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.