okay
so we have our phi_in our plane wave
going in
phi_r our plane wave reflecting and
phi_t
our plane wave transmitting we've solved
for the amplitudes for reflection and
transmission
to solve for the probabilities we need
to use our probability current density
we derived
in a previous video so the general form
looks like this where psi
is the time dependent wave function
in fact for all our plane waves the
time dependent
solution takes the following form
that is it's just the time
independent solution that we've been
working with
multiplied by a winding complex phase
where all the complex phases take the
same form here they wind at the same
rate given by the energy
of the particle and so we get the
expression for our probability current
density
for our time independent wave
functions of this form
that is pretty much the same form but
with phi in place of psi
so let's use this to calculate the
probabilities rather than the amplitudes
of reflection and transmission in the
two regions in another worked example
okay so let's look at the probability
current densities
in the two different regions
so from the board we had that j(x)
is equal to i h bar over 2
m phi d by dx
phi star minus phi star
d by dx phi
in region one
or rather it's actually the
reflection
part we'd like so we'd like j_reflected
is equal to as a function of x
we're going to insert phi reflected into
here
and remember phi so we've got phi_in
is equal to
e to the i k x
phi_reflected as e to the minus
i k x and that's multiplied sorry i
missed the prefactor by r
and phi transmitted is e
to the i k prime x
multiplied by t so if we stick phi_r
into here we find
i h bar over 2m out the front
let's write that in full for the
first one so we have
r e to the minus i
k x d by d x of phi star so phi star
that's d by dx
of r star
e to the plus
i k x
minus phi star is r star
e to the i k x and
d by d x of r e
to the minus i k x
and this equals i h bar over 2
m the r is a constant
so the derivative doesn't
change it
so we have: ... and that's true for both
cases so we're going to have a modulus r
squared out the front here
d by dx acting on e to the ikx brings
down an ik
so we get i k and then and it just
leaves it e to the ikx still
the e to the ikx cancels with the minus
i kx
and so that's all we get and then we
would have a minus here
e to the i k x d by d x on e to the
minus psi k x brings down a minus i
k minus and minus is plus so i actually
get a plus
ik and so we find that j_r
is equal to that's 2 i k
here 2 cancels with a 2 here we get
minus h bar
over m modulus r
squared k
if we'd have done the same for
phi_in
we would just have found that k goes to
minus k as you can see here and we lose
the modulus r squared
so we find that j_in
is equal to plus h bar over
m and we still get the k here
let's take a quick look at what we've
got here sorry let's move that over
slightly so you can see it
good so look at j_in
we have h bar k over m but remember
hbar k
from our de Broglie relation is just p
the momentum
and so this is p over m but p over m it
should just be our velocity
right classically momentum would be mass
times velocity and so
we can kind of think of this as the
velocity
or related to the velocity of the
particle remember it's a probability
current density
so it tells us how the
probability to find the particle evolves
with time but when we look we always
find the particle in one place
so the probability of reflection
let's call it R
is let's say it's defined to be
it really is this is just a sort of
physical way to motivate it it's
the probability current density for the
reflected wave
divided by the probability
current density of the in-going wave
and so in this case we get divided by
the hbar
k over m and we just get minus
r squared and sorry it should really be
the modulus of this because it's a
probability
the so we have the modulus of this
and so we get |r|^2
the minus sign of course is coming about
because j is a vector quantity
so the in-going wave has positive
velocity if you like positive current
probability current density
so the reflected wave must have a
negative one because it's a vector
quantity
so you can think of k as being negative
in this case
if we do the same thing for
phi_transmitted
we find that j_transmitted
is equal to: ... so it's rightgoing
you can do this explicitly i'm just
going to
state the result so we're going to get
a plus h bar over
m but we don't get a k we get a k prime
and we get a modulus
t squared much like we have the modulus
r squared for the other case
and so the probability of
transmission which is
defined to be j transmitted over
j incident is equal to ...
divided by h bar k over m and we get
modulus t squared k prime over
k and again sorry it's really
technically the modulus of this just in
case there's any
sign coming in there okay
so it's not just modulus t squared
the reason for the k prime of the k you
can think of it as the particle having
different velocities in the different
regions
and so you're saying more likely to find
a particle
if it's moving slowly in a region is
one
loose way to think of what's going on
here so we have the probability
for reflection and transmission
and in fact we can substitute back in
the energy E and V_0
into this expression if we want
to get it back in terms of
of the energy of the particle and
V_0
okay so let's take a look now at the
case that
E < V_0 and we'll switch
back to the board for a second