Now, we're going to put to work the main equation of Quantum Mechanics, the
Schrodinger equation, and use it to solve a very interesting problem that will also
illustrate why quantum mechanical effects are not important in our everyday lives
while being essential when we're deal with atomic link scales.
The question we're going to ask is what happens if when enforced a particle like
solution in our Quantum Mechanical problem.
So basically, what it means is that we postulate, let's postulate that the t
equals 0, at initial moment of time, t equals 0, we have the wave-function such
as this, which is called Gaussian wave-packet.
And this wave-function it, it basically describes the particle which is localized
in space around 0 so it's a typical Gaussian around 0 and this spread out of
this Gaussian is a form of g, which is the parameter of the [unknown] in some
coefficient is not very important at this stage.
Now, the calculation that we are going to present is essentially to the evolution of
this initial condition with time, under the action of the Schrodinger equation,
and it's a big technical question. So, for those of you who are not
interested in technical details, I would like just to present the final answers so
that you can understand the bottom line without going into technicalities.
And so, the final answer is here, so here, we have the wave-function as the function
of time models of it squared. So, we describe initial condition
particles and as we will see, this psi squared is, has the same Gaussian form as
the regional wave-function. But the difference here is that the
parameter d, which describes its spreading, it increases with time.
And it increases with time as 1 plus t squared over some tau squared, where tau
squared is this typical time scale at which the spreading occurs.
So for now, look at this dependence of psi squared on x.
And as time goes by, we will see a more and more uniform density.
So eventually, it will be almost flat. Now, this basically implies that the
particle light solution, this type of wave [unknown] is unstable in quantum
mechanics. So but the important thing is the time
scale itself that we, at which this spreading occurs.
And let us estimate this time scale for two sort of diametrically opposite cases.
So, one case will be the case of an electron and microscopic elementary
particle with a mass of about 10 to minus 27 grams and let's assume that it's
localized on atomic length scales [unknown] 10 to minus eight centimeters.
So, if we plug in the numbers, what we're going to, to see is the typical time scale
which the spreading of the localization radius, if you want of this electron
doubles is in the order of 10 to the minus 16th seconds which is an extremely short
time scale hardly observable in any experiment.
Now, if we take on the contrary in a microscopic object, let's say, a typical
human being, which is also described by Laws of Quantum Mechanics so the
difference will be quite noticeable. So here, while the mass, of course, of the
microscopic object here will be around, let's say, 50 kilograms, and let's say
want to be localized on a distance of total 1 centimeter.
Now, if we now plug in the numbers into this equation, what we're going to see is
that the typical time scale at which sort of quantum delocalization of human being
recursed is a [unknown] 10 to, to, to 30 to plus 30 seconds.
So, to put this number in perspective, so this is equal to the 10 to the 23 years or
10 to the 13th, 10 trillion lifetimes of the universe years.
So basically, it's in some sense, meaningless number.
Sp, it's a number which never, which is never relevant in any realistic
circumstances so we should not be worrying too much about being quantum delocalized
due to quantum mechanical effects. And if we now look at different time
scales for various objects that we're dealing in, in our everyday lives, we're
going to see that essentially, for all these objects, even the smallest ones, we
can safely say that quantum mechanical effects are completely irrelevant.
But at the moment, we're going to, we're going to go to a fundamental microscopic
run scales is going to change very significantly.
Now, I'm going to actually derive the main result that they presented in the previous
slide about the revolution of the Gaussian wave predicate.
And as I mentioned to you, this derivation is a bit on the technical side so those of
you who are not really interested in this technicalities may just skip it towards
the end of this lecture. But those who want to actually learn the
technical part of Quantum Mechanics, which is the main part of Quantum Mechanics,
should pay attention perhaps to derive it on your own.
Now, the main element of the solution is the decomposition of the original the
initial condition of this wave-packet Into the plane-waves.
So, the plane-waves are presented here. So, sort of as an intuition as to why we
want to do something like this, we should recall that the Schrodinger's equation was
deduced or derived from almost by construction to describe waves rather than
particles. So, the initial condition that we can
choose is in some sense, an arbitrary function.
So, there's actually no reason why it would be a convenient solution to, to
anything. So, while the plane-waves are indeed the
solutions, and so by the composing this function or any initial condition for this
matrix, this plane-waves, we actually simplify things a lot, as we will see
later. From the mathematical point of view, this
decomposition is nothing but a Fourier transform of this Gaussian wave-function,
into the into the harmonic function, into this cosines and sines.
