Hi!
This episode, we're going to be looking at something that
comes up a huge amount in physics,
especially quantum physics and thermodynamics,
namely, the exponential function e to the x.
Now, don't be frightened. You've already seen it.
I just didn't call it by its name.
It'll help out quite a lot with the notation in the next few episodes
It's sometimes written as exp of x,
usually to avoid having a ton of stuff jammed into a tiny superscript.
It's just another way of writing exactly the same thing.
You might have heard of e as an example of an irrational number,
about 2.718,
and that by putting any number to the power of a whole number
is the same as multiplying that first number by itself,
the second number of times,
and while that's sort of true,
it doesn't really matter here, and confuses things quite a bit.
We're going to start from a more relevant definition of e to the x,
which is that:
e to the zero equals one
and for all values of x, the rate of change in e to the x,
with respect to changes in x, is itself.
In other words, it's derivative is also e to the x.
This may seem like a strange definition,
but there is exactly one function that satisfies these two criteria,
even supposing we don't assume anything about what an exponent is,
and it's actually how e to the x originally came to prominence.
By the first criterion,
the value is one when the exponent, x, is zero,
and by the second criterion,
at that point, the slope must also be one.
As x increases, e to the x increases, initially at the same rate as x,
but as its value increases,
by the second criterion, its slope must also be increasing.
If we take a huge number of tiny steps along,
each time, getting the value, and then updating the slope,
to figure out how fast to go up on the next step,
we can trace out the function.
As the size of steps approaches zero,
the values we will get approach the true values.
We can do the same thing in the opposite direction, to get values for when x less than zero,
just moving down and updating the slope, instead of moving up.
This doesn't work with any other number as the base of the exponential, either.
For example,
the derivative of 2 to the x won't be 2 to the x.
It will be off by some constant factor.
For noise-free quantum physics,
the exponential function is most
commonly used in a very particular way
and we've already seen one we just
weren't calling it that X can be
anything
so if we just replace all axes with
another value let's say five x the
equation should still be true and we get
this this says that the rate of change
in each of the five x with respect to
changes in 5x is equal to eat at the 5x
what's d 5x though
well suppose we start with some value of
x let's call it X 0 and then we add some
amount DX to it
how much did 5x change well 5x starts
out as 5 x 0 and after increasing X it's
five times x0 plus DX or five x 0 plus
5dx that means 5 x increases by 5dx when
x increases by DX
in other words d 5 x equals 5dx that's
just a long-winded way of saying that if
you multiply two numbers by five
the difference is five times larger than
it was
replacing the 5x with 5dx gives us this
so then we can multiply both sides by 5
and get this the number five was
arbitrary so this works for multiplying
X by any value a even if it's a complex
number the case that comes up over and
over in quantum physics is where a is an
imaginary number
like I or negative 7 I but what does
that mean what is e to the power of an
imaginary number
let's suppose a is I since multiplying
by i rotate something a quarter turn in
phase this says that as X changes each
of the IAC's always changes in a
direction whose phase is a quarter-turn
different from the value of e to the ikx
we know that at x equals 0 e to the I
acts will be e to the 0 which is one and
as we saw in previous episodes if you
only ever change something in a
direction that's a quarter-turn
different from its current value it
doesn't change the magnitude it only
rotates the phase at a rate equal to the
magnitude of the derivative which in
this case is 1 Radian per meter since
the magnitude of e to the I ax is 11 x
equals 0 and it never changes its always
magnitude 1 and the phase rotates at one
Radian per meter everywhere
this means that each of the I acts for
any value of x has a phase of X radians
and magnitude 1 if we have e to the ikx
the magnitude of the derivative is k +
the derivative is still a quarter turn
from each of the ikx so the magnitude
would still be one everywhere that the
phase would rotate at k radians per
meter everywhere at position acts the
phase would be KX radians
as you can see each of the ikx is
exactly the starting wave from last time
an interesting property of this is that
since we can reverse the phase rotation
by replacing I with- I and the complex
conjugate of a number just has the phase
negated we know that each of the
negative ikx must be the complex
conjugate of e to the ikx since the
magnitudes of both each of the ikx and e
to the negative x KX r1 and their
complex conjugates of each other
multiplying them together gives the
constant value 1 e to the ikx describes
the initial state of the wave we looked
at last time but we also observed that
the phase changed over time as the wave
move forward since the phase-change by-
h-bar over 2 m times K squared every
second multiplying by time t in seconds
gives the total phase change after time
T combining this phase with the phase KX
from the initial wave means that this
wave function at position X and time T
can be written like this that's super
script is a bit awkward to read so we
can write it using X which means exactly
the same thing as e to the power of the
thing in the parentheses but is easier
to read in this case I sometimes write
at one way and sometimes the other way
depending on how bulky the exponent
books you can double check that if you
plug in 0 for the time one of the terms
disappears so it simplifies back to our
initial wave as i mentioned at the
beginning of this episode this notation
looks a bit complicated but just
remember that whenever you have eaten
the I times anything it has magnitude 1
and the phase is whatever the is x in
this case the phase changes at a
constant rate as you move along X and it
changes at a constant rate as time moves
forward so we have exactly the simple
wave moving forward that we saw last
time
it's just written in a different way
that can be used consistently in other
equations if you remember back
we also found an episode 5 that the
second derivative of the initial wave is
negative K squared times the wave itself
or in this notation the second
derivative of e to the ikx is just
negative K squared times e to the ikx
you can find the same thing by taking
the derivative twice and realizing that
you can take any constant factors from
inside the derivative to outside because
just as we saw before the tip of an
arrow that's k times as long
we'll have to move k times as fast to
rotate the same number of radians per
meter and if the arrow is rotated by a
quarter turn the direction it's moving
will also be rotated by a quarter-turn
if we have multiple variables and we're
only taking the derivative with respect
to one you can just treat the others as
constant values instead of variables for
example these are the derivative with
respect X and the derivative with
respect to t for the time-dependent
solution i glossed over some details
here but I wouldn't worry too much about
them right now we've been using
lowercase K for the angular wave number
in radians per meter when people have an
angular velocity in radians per second
namely how fast something is rotating as
time progresses it's often notated using
the lowercase Greek letter Omega this
wave is rotating at negative H bar over
2m times K squared radians per second so
that would be the angular velocity for
this way we'll be using this notation a
lot in the next few videos because it
will make it much easier to reason about
these waves mathematically when it
starts becoming difficult to visually
follow everything that's going on
see you later
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