Schrodinger Wave Equation - Atomic Structure, Physical Chemistry Video

this video is a review of the
Schrodinger equation and particle in a
box chapter in the quantum chemistry in
spectroscopy playlist so we start by
using the classical wave equation to
define the Schrodinger equation which is
our quantum wave equation which we solve
for any system once we have our
potential energy specified V of X so at
every value of x we know what potential
energy our particle experiences then we
can solve for our wave function Phi of X
and get a total energy e
for a particle of a given mass M so our
Schrodinger equation is minus h-bar
squared
Planck's constant over 2 pi squared
divided by 2 times the mass times the
second derivative of the wave function
with respect to position plus the
potential energy times the wave function
equals the total energy times the wave
function again so we use operators here
where an operator is acting on a
function to give some new function or in
general do something to an object to
create a new object in quantum mechanics
every possible physical property we
could measure position momentum energy
etc has a has a quantum mechanical
operator associated with it for a
position that's multiplying by X for
momentum it's multiplying by minus IH
bar and taking the first derivative with
respect to X so there's a special kind
of operator and function relationship
called eigenvalues so whenever an
operator acts on a function giving us a
constant times the same function the
function is an eigenfunction of that
operator and the constant is called the
eigenvalue this is important
specifically because the Schrodinger
equation is an eigenvalue equation where
the hamiltonian total energy operator
acts on our eigenfunction the wave
function to give the eigenvalue the
total energy so the wave function is
interpreted as a probability density
function the wave function times its
complex conjugate times DX is the
magnitude of the square wave function
times DX that's the probability that our
particle is between the positions X and
X plus DX so provided DX is sufficiently
small we then apply the Schrodinger
equation to our first model system the
particle in a box where the potential
energy is infinite outside the box and
zero inside of it we get that our wave
functions are a bunch of sine functions
just as they were for the vibrating
string and we get that the energies of
our our quantized states are H squared N
squared over 8 ml squared where our n
here is a quantum number starting at 1
and going up from there and L is the
length the Box M is the mass of the
particle we then apply the particle in a
box model to the Pauline uv-vis spectra
where the difference in energy between
two states is equal to the Planck's
constant times the frequency of a given
Photon
so we show there in that video that the
wavelength of the photon absorbed is
approximately proportional to the number
of double bonds or the length of our
system as correctly predicted by the
particle in a box model we can then use
what are called expectation value
integrals to calculate things like the
average position and momentum of the
particle in a box depending on our
quantum number so we can calculate
average position momentum position
squared and/or momentum squared we can
calculate the uncertainty for a given
particle by for a given property for a
particle by the following type of
expression where we compute that
expectation value integral for the
operator minus its average value squared
acting on the wave function and then
taking the square root of the final
result when we do this for position and
momentum from these values we get a
value which is an agreement with the
Heisenberg uncertainty principle that
this product must be greater than or
equal to H bar over 2 extending this to
three dimensions we have the energy is
now a function of three quantum numbers
one for each dimension each starting at
1 and going up as an integer or if H
squared over 8 m and x squared over LX
squared plus n y squared over ly squared
plus n Z squared over LZ squared length
in each dimension
and heats individual quantum number and
the wave function then is a product of
the wave functions from each individual
dimension and then our final concept to
discuss is degeneracy which is how many
energy levels how many states have the
same energy
when you're the only state with a given
energy you're singly degenerate when you
have two you're doubly degenerate etc
things where you have five equal energy
levels will be called fivefold
degenerate and generally this degeneracy
arrives due to some type of symmetry in
our equation so links to each individual
video in the on screen annotations and
in the description below