In the previous video, we went over a
rather technical mathematical derivation
of the continuity equation, and this
derivation involved in a rather essential
way manipulations with various operators.
The appearance of operators is actually
not specific to the particular problem of
continuity equation but they appear
actually throughout quantum mechanics and
play a very important role generally in
the mathematical formulas of quantum
physics.
In this short video, I would like to, make
a few general comments about operators and
relate them to quantum observable
quantity.
So, in order for me to trace the origin of
the operator objects in quantum theory,
let us go back a little bit and remind
ourselves the way the Schrodinger equation
was derived.
So I use derive here in quotes because
there is no really rigorous way to derive
the Schrodinger equation as a fundamental
equation of quantum theory.
But we sort of followed the logic that the
early practioneers, such as Schrodinger
used.
And the first piece of information which
was essential was the fact, the
experimental fact that quantum electrons
and other quantum particles may exhibit
wave-like properties.
And in, in order to explain this we sort
of introduced, in that profession, a
plane-wave function which would explain
the date in session.
And in this wave function we also
introduced the, the momentum and the
energy of the electron which, according to
another piece of information that is known
from the theory, must be related to one
another using this, this version relation
So basically kinetic energy of a free
particle mv squared over 2 or p squared
over 2m.
Now the equation itself was constructed in
order to ensure that the plane wave
finction would come out as its solution
and this expression relation would be
respective and so this is what equation
actually look like.
And it evolved the second derivative with
respect to the coordinate and the first
derivative with respect to tau.
Now the next step and this is really the
first time we sort of thought of
introducing an operator was to write the
equation in different way.
So we had the sort of universal left hand
side which is just i h bar.
The derivate with respect to time and the
right hand side was an, sort of energy
operator, the [unknown] where we associate
it with the momentum that would have
appeared in the classical theory.
This guy, which is minus i h bar gradient
Which indeed is an operator that converts,
well that acts on the wave function.
And the last step in deriving the gain
includes the Schrodinger equation was to
generalize this energy operator, well
called Hamiltonian from being the free
energy operator which is just p squared
over 2m to kinetic energy plus potential
energy, and this is the fundamental
Schrodinger equation, which remarkably
describes all of the relativistic quantum
physics, including and describes atomic
spectra and well even many particle
systems, theory of metals, theory of
superconductivity.
Everything is contained in a compact way
in this remarkable equation.
And it's also sort of remarkable that you
can get to it from very few simple
experimental facts, by adding small pieces
together and getting to this fundamental
result.
Now, but in doing so, we were forced,
indeed.
To, introduce, well at least three
operators.
So the first one we have seen already many
times, it is the momentum operator minus
IH bar, radiant, and by the way I should
mention that the minus sign Is really sort
of a convention which is sort of must be
consistent with the convention we choose
to describe a plane wave in a sense.
It could have been plus, but this is the
convention that we use.
So apart from that we also introduce the
potential energy which enters this H hat.
And in this set-up, the potential energy
naturally appears as just multiplication
operators.
So, we can think about coordinate
operator, multiplication operator that,
well there should be actually a vector
here multiplication operator simply
multiplies the wave function.
And finally, the kinetic energy itself
sort of the first step, was appeared here
and it was the second[INAUDIBLE] with
respect to spacial coordinates.
Now it turns out that the appearance of
operators was not unique just to this sort
of derivation of the Schrodinger equation,
but sort of the popup throughout quantum
theory.
Whenever we deal with any analog's of
classical properties.
For example, when dealing with the angelo
momentum that classical theory is R cross
P, so all we have to do to make it into an
operator, is to put a hat actually on, on
top of the corresponding quantities.
So p here would be replaced with minus i,
h bar a gradient, according to this, and r
would be just multiplication operator,
etc.
So we can always find a proper
generalization of classical properties.
So that's great we established that
operators do appear in the mathematical
formalism of quantum theory.
And from the mathematical point of view it
may be quite interesting.
But what physical significance do these
operators really have.
Unlike their classical counterparts, which
can be measured directly in a classical
experiment, let's say we can measure
momentum, every momentum position, the
operators cannot be measured per se.
They are operators.
So how do we associate an experimental
observable with them?
So in order to answer this question, we're
going to use a sort of method that often
times is used in science by theorists
answering the question in a simple,
special case and then generalizing the
results to a more general category of
object.
So and the special case that we're going
to consider is the measurement of a
coordinate of a quantum particle.
So, the Born rule tells us that the, the
absolute value of the wave function
squared uses the probability distribution
function.
Of different positions of a quantum
particle.
Now in general, if we have a probability
distribution function let's say F capital
F of x of some random variable x.
So if we want to find, let's say the mean
value the expectation value of this x What
we're going to do, is we're going to
average X, integrate X over all possible
values weighted with this probability of
distribution.
So by, in full analogy with this, we can
define the mean value of a, a
three-dimensional coordinate by
integrating the wave function, absolute
value of wave function squared.
With the coordinate over the volume.
Over the basically all three-dimensional
space, there's a square here.
So and this is an actual definition of the
mean position.
So, we can rewrite this definition as so,
because psi, absoluste value of psi
squared of course.
You simply si star times si of R.
So now comes this generalization of this
definition of the expectation value of a
coordinate to a general case of an
arbitrary quantum mechanical operator.
So lets say we have a classical property X
lets say momentum, angular momentum
coordinate energy, whatever.
And let's assume we know the quantum
mechanical operator associated with this,
some x hat.
So the expectation value of this quantum
observable X is going to be calculated in
full analogy with this equation by simply
replacing the coordinate operator in this
equation with the operator X, whatever it
is.
So for example if x happens to be so let's
say, if x happens to be a momentum, then
we're going to simply write it the
expectation value of momentum is going to
be an integral of v star minus i h bar
gradient of psi, et cetera.
So it turns out that this conjecture
actually works.
In that it is consistent with the
experimental data and the experimental
data implies that if we, let's say we have
and example of quantum systems, each
described by wave function psi of r, and
if we measure, this quantum, some property
x of this quantum systems we're going to
get different results.
But the average value of this measurement
is going to be consistent with this
definition.
With this definition of, the expectation
value.
And, so, it becomes a very important basic
principle of quantum mechanics, that I
read right here.
First of all, that physical observables
are associated with what I didn't discuss,
self-adjoint, operators, acting on the
wave function, with the expectation value
defined as above.