Find an expression for the Hamiltonian, the Green's Function in
Electrodynamics and the time independent Schrodinger Equation.
Derive a force equation from each one
Extract the radial part of the Schrodinger. wave equation in
spherical coordinates for a hydrogen like atom. Use the methods of
eigenvalues. Plot the results with the radial wave function as a
function of the distance from the nucleus r.
Use the Schrodinger equation solution of the H atom
corresponding to its wave function for the 3dxy orbital to
explain why this orbital has no radial node.
Questions to consider:
(j) What is the value of the wave function and thus the radial part
of the function at a node?
(ii) What factor of the radial part of the wave function,
containing r, can equal your value in (i) and thus allow
you to obtain a value for r?
1. Write the Schrodinger equation for particle on a ring, and
rearrange it until you have the following: ? 2? ??2 = − 2?? ℏ 2 ?
…
a) Assuming that ?? 2 = 2?? ℏ 2 , (where ml is a quantum number
and has nothing to do with mass), show that the following is a
solution for the Schrodinger equation you obtained: ?(?) = ? ?
????…
b)Now think about bounds of variable ?. Using that argue that...
Set up the