In this video, I'm going to elaborate a little bit on the physical meaning and
interpretation of operators in quantum mechanics.
And, in particular, we're going to discuss a very important class of problems that
appear throughout quantum mechanics. That is so-called eigenvalue problems.
To see how they, what they are and how they arise, let me first write down the
Time-Dependent Schrodinger equation. We can only go Time-Dependent Schrodinger
equation, but with specifically with the Time-Independent Hamiltonian.
So physically, this means that the potential in which our quantum particle is
moving, is not time dependent. And actually most problems we are going to
see are, fall into this category. Which is a very reasonable assumption, so
that neither with the mass nor the potential are time-dependent.
So, in this case, it turns out there is absolutely no need to deal with this
complicated differential equation, partial differential equation that involves the
time derivative. So, the time dependence can be sort of
factored out of the equation in a very straight forward way, and this is
accomplished by performing separation of variables.
So basically, what we can say is let's look for our solution in the full wave
form. So, the full, the full wave function is
going to be a product of some time-independent piece, psi of r and this
exponential which is sort of left over from the plane wave.
So, this psi of r does not have to be a plane wave if the potential is exists but
it, it is not a plane wave. If there is a potential, but the time
dependence sort of remains the same. And so, if we plug in this uh,[UNKNOWN]
into the Schrodinger equation, so we're going to have in the left-hand side in the
derivative with respect to time psi of r, e to the power minus i energy t over h
bar. So, I should mention that e of course
corresponds to the energy. And so, since you know, this lower case
psi of r is not depend on time, we don't have to differentiate this part.
We're going to differentiate just the exponential.
And just essentially by construction, we're going to get, in the left-hand side
at e psi of r and its exponential. On the other hand, the right-hand side
since there's nothing time-dependent there simply reads the Hamiltonian acting on the
position dependent part. And the, the exponential is also here but
it's sort of irrelevant in the sense that the Hamiltonian doesn't see it.
So, the Hamiltonian only involves the derivative with respect to spatial
coordinates or the potential, which is just multiplication operations.
So, we see that both sides of this equation contain this exponential, and so
we can just cancel this exponential and we are left with let me just switch the order
in which this guys appears. So we are left with a full wave equation,
H[UNKNOWN] psi or r is equal to e times psi of r.
And, this is what is known as Time-Independent Schrodinger equation, a
stationary Schrodinger equation. So, this category problems, where an
operator acting on a vector in some space, reproduces the same vector multiplied by
some number. So this is category of problems in math
are called eigenvalue problems. And in the context of this mathematical
theory psi here represents what's known as eigen vector, let me write it.
And e is a, an eigenvalue of the corresponding operation.
This operation is Hamiltonian. Now, let me tell you from the outset, we
shall discuss it in more details later that the physical significance of these
eigenvalues is in that they determine the possible outcomes of experiment that
actually measure in this case energy. So we know that, for instance in classical
physics, if I were to have some potential well.
So, let's say this is my V of x, and there is some potential.
Well, just one example of problems we're actually going to study.
So, if I have a classical particle, I would be able to put a classical particle
with any energy and this round, we'll just oscillate between the two turning points
if there is no friction. And I can do so for any value of energy.
It turns out that in quantum mechanics, it's not necessarily so.
So, well, it's not so. And the, the range of available energy is
limited in this case. So, what I would have is, is discrete,
so-called discrete spectrum by which I mean a possible solution to this equation.
So this levels, let's say E1 or E2 are going to come out of this eigenvalue
problem measure. So we see that an eigenvalue problem for
the Hamiltonian arises naturally from the original Schrodinger equation.
And as I should becomes a very important independent, in some sense, equation of
quantum theory. And actually, oftentimes, when we say
Schrodinger equation, we actually refers to this H of psi equals[UNKNOWN] psi and
not to the original along with the time derivative.
And furthermore so eigenvalue problems also appear for other operators in quantum
theory. So, it's very important.
And we should know at least a few basic things, basic properties, of say
uh,eigenvalue problems and the operators involved in this eigenvalue problems.
So we don't have time now to go into comprehensive theory of this.
Mathematical theories could be, in principle, a subject of a year long
course. So therefore, I will just emphasize a few
things that are really crucial for the future, and also introduced some notations
and terms that are important. So, one such important thing is the notion
of a Hermitian operator, that is defined by the[UNKNOWN] relation.
So, let's imagine we have an operator A acting in a space of functions psi of r.
So I'm not going to define precisely mathematically what this space is, but
I'm, you know, we should just think about wave functions that describe the
properties of our quantum particles. So one can define Hermitian-adjoint
operator, A dagger to the operator A by the following relation.
So, the left-hand side of this equation is essentially identical to what we saw in
the previous video when we defined an average or an expectation value of an
operator A. It's an integral of an entire space of psi
star. Let's see how a wave function times the
action of our operator A on psi. And the operator A dagger sort of, is
moves from the right to the left. And now acts on the function which is
ah,[UNKNOWN]. And so, if the two expressions are equal
for any psi of r, then we have found the Hermitian-adjoint operator.
