In the first lecture, we guessed in, in some sense the form of the Schrodinger
equation relying on a few very basic, experimental facts, such as wave particle
duality and the dispersion relation of a free electron.
But we didn't talk too much about the meaning of the main object that appears
in this equation the wavefunction, which was introduced in an ad hoc fashion.
So today, I'm going to talk about the actual physical interpretation of quantum
theory formulated this week. And we are also going to introduce the, a
few very important standard notations and conventions that are going to be used
throughout the course and also that are routinely used in the standard literature
on the subject. So, here is the Schrodinger equation,
once again we're going to see it very often in this course.
And I'm also showing here the, a part of the first page of the original paper by
Schrodinger, published in December of 1926.
And it's actually very interesting how Schrodinger came up with this work and
this equation. So, apparently the story started back in
1925 or so, when he was working under Debye, and Debye had just read a paper
about de Broglie, where de Broglie was introducing his wave particle duality
ideas. So, he got interested in these ideas and
suggested Schrodinger to give a seminar on de Broglie's work.
So apparently, Schrodinger actually dismissed this at first, saying that he
didn't even want to think about such a silly theory.
But he had to give in because well, Debye was effectively his supervisor.
And so, he was looking into de Broglie's work, trying to present it in a more
mathematically sophisticated form. And in doing so, he came up with the
Schrodinger equation essentially, as we now know it.
And which brought him the worldwide recognition and a Nobel Prize in Physics
in 1933. So, another important thing that
Schrodinger did was that he used his equation to solve a very important
problem of the charged particle and the cool-down potential, which essentially
describes a quantum hydrogen atom. And he found the energy level structure
which is which was consistent with the with de Broglie's atom.
And so, actually, Charles Clark is going to talk about the the solution later in
this course but I'm just going to mention here that this was indeed very important
and it was a clear smoking gun that Schrodinger was on the right track.
But according to Debye at least quite interestingly that Schrodinger didn't
really quite understand the true meaning of his own work.
He actually dismissed about the importance of it at first.
So, I don't know if it was just Debye being jealous or it, it was true.
But certainly if you actually read the original paper by Schrodinger, you don't
see too much insight into the true physical interpretation of his own
equation. And the correct interpretation of
Schrodinger's equation was developed very shortly after Schrodinger's work by Max
Born. perhaps, actually, I should say and note
the correct interpretation but the commonly accepted interpretation because
scientists actually have been arguing about the interpretation up to these
days. And some are still not convinced that the
Born interpretation and the Born, Born rule that I'm going to present is the
only a correct view of Quantum Physics. But certainly, this Born rule is the
cornerstone of a, a standard quantum theory, and it is indeed consistent with
all experimental data as we know it at this stage.
Now before formulating this rule I would like to make a few general comments about
Quantum Physics and make, make them in contrast to classical Physics.
So, in classical Physics, if we have a classical system, let's say, this is a
closed classical system with some closed classical particles moving around.
And if we know everything about the system, let's say,we know all the
coordinates of all the particles of a certain moment of time, and all the
velocities at a certain moment of time. Then classical theory predicts with
absolute certainty, every, the result of any conceivable experiment that is going
to happen in the future, which is determined by, let's say trajectories of
particles at any subsequent moment of time.
And these deter, determine all possible outcomes of all possible experiments.
Now, this is the classical system. So, the truth about the quantum system,
so if you look at the, the actual association with quantum systems, so if
you have a quantum system like that, say, with some wavefunction of psi.
So, even if we know everything possible about this quantum system, that is, if we
know psi, and if we know all the forces, everything about this system, we still
cannot possibly predict with certainty, the outcomes of well-posed experiments.
Let's say, if we have some detectors here, and they measure well, let's say
detect electrons or some other particles. So, even if we know everything about the
quantum electrons, we still cannot predict for sure which detector is
going to pick up electrons at certain moment of time.
And so, this uncertainty and is intrinsic to Quantum Physics and there is no way
around it. And so, Born, Born realized it and he
also found a way to quantify his uncertainty using the wavefunction.
And here's what we know now is the Born rule, is written here, so basically he
proposed, and afterwards it was confirmed.
again, by comparing with experimental data, that the absolute value of the
wavefunction, as related as certain position in space, and a certain moment
of time gives a probability density of finding a quantum particle described by
this wavefunction in this this position and at this moment of time t.
Well, the probability itself is going to of psi, the absolute value of psi squared
times the volume the elementary volume in the vicinity of a certain point.
This work of Max Born turned out to be extremely influential and groundbreaking
and he eventually received a Nobel Prize for these ideas in part in 1954.
As a matter of fact, he should have received this award much earlier.
He was nominated by Albert Einstein actually back in the 30s.
but he didn't get it due to some political reasons.
In but in any case, this probabilistic interpretation was and actually remains
very remarkable feature of Quantum Physics, which is also a source of
confusion often times. And I'm actually supposed to tell you,
now about this probabilistic property being a mystery of Quantum Physics and
something which we will never be able to understand.
But I would actually argue otherwise that there is, in fact, nothing mysterious
about it. If, by mysterious, we mean something that
is not consistent with our everyday intuition, everyday experience, so
actually uncertainty is a part of our everyday life whether we are dealing with
a sporting event or any other event in our life, we can predict with certainty
if we know everything how it's going to turn out.
And so I don't see a reason to demand from size deterministic answers.
And nature just doesn't work in this way, and so we should just accept this fact
and get used to it.