5. For an electron bound to a single proton (in what we call a hydrogen atom):
a) What is the energy of the electron ground state?
b) What is the kinetic energy of the electron ground state?
c) What is the potential energy of the electron ground state?
d)
What is the ground state wave-function? Write this in the form
\(e^{-r/b}\) with a normalization factor also written in terms of b.
(Subsume hbar, m and e into the parameter b.) What are the units of the
parameter b?
[Note: we are using b here because we will use a for the lattice constant in crystal lattices.]
e)
In terms of x, y and z, what is r? Graph r as a function of x for
y=z=0. Graph the ground state wave-function as a function of x for
y=z=0.
f) What is the "size" of the ground state? Discuss here how one might quantify that.
6.
a) How many first excited states are there for the hydrogen atom? How
many linearly independent first excited states are there for the
hydrogen atom?
b) The \(\psi_{n,l,m}\) first excited states that
you learned about in physics 102 can be written as a linear combination
of a state proportional to \(xe^{-b/2a}\) and 3 other states. What are
the 3 other states? Which one is very different from the others even
though it has the same energy?
[Extra credit: comment here on what
symmetry leads to this extra degeneracy, and, also on what symmetry
leads to the 3-fold degeneracy.]
c) Express the \(\psi_{n,l,m}\) first excited states in terms of your 4 states from part b.
7.
Use 3 of your 4 first-excited states from problem 5 to construct sp2
first excited states. Visualize and discuss the essential nature of
these sp2 states (and the leftover state). How are these important to
life on earth?
8. Use all of your 4 first-excited states from problem 5 to construct sp3
first excited states. Visualize and discuss the essential nature of
these states. How are these important to semiconductor physics and all solid state devices (phones, computers...)?
Do anyone have an idea for 5f?
ReplyDeleteI would say the "size" can be measured by how far the wave function extends radial outward form the center? for the ground state Y100 it is just spherically symmetric and falls of rather quickly so we can say its "size" is that of a sphere of radius a0=bohr radius. Have to do the calculation but I believe for the ground state that is right.
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