And an important class of operators that appear in quantum mechanics are called
Hermitian operators. And they are characterized by the fact
that their Hermitian-adjoint is equal to the operator itself.
So the, they are called Hermitian. The importance of the Hermitian operators
manifests itself, if we look at their properties, the properties of their
eigenvalue problem. So in general, if we have an eigenvalue
problem for a Hermitian operator, there is no guarantee that its eigenvalues are
going to be real. So in the context of quantum physics, this
would lead to a bit of a problem because well, let's see if we have a wave function
which is described by, let's say, psi sub a.
And we want to take, to measure the expectation value of this, the quantity
corresponding to this operator A. So, we will calculate this average value
and we're going to get, if A is complex, we're going to get an expectation value
which is complex. And it's sort of, in most case, it doesn't
make sense. So, for example, if we measure energy of a
particle, we certainly don't want to get i the measuring constant i.
We want to get a reasonable number that has sort of it's counter part in classical
physics, which does not normally account these number.
So what I'm saying here is that if we want to describe physically reasonable
quantities, so we want these operators to give rise to real eignevalues.
So, this would make sense. And the Hermitian operators have this
property, exactly, this property so the eigenvalues of the Hermitian operators are
all real. So, another crucial important property of
the eigenvalue problem for a Hermitian operator is that its eigenvectors, these
solutions psi sub a of r to this equation form a basis in an appropriate space.
So that is the space where our functions leave.
So for instance, in the context of quantum physics, these are going to be wave
functions. And of course, I should mention that the
statements I'm making are not mathematically rigorous statements.
So, if you want to see a precise definition or a theorem, so there is, of
course very rich mathematical literature on the subject.
And I would encourage you to look into this, but at this stage, I just want to
give you an idea about the properties. And so, the property that I'm presenting
here is that essentially, if you have any relevant function, psi of r, which has
nothing to do with our problem and this function sort of lives in this space where
our operator acts. So then, a set of all eigenvectors of this
operator, which in principle, can be either a discrete or continuous.
In which case, I would have written an integral, even both.
So, but in any case, any such wave function can be represented as a linear
combination of the eigenfunctions of the operator A.
And this has a very consequences as a set in quantum physics.
Now going closer to quantum physics and further away from sort of mathematical
statements, let me just summarize things that really are important in the context
of quantum mechanics. So, the first statement here is that
physical observables, essentially things that we want to measure in quantum
mechanics, are described by the Hermitian of operators, sometimes also called
self-adjoint operators. So, I should mention that there exists a
subtle difference, actually a mathematical difference between the Hermitian and
self-adjoint operators. But it involves rather complicated math.
And in the framework of this course, we can just consider Hermitians and
self-adjoint to be the same thing. But if among you there's a math aficionado
who wants to know the difference you know, ask this question in the discussion forum
and I will be happy to elaborate on this. But anyway, so if we have identified a
quantity of interest, our observable A, and identified an operator associated with
this observable. So eigen, then, then we are, we, we need
to solve the eigenvalue problem for this operator.
And the available eigenvalues, they determine possible values that actually
can be measured in experiment. So, also let me reiterate a very important
fact which was discussed on the previous slide.
And we're going to talk a little bit about this in the next video.
Namely, that these solutions to this eigenvalue problem, this functions psi of
a of r, form a basis in the space where our wave functions leave.
Which implies that any wave function describing the physical state of a system
can be expanded as a linear combination of this eigenvector.
So, the last comment, but not the least important, that I'm going to make relates
to the interpretation of these coefficients, c sub a, the complex numbers
that appear in this expansion. As a matter of fact, actually, this is
probably the most important take home message that I would like you to take out
of this slide. So imagine that we have an ensemble of
identical quantum systems, which exist in exactly the same quantum stage described
by some wave function psi that we expanded in our bases of these eigen states.
And let's imagine that we're going to form, let's say three identical measures
of the identical quantity a. So, we want to measure a in exactly the
same way with exactly the same kinds of systems.
So, the weird thing about quantum mechanics is that even though everything
is the same about these systems, we are actually going to get, strictly speaking,
three different results. So, there is a chance some of them will be
the same bot. There's no guarantee at all.
And the probability of finding a system in a certain state, which is sometimes is
referred to as a collapse of the wave function into certain eigen state.
So, this probability, let me call it w sub a, is equal to the absolute value squared
of the coefficients in this expansion, so this is the probability.
So in this, in this example again, so what, what we can see for sure is, the
only thing we can say for sure is that all these measurements are going to result in
a value of A that belongs to the spectrum, that is, to the set of available
eigenvalues of the operator A. But which particular value is going to be
measured, we don't know. We only can say which is more likely, and
we can quantify this uncertainty. But this is the maximum we can do.
And so, this principle is actually very important in quantum mechanics and we're
going to use it later in the course very often.