Eigenvalues and Eigenfunctions - Atomic Structure Video Lecture - Chemistry

detour number two before we start
solving the Schrodinger equation for a
specific case is to look at AG in values
and eigen functions so we showed in the
previous video that an operator is
something that acts on a function and
gives back a new function so if an
operator acts on a function and what you
get back is a constant times the same
function as before then we have an eigen
value equation so the operator acts on f
of X which returns Omega a constant
times the same function f of X f of X is
then called an eigen function of
operator oh and Omega is the eigen value
of that eigen function for that operator
so what we have in a lot in quantum
mechanics is called the eigen value
problem that's given an operator find
the set of eigenfunctions and
eigenvalues for that operator this is
very important because the Schrodinger
equation is an eigen value equation we
can factor our shorting our equation
from previous videos thus as thus sorry
we can factor it like this negative H
bar squared over 2 times the mass times
the second derivative with respect to X
plus the potential energy as a function
of X all of that times the wave function
sigh of X equals the total energy e
times the wave function again sigh of X
so in fact the Schrodinger equation is a
an eigenvalue equation where we have
this hamiltonian operator or a total
energy operator H with its kick with a
little cat carrot hat at the top equals
negative H bar squared over 2m second
derivative of X plus the potential
energy so H sy equals e sy is the
shorthand of the Schrodinger equation
the Hamiltonian operator acting on the
wave function gives the total energy
times the wave function again
so given a Hamiltonian operator we must
find the wave functions and total
energies that correspond to that
Hamiltonian and we will that's how we
solve the Schrodinger equation okay so a
lot of these operators that come up as
promised from the previous video we have
e the total energy which is represented
by the Hamiltonian operator H negative H
bar squared over 2m second derivative
with respect to X plus potential energy
this is the most important operator in
quantum mechanics you 100% should
memorize this entire operator right now
to save yourself much trouble in the
rest of this course
100% memorize this equation now the
components of this are two other
operators h equals t plus v the kinetic
energy operator which is negative h bar
squared over 2 times mass times second
derivative with respect to X now that
has to be kinetic energy because the
other part is the potential energy so
this V of X here is the potential over X
so we have our potential there the
potential energy operator is just
multiplied times V of X so that means
that the rest of this over here had to
be the kinetic energy because kinetic
plus potential equals the total so
that's why our kinetic energy operator
is this but we can do better than that
we can actually find sub components of
that so kinetic energy is one-half MV
squared or P squared over 2 n momentum
squared over 2 times mass now look this
kind of looks like something over 2
times mass it's negative H bar squared
times the second derivative with respect
to X and this is the momentum squared
operator and not only that but this is
all things that are squared things that
are done twice the momentum operator for
momentum in the X Direction is the
square root of this it is and it's the
negative square root of it it's negative
IH bar d DX so if you want to know the
momentum of a particle you take you
multiply times negative IH bar and take
the first derivative with respect to X
and then the much more simple case for
position the value of X is the X
operator is just multiplied times X and
X square the position squared operators
just multiply times x squared so these
are the very common operators that
you'll see all the time in quantum
mechanics there are others as well as I
said every property which you can
measure in classical mechanics has a
quantum operator associated with it
but the most important operator being
the Hamiltonian operator which is a sum
of the kinetic energy and potential
energy operators giving us the
Schrodinger equation which is an
eigenvalue equation h sy equals e sy