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.