hello the behavior of particles in
quantum mechanics is governed by what's
called the schrodinger equation
written here in its time-dependent form
so we sometimes call this
the time-dependent schrodinger equation
usually abbreviated to TDSE for short
it was written down in 1925
by this man Erwin Schrodginer as you can
see a very well-dressed gentleman as
they
were back in the 1920s but you know
what, it's the
20s again so let's see if we can't
conjure up a bit of that 1920s chic
in this video
it worked. Magic! OK.
So the schrodinger equation when
schrodinger originally wrote it down
was intended to be a classical
description of classical waves
this quantity psi which we call the wave
function was thought of
as something like the wave function
which would describe say water waves
in which case it would take some value
which would tell us about the height of
the water wave
as a function of position and time
in fact as we'll see while there is a
good description of
the wave function as something like a
classical wave, in fact quantum mechanics
goes beyond classical physics as we now
of course know. So before we
proceed let me just make a couple of
notational points
when i write d(psi)/dt like this, this
is a partial derivative with respect to
time
holding all spatial coordinates constant
so we can also denote this in the
following way
so we denote this
stating explicitly that
the positions are all constant
sometimes i'll denote this
in a simpler manner in the following
form
where partial subscript t is just
shorthand for
d by dt acting on psi and sometimes i'll
use
newton's notation psi dot
similarly a partial derivative with
respect to x
implies that we're holding the other
variables constant so time
y and z which can again be abbreviated
just partial subscript x acting on
psi
and sometimes we'll write this as psi
prime
where this last notation will only
really be used in one dimension where
it's unambiguously a derivative with
respect to x
so there are a couple of key
assumptions we're going to make
throughout this course let's just wipe
this board off and write them down
when i say wipe the board off of course
i mean magically clear the board
so let's write down some key assumptions
the first is that the schrodinger
equation
is non-relativistic. It's going to
describe the behavior of some massive
particle
in the non-relativistic limit. There are
relativistic extensions to it
but we're not going to consider those in
this course
the second is that we're only going to
describe single particles.
Again there are extensions to the
schrodinger equation which will allow it
to deal with multiple particles
but that's not going to be the focus of
this course. The wave function psi
always governs the behavior of a single
particle
okay so let's take a closer look at
what's going on in the schrodinger
equation let's clear the board again
so let's have the schrodinger equation
back
so we'll look at properties of the wave
function psi in future videos
but for now let's just note that it's a
complex-valued function
and it has out the front of it an
arbitrary
global complex phase.
It's a mathematical redundancy built
into the the wave function
it has this global phase which can be
changed arbitrarily.
But the relative phase between two
different wave functions
is important and we will see that later
on in the course
but for now there is this redundancy
built into it.
We've got
the imaginary unit here i
square root minus one. Imaginary
numbers play an important role in
quantum mechanics.
hbar here is the reduced planck's
constant h over 2 pi
with units of energy multiplied by time
and then we have this quantity here H
which
is called the hamiltonian let's write
that down
so h is defined as follows
it's a differential
operator we have this nabla squared in
here
which okay we can expand as follows
so it's just the partial derivative
with respect to x
squared plus that of y squared and that
of z squared remembering the notation
from the last board and if this looks
confusing just remember this is always
acting on psi so this term here
(d/dx)^2 acting on psi is really just
psi
double prime. Okay.
So it's called the Hamiltonian
and it's what's called an energy
operator for the system
so the hamiltonian acting on psi
is going to return the energy of the
system as we'll see
in a second in general
specifying the hamiltonian specifies the
entire quantum problem we want to solve
so when we
write down a quantum mechanical
description of a system we just need to
write down the hamiltonian
and in fact this first term is fixed
what we what need to specify is the
potential of the system which is much
like what you do in classical mechanics
as well
so the potential we will assume in
this course is time independent
it doesn't need to be but we won't
consider
time dependent potentials here. So
let's take a look at attempts to
deal with
how to solve this equation
so the time-dependent schrodinger
equation
where i've written it with psi dot this
time is a separable
equation what we mean by this
is that we can substitute the
following anzatz
we can say that psi(x,t)
is equal to:
-- and let's treat it in three
dimensions in general -- is equal to
phi(x) only
and T(t) only
so these are two separate functions one
of which is only a function of time
and one of which is only a function of
position when we substitute
that in
we see that the equation reduces to the
following form
where the phi has pulled through the
time derivative
because it's not a function of time and
this is a partial derivative
the partial derivative of T(t) with respect
to time
is actually a total derivative because
it's only a function of time
and over here the hamiltonian because the
potential is time independent
hamiltonian only acts on the spatial
coordinates so we can pull through the t
over here rearranging slightly we get
two different equations
one purely in terms of t and one purely
in terms of x
so we've turned what was a partial
differential equation in terms of both x
and t
into two separate equations one is an
ordinary differential equation in terms
of time
and the other is potentially still a
partial differential equation in terms
of positions
so the fact that these two are equal for
all times
and all positions means they must both
be equal to the same constant
and we're going to suggestively call
that constant E
suggestive because it should remind us
of an energy
so let's number these equations let's
call this one (i)
and this one (ii)
we'll move this up to the top of the
board and we'll deal with (i) and (ii)
separately
so (i) simply rearranges
to the following form
H acting on phi is equal to
E multiplying phi and this is what's
called the time
independent schrodinger equation because
phi here is now only a function of
positions not of time.
Let's put a box around it in general
this one is very tricky to solve in fact
in general it's impossible to solve
remember the hamiltonian here contains
all the real information in the problem
in the form of the potential
so in general we need to solve this
differential equation
in terms of boundary conditions which we
specify and
you can see that it's an eigenvalue
equation
in that we need to solve this for
both
the eigenfunctions phi(x) and the
eigenenergies E
which will be the energies of the
particle let's take a look at the second
equation there
where i've done our favorite trick of
multiplying through by
dt. We can then integrate both sides
to give the result:
that is, the time evolution of t is simply a
phase winding. The winding of the complex
phase
so sticking our anzatzes back together
again remember psi(x,t) the wave function
is the product of T(t) and phi(x) we find that
if we can solve the time independent
schrodinger equation which by the way we
sometimes call TISE
then for free we get the time evolution
of the wave function
where phi(x) is the solution to the
TISE so while this is in general
impossible to solve there are a set of
very important cases which is possible
to solve
and those are the ones that we'll be
looking at in this course
okay thank you for your time it must have been
very hot in the 1920s if this suit is
something to go by
I think I'll turn back into my normal
clothes
excuse me Geoffrey sorry about that
so i'll see you in the next video where
we will take a look at some particular
solutions to the schrodinger equation
thank you