M2L4e.txt

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The interesting thing about the length scaling
is-- and I just want to draw your attention to it,
because it can be confusing-- that in hydrogen we
have not only one length scale but two length scales.
We have mentioned one of them already,
which is the energetic length scale, 1 over r; 1 over r
is the Coulomb energy.
Because of the Virial theorem it's
proportional to the total energy,
and that's what you know-- what you remember-- when you wake up
in the middle of the night out of deep sleep,
that the energy of hydrogen is 1 over n square.
So therefore this is a0 over n square.
However, if you look at the wave function of hydrogen,
you factor out when you solve the radiant equation,
you factor out an exponential.
There's sort of polynomial and then there is an exponential ek
decay, and the characteristic lengths
in the exponential decay of the wave function is na0 over z.
So therefore we have-- when we talk about wave functions
with principle quantum number n, there
are two length scales, 1 over rnl scales with n square,
but it's the characteristic length
scale in the exponential part of the radial wave function scales
with n and not with n square.
And it is this exponential part of the wave function
which scales with n, which is responsible for the probability
to find the electron s in nucleus which,
as I said before, the z scaling is simple,
but the n scaling is not n to the 6, its n cube.

And this is really important, and this
describes the scaling with n for everything which
depends on the presence of the electron s in nucleus.
One is to quantum defect and the other one
is the hyperfine structure.
Let me just give you one more scaling.
I've discussed now what happens for 0 angular momentum,
for finite angular momentum states,
Psi is proportional to r to the l, so therefore if you ask what
is size square, it scales with 2l and, at least for large n,
the n scaling is, again, 1 over n cube.
OK.
That's what I wanted to present you today.