematical description of an entity 
which is a particle like entity. 
And by decomposing it into a Fourier 
transform, we represent it as a linear 
combination of waves. 
So finally I would like to use the wave 
particle duality for electrons now in 
order to guess, if you wondered, to 
derive in quotes the wave equation that 
governs their quantum wave properties. 
So here we're relying on on experiments 
of the type performed by, let's say 
Dennison and Germer, lacking the fraction 
where we clearly saw that the data can be 
on the studio but we assume that the 
electrons behaves behave as waves. 
And so let's assume that this is so, 
let's assume that the free electron is 
described by some wave function, psi, so 
the precise physical meaning of this 
function will be discussed later about 
the course, but this stage let us just 
assume that this is so. 
So just like in the case of photons we 
have an electric field which had the form 
of a plain wave. 
Here we have some electron wave function 
which has the form of a plain wave. 
With some momentum P and energy epsilon. 
But, unlike photons where energy scales 
with momentum linearly, so, we know that 
for that non-relativistic electrons, the 
kinetic energy is basically m v squared 
over 2 or a p squared over 2m. 
So, what we, what we have to demand is 
whatever equation governs the quantum 
properties of electrons. 
It must give us this functional form to 
describe a free electron. 
And this spectrum as a constraint on this 
solution. 
And so, basically we can construct such 
an equation by hand, that gives that has 
this properties. 
And this is what we can do, so we can 
write this equation. 
Now here if we plug in this guy into this 
equation, so the first derivative is 
going to give us minus h squared over 2 
m, m here is the mass of course. 
The second derivative is going to give us 
minus 1 over h bar squared p squared. 
And the second term here is going to give 
us, minus i h bar d over d t, is minus i 
h bar epsilon. 
And we will have the same playing with 
multiplying both terms, so we must demand 
that this that this guys are equal to 0. 
Okay, so a lot of things cancel out here, 
so we have P squared over 2M here, and we 
have a minus minus, so basically minus X 
on here. 
And indeed, were, we reproduce this 
desired spec. 
Now, this constructional course is 
nothing but a guess. 
But we can, it's a very convenient guess 
in that we can generalize this equation 
just by writing it in a slightly 
different form. 
So let me do so. 
so what we're going to do, we are 
going to put the, time derivative in the 
left hand side. 
So basically this time derivative is 
going to be on the left hand side, i h 
bar d over d t psi. 
And eh, this guy, we're going to 
interpret it as a energy function. 
So after all, p squared over 2m is just 
the energy of a free electron but what we 
are going to do, we're going to put here 
some energy function, otherwise known as 
Hamiltonian. 
So this guy is very important object, 
it's a Hamiltonian. 
And we're going to postulate that this 
Hamiltonian is essentially kinetic 
energy, P squared over 2m plus some 
potential energy, P of r. 
So our P here is an operator minus i h 
bar radiant acting on whatever it is that 
we have here so this wave function. 
So and this the canonical Schrodinger 
equation that basically contains most if 
not all non-relativistic physics. 
So of course the way we derived it, in 
quotes, is not really a proper 
derivation. 
It was more like a guess. 
And in some sense this is what 
Schrodinger did. 
well, almost hundred years ago now. 
And, the reason we actually believe that 
this equation is a true equation that 
describes quantum physics, is because 
when we use it, we actually solve 
problems where, apart from just free 
particles, where who have In 
non-trivial[INAUDIBLE] potential lensky. 
Let's say in the case of an atom, or in, 
in the case of sketrinkle particles. 
So this equation produces theoretical 
results which are perfectly consistent 
with the experimental observations. 
So there is no question now, now these 
that this equation indeed, it describes 
quantum reality. 
But the way that people in the early days 
that got to this equation, was very very 
non-trivial.