Hi, everybody.
In this video, we will continue to talk about ...
how quantum mechanics works.
In the previous video, we introduced the ...
Schrodinger equation and we introduced the idea of wavefunctions.
But we still didn't really get down to the rules of how ...
quantum mechanics operates.
In other words, if an object is in certain ...
wavefunction, we didn't really talk about ...
what happens when you try to measure a certain property,
like position or momentum or energy.
In this video, we're going to talk about the ...
postulates, the rules that govern quantum mechanics.
And there are five of them.
The first postulate is something we've kind of ...
talked about already, the idea that the position and ...
the momentum of an object can be fully described by its wavefunction.
What we haven't really talked about yet is that there are ...
certain conditions that dictate whether a wavefunction is acceptable.
To summarize these conditions,
the wavefunction basically has to be a mathematically ...
well-behaved function.
Because the Born interpretation of quantum ...
mechanics tells us that the square of the wavefunction is ...
proportional to the probability of finding a particle there,
we can infer that the wavefunction has to be normalizable.
In other words, we have to be able to ...
integrate this probability to 1 over all space.
And also, the wavefunction has to be single-valued.
Because the Schrodinger equation has a second ...
derivative in it, and we have to be able to ...
take the second derivative of any wavefunction,
we can infer that the wavefunction and its first ...
derivative both have to be continuous.
In other words, the wavefunction has to be ...
continuous and smooth.
Probably the easiest way to wrap our minds around all of ...
these conditions
is to look at some examples of wavefunctions that are not ...
acceptable by this postulate.
If we go through these wavefunctions one by one,
you can see the wavefunction A is not continuous,
wavefunction B is not smooth,
wavefunction C isn't single-valued,
and wavefunction D can't be normalized because it goes to infinity.
So all of the wavefunctions that we'll come across in this ...
course have to be well-behaved:
continuous and smooth and single-valued and normalizable.
The second and third postulates of quantum ...
mechanics are pretty dense, and they really start getting ...
into the rules of how quantum mechanics operates.
The second postulate tells us that every observable ...
property, whether it's position or ...
momentum or energy, corresponds to a linear,
Hermitian operator.
And the third postulate tells us that the only possible values ...
for us to measure when we take a measurement of an ...
observable property are the eigenvalues of the ...
corresponding operator.
There's a lot to unpack here.
First off, we should define what we ...
mean by an "eigenvalue" and by an "eigenvalue equation".
An eigenvalue equation is one that has the following form:
some operator applied to a function,
is equal to a constant
times the same function.
In this type of equation, the function is called the ...
eigenfunction and the constant is called the eigenvalue.
This is a very general type of problem,
and it's one that we'll see all the time in quantum mechanics.
And actually, we've already seen an ...
example of an eigenvalue equation.
The Schrodinger equation itself -- H psi = E psi -- is an ...
eigenvalue equation.
The Hamiltonian operator, which is the energy operator,
applies to some wavefunction psi,
is equal to a constant, the measured energy,
times that same wavefunction psi.
The Schrodinger equation is the quantum mechanical ...
equation that's specific to energy measurements,
but there are operators for every property.
There's an operator for position,
and an operator for momentum, etc.
This slide shows us the one-dimensional operators for ...
position, momentum, and energy.
You can see that the energy operator is the Hamiltonian ...
operator that we've seen already,
which can be split up into kinetic energy and potential ...
energy, where the kinetic energy term ...
is -hbar^2 / 2m times the second derivative,
and the potential energy depends on the chemical environment.
The position and momentum operators are both pretty ...
simple in one dimension. The position operator just ...
says that we multiply x by our wavefunction.
And the momentum operator is -i hbar times the first derivative.
In each of these cases, and when we take any ...
measurement in quantum mechanics,
if we have an object in wavefunction psi and we want ...
to predict the outcome of a measurement,
all we have to do is apply the corresponding operator to the ...
wavefunction and find the eigenvalue.
For example, if we apply the Hamiltonian ...
operator to a wavefunction psi and the result is 5 times the ...
original wavefunction psi,
What that tells us is that if we were to take a measurement ...
of the energy of an object and wavefunction psi,
then we would 100% of the time measure that the ...
energy of that object is 5.
The measured value of that property is equal to the ...
eigenvalue of the operator that corresponds to that property.
The second postulate also had one more big math word in it.
It said that every quantum mechanical operator is what's ...
called "Hermitian".
The definition of what it means to be a Hermitian ...
operator is shown here.
It's an operator for which, if you take this integral shown ...
on the slide and you switch the order of the two functions ...
psi i and psi j,
the value of the integral stays the same.
I don't want to dwell so much on this formal definition,
but for our purposes, there are two important ...
consequences of the fact that every quantum mechanical ...
operator is Hermitian.
First of all, for a Hermitian operator,
all the eigenvalues are real.
This is important because as we said,
the eigenvalues are the values that we actually ...
measure when we measure, say,
the energy, the position,
or the momentum of an object.
And when we take a measurement of a physical ...
property, we would hope that the values ...
that come out are real.
The fact that the operator is Hermitian guarantees that they are.
Another property of Hermitian operators is that all of the ...
eigenfunctions are orthogonal to each other.
That basically means that the functions are perpendicular to each other.
The definition of orthogonality is shown at the bottom of the ...
slide. If you take the integral of two ...
functions multiplied together over all space
and it comes out to be zero, then those two functions are orthogonal.
As we'll see later on, it's important that all of the ...
eigenfunctions of a given operator,
of a given property,
are orthogonal to each other.
That's why, for example,
all of the atomic orbitals of a given atom are orthogonal to each other.
Since we'll be seeing a lot of eigenvalue problems in this ...
course, I want to give us some practice.
So let's consider the following operator A,
which we're defining as the second derivative of a ...
function with respect to x.
If we're looking at this operator A and each of the ...
following functions f(x), let's answer these two ...
questions. First of all,
is each function an eigenfunction of this operator A?
And second of all, if it is,
what is the corresponding eigenvalue?
I'll give you a few seconds to think about this,
and if you'd like more time, I would encourage you to hit ...
"pause" and take whatever time you need.
So the question we're really trying to answer here is,
if you apply this operator A to a given function,
is the result a scalar multiple of the original function?
The first function were looking at is f(x) = x^3.
The second derivative of x^3 with respect to x is 6x.
And if you compare 6x to x^3, they're not scalar multiples of each other.
So what that means is that x^3 is not an eigenfunction
of this second derivative operator.
If we consider the second function,
f(x) equals e^3x,
and we take the second derivative of that,
what we end up with is 9e^3x,
which you'll notice is just 9 times our original function.
So that means that yes, e^3x is an eigenfunction of ...
this operator and the corresponding eigenvalue is 9.
And finally let's look at the sine function, f(x) = sin(x).
If we take the second derivative of sin(x),
we end up with -sin(x), which is just -1 times our ...
original function.
So yes, sin(x) is an eigenfunction of ...
this operator, and the corresponding ...
eigenvalue is -1.
So far, we've talked about the ...
outcome of measurements when a wavefunction happens ...
to be an eigenfunction of a given operator.
To reinforce this, let's think about this particular example.
Suppose we have a property A for which we have to ...
wavefunctions that are normalized eigenfunctions.
The function shown in blue, which we'll call psi a,
has the property that if you apply this operator to psi a,
you end up with 1 x psi a.
And the function shown in red, which we'll call psi b,
has the property that when you apply the operator to psi ...
b, you end up with 7 x psi b.
So, if you measured this property ...
A for particle and state psi a, what do you think its value would be?
And the same goes for a particle in state psi b.
What do you think its value would be?
Hopefully, you remembered that the ...
outcome of these measurements would just be ...
the eigenvalues corresponding to each eigenfunction.
If you measured property A for a particle and state psi a,
you would with 100% probability measure a value of 1.
And if you measured a particle in states psi b,
you would again with 100% probability
measure a value of 7.
But an object isn't necessarily always in an eigenfunction of ...
a given operator.
Now let's think about a different possible ...
wavefunction, which we'll call psi c,
that itself isn't an eigenfunction of operator A,
it's a sum of two eigenfunctions.
We're defining psi c as 3 x psi a + 1 x psi b,
and you can see a picture of what this function would look like.
So the question is: What do you think would ...
happen if we measured the value of property A for a ...
particle and state psi c? What values might we see?
Welll, one important thing to ...
remember is that the third postulate of quantum ...
mechanics tells us that the only possible outcomes are ...
still the eigenvalues corresponding to the ...
eigenfunctions of property A.
So because psi c is just a sum of two eigenfunctions,
psi a and psi b, the only possible values that ...
we might measure are the eigenvalues corresponding to ...
psi a and psi b. So that's 1 and 7.
So you may be wondering, which is it? Is it 1 or 7?
And this is really fundamentally the strangest ...
thing about quantum mechanics.
Quantum mechanics tell us that we can only think in ...
terms of probability.
If an object is in state psi c, and we measured the value of ...
property A, we might measure the value ...
1, we might measure the value ...
7, but we can only know probabilities.
And those probabilities depend on the coefficients of ...
each eigenfunction within psi c.
In this case, you may notice that psi c has ...
more psi a character than psi b character.
It's three times psi a plus only one times psi b.
And as a result, we're more likely to measure ...
the eigenvalue corresponding to psi a.
In other words, we're more likely to measure ...
the value 1 than we are to measure the value 7.
In general, this gets us to the idea of ...
"superpositions". A valid wavefunction doesn't ...
have to be an eigenfunction of a given operator.
It can be a sum of eigenfunctions,
which is what we call a superposition,
which isn't an eigenfunction itself.
So in general, we can define a wavefunction ...
psi as some coefficient times the first eigenfunction,
plus some other coefficient times the second ...
eigenfunction, etc.
And what we've just learned is that,
first of all, the only possible measured ...
values, if we measure the value of the ...
property, are the eigenvalues ...
corresponding to psi 1, psi 2, etc.,
the eigenvalues corresponding to the ...
eigenfunctions of the operator.
That's what the third postulate of quantum mechanics tells us.
And if our superposition psi is normalized,
it turns out that the probabilities of measuring ...
each outcome are proportional to the squares of ...
the coefficients: c1^2,
c2^2, etc.
We haven't really demonstrated this last point ...
here, but we'll spend more time on it in class.
We're almost done. There are just two more ...
postulates of quantum mechanics that I want to ...
briefly mention.
The fourth postulate tells us that if an observable property ...
is measured repeatedly for an object in wavefunction psi,
then the average value
Is given by this integral:
the integral of the complex conjugate of the wavefunction ...
times the operator applied to the wavefunction,
integrated over all space.
We just saw that in general, the value of a measurement ...
might be different each time.
But if you took that measurement a large number ...
of times, then the average is given by this integral.
This type of integral is called an "expectation value",
and we'll see plenty of examples of it in class.
And finally, the last postulate of quantum ...
mechanics tells us that a wavefunction evolves in time ...
according to the time-dependent Schrodinger equation.
The forms of the Schrodinger equation that we've seen so far,
and really all the forms that will cover in this class,
are the time-independent Schrodinger equation.
So I include this last postulate mainly for completeness.
We won't talk about it much.
So that's all for now. We've learned in this video ...
more or less how quantum mechanics works.
And in the next video, we'll talk about a few of the ...
strange philosophical implications of quantum mechanics.
