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05-4 Definite Uncertainty
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401 Avatar4-MoreUncertainty
DIANA: Schrödinger’s Equation is really cool!
CHAUCER: Yes it is, Diana.
KEVIN: I agree, but can we talk a little more about Heisenberg’s Uncertainty
Principle? Isn’t there more to consider than the approach we used to describe it?
CHAUCER: Certainly Kevin – Jeeves, let’s have a closer look at that please.
405 More Uncertainty
JEEVES:
Werner Heisenberg did more than make a QUALITATIVE statement when he
said that a particle’s position and momentum could not be known simultaneously
…
He wrote an equation that QUANTIFIED the relationship. Let’s see how that
information can lead to an easy understanding of the hydrogen atom described in
exquisite detail by the Schrödinger Equation.
Without going into too much detail let’s look at a proton and an electron.
Since the electron has a very tiny mass, it can occupy a very large region of
space.
Conversely the proton has a large mass, 2000 times that of the electron, and
therefore it occupies a very tiny region of space. The result is the Quantum
Mechanical hydrogen atom…
A tiny massive nucleus surrounded by a much larger cloud representing the
electron.
If we look at a simple graph relating the probability of finding the electron in a
shell at a given distance from the nucleus, we find that as we travel outward from
the nucleus, the probability increases at first as the shell expands. It reaches a
maximum value and then decreases again as the electron cloud thins to almost
nothing at large distances.
Amazingly, the radius where the probability reaches a maximum is precisely
equal to the radius of the first allowed orbit of Niels Bohr ‘s model of the
hydrogen atom.
And it’s energy is exactly equal to the energy of an electron in this orbit in the
Bohr atom.
So this is a VERY good picture of a hydrogen atom with the electron in the lowest
energy state. The electron occupies a cloud instead of an orbit, but it spends
most of its time at the radius predicted as an orbit by the Bohr model. AND it
spends most of its time possessing the energy that an electron in that orbit would
have.
410 Excited States
But of course, the atom is not always found in this lowest energy state. As there
are other orbits allowed in the Bohr model, there are other, higher energy states
in the quantum mechanical hydrogen atom.
These states are defined primarily by the quantum number “n” that we talked
about earlier. And for each state, the electron has a different energy which
results from the shape of the electron cloud.
For n=1, called the ground state, the shape is a symmetric cloud…the same in all
directions.
For n=2, the shape can take on two forms although both shapes have the same
energy…
One is a double spherical cloud – one sphere inside the other.
While the other shape for n=2 is in the shape of a dumbbell
For other values of n, the shapes can be pretty strange…
Like this torus PLUS dumbbell shape.
415 Spectral Lines
An electron in the lowest energy shell in an atom can be struck by and absorb
the energy of a photon giving it enough energy to jump to the next energy shell.
And the reverse process allows the electron to jump back down into the lowest
energy shell and emit a photon.
The color of the photon depends on the energy difference between the two
shells.
This explains the spectral lines that identify an element. Since white light
contains all the colors in the spectrum, when we shine white light on a sample of
an element under the right conditions, The atoms absorb all of the photons that
allow their electrons to jump to other energy shells.
SO the absorption spectrum is all the colors in white light minus those that match
the difference in energy shells within the atom.
And when those electrons spontaneously jump back down to the lowest energy
levels, that emission spectrum contains only those lines that match the difference
in energy shells within the atom.