And we know very well how to do this decomposition but before we before we go
to it, I would like to remind you of the following identity for Gaussian integral
which we, which actually we're going to use at least three times in this
deliberation, in different topics. So, if we have an exponential of a
quadratic function with some coefficients alpha and beta here, so the integral from
minus infinity to plus infinity is well-known and it's given by this
equation. So, we're going to just use it later on.
Now the reasons UNKNOWN] in this, this stage of the derivation is because the
Fourier transform or in other words this, this Fourier harmonics which in the
context of Quantum Mechanics are called wave-function in the momentum space are
determined exactly by this type of Gaussian integral.
Because what we have to do in order to do in [unknown] on this phi of p, we have to
take avoid the integral of this of this exponential e to the power of minus x
squared over 2d squared with, with this e to the power minus i over h bar px.
And if we look at this integral, so, and compare it with this Gaussian expression,
we will, we will see that alpha here is, is equal to 1 over 2 d squared.
And beta is equal to minus i h bar p. So if we use now this formula, so beta
squared is equal to minus p squared over h squared and divided by 4 alpha will give
us the Fourier expression. So, if I completely ignore the overall
coefficient, it's not really important for the following.
So, now let me just erase this and get rid of this Gaussian integral.
So now we're going to interpret, actually, this Fourier transform in an interesting
way. So the reason we have constructed this
initial condition in this form, was because we wanted to localize our particle
sort of in the certain length scales of order d.
So, the uncertainty of our initial condition in, in, in the real space is of
order d, but by looking at this sort of wave-function in the momentum space, we
see that the uncertainty in momentum delta p is a [unknown] h bar over d.
So, it's inverse proportional to the, a localization distance in real space.
And so, this sort of dual relation, so the more it's localized in real space, the
less it's localized in the momentum space and vice-versa is a particular
manifestation of what is known as the general Heisenberg uncertainty of
principle, which is which is written here. So, that in, in general, whatever wave
function you have, whatever quantum state you possibly can construct, there is
always a constraint that the uncertainty in position, then the uncertainty in
momentum is larger or equal than h bar, h bar being the plane constant.
So now, we're in the position to solve the main technical problem that we formulated
in the beginning called this, this segment that is to solve the Schrodinger equation
the free Schrodinger equation here. So, the [unknown] here is just p squared
over 2m with initial condition written as so.
So, instead of writing it as a Gaussian real space, we write it as a linear
combination of plane -waves that was basically the Fourier transform in the
previous slide. Now, in order to write the time-dependent
wave-function we need to take into account two simple circumstances.
The first circumstance is the fact that the Schrodinger equation, the fact that
the Schrodinger equation is a linear equation.
Which implies that if we have several solutions to this equation, psi 1, psi 2,
psi 3, etc. Their sum is also a solution, with the
initial condition being simply the sum of the corresponding initial conditions.
Now, the second circumstance is we have to recall that a plane wave written as so is,
is an, is an eta solution to the Schrodinger equation.
In some sense, we have constructed Schrodinger equation such that it would
give us this guy as a solution. And here, the epsilon of p is the energy
of the electron, the energy of free electron and simply equal to p squared
over 2m. Now if we take a look at the initial
condition again written in this form, we see that it's an integral over momentum,
but integral is a sum, in some sense which just happens to be in infinite psi.
So, we can use the fact that a Schrodinger equation is a linear equation and write
psi of x and t, the full solution as so. So basically, all we have to do is to
replace this plane-wave at, at t equals 0 with the plane-wave at a finite time with
epsilon of p being p squared over 2m. And so, to write now the wave-function in
a more convenient way so what remains now is to calculate the Gaussian integral over
momentum. And it can be done again using the same
identity that we used in the previous slide but now, the parameter alpha here is
going to be d squared over 2h squared plus i, it divided by 2m h bar and the
parameter beta now is ix over h bar. So again, if we use the same identity,
what we get is the, the following expression for the wave-function.
We see that this expression is an intrinsically complex function but if
we're interested just in determining the density of our particles or where the
particle is located, so the only thing which matters, as I mentioned already and
we'll discuss it in more details a little later, is the absolute value of the
wave-function squared. So this absolute value squared can be
calculated. Again, we don't, we don't worry too much
about the overall coefficients when we look at what is what appears in the
exponential. So, we can write this we can write this
density as an exponential, basically from here, minus x squared over d squared, but
there is also another term, which is 1 over 1 plus t squared or tau squared, and
tau here is exactly the time scale we discussed in the beginning, which is mass
times the delocalization length, in some sense, d squared divided by h bar.
So this completes the derivation of the main result that we discussed earlier and
also completes the first lecture. Thank you very much and I will see you in
class later in the week.