hello in this video we're going to take
another look at the harmonic oscillator
which we looked at in the previous
video but this time we're going to look
at it using
operator methods which is a lot more
exciting
okay so here is the the time
independent Schrodinger
equation so let's write the kinetic
energy term as p
squared over 2m with p the momentum
operator
plus half m omega squared x squared
where x is the position operator
and it's acting on eigenstates phi_n and
giving eigen energies E_n
and again the phi isn't really serving a
purpose here let's just label this ket
as |n>
and bear in mind that the wave function
we solved for
in the previous video which we call phi_n(x)
here that's just given by the x
projection
of these states we're about to solve but
let's keep working with
the ket formalism okay so
a very convenient trick that we can
use in this problem and actually many
problems is to define the following
operators
so we define an operator a^dagger to be
this it's a linear combination of the x
operator and p operator
and its Hermitian conjugate given by
this
remember that p and x are both
Hermitian themselves so
they return to themselves under Hermitian
conjugation so these two operators have
various names
they're called ladder operators
raising and lowering operators where
this is the raising and this is the
lowering
and creation or annihilation
operators
where the last term will hopefully
make more sense by the
end of this set of videos so we can
see straight away that these do not
correspond to physical observables
because these operators are not
hermitian a and a^dagger
are different operators but they are
nevertheless useful
so let's put them to that use now as
follows
so first consider what we get when we
have a^dagger
it must be given by this we can
multiply this out
so we're going to get four terms we'll
get this term
this term this term and this term just
like multiplying out any quadratic
and this gives us the following
where we've been careful to keep track
of the non-commuting operators x and p
and in fact this expression here is
simply the commutator of those two terms
but we know that by definition our
canonical commutation relation
is that the commutator of x and p is
i h bar multiplying the identity
operator
and so multiplying through we find the
result
multiplying both sides by
h bar omega we find this
so this bit is just our time
independent schrodinger equation this is
our hamiltonian
we've got this extra term added here so
rearranging a bit we find the following
result
for our hamiltonian and for our
time-independent schrodinger equation
well we can actually drop the identity
operator now because this
is just the number one
and so this is
then our time independent Schrodinger
equation
so let's consider this commutator it's
given by this
just inserting the forms of a and
a^dagger of course
x is going to commute with x p with p so
we only get the
x with p commutator terms and so it
reduces to the following
where this cancels with this this
cancels with this
and p commutator x is minus
i h bar times the identity operator and
so the
end result is this the commutator of
a^dagger is the identity operator all
right that's good
and the really important result is
when we now check the commutator with
the hamiltonian
so the commutator of the hamiltonian
with a^dagger
is given by this because remember the
hamiltonian is just h bar omega
multiplying a^dagger a plus a half times
the identity operator
by definition this commutes with
everything so this disappears
so we're left with this commutator to
evaluate but this is quite
straightforward because we can just
multiply
we can just expand it that's just
writing the commutator out
and we notice that we have an
a^dagger on the left of everything so we
can write it as follows
this is just the commutator of a with
a^dagger
which we know is just the identity
operator
and so this is our result similarly for
h with a
we find just minus h bar omega times a
okay so that's all well and good but now
we can use it to actually derive the
full set of solutions for the energy
eigenstates and the energy eigenvalues
so let's do that now
actually let's do it to the next video
what are you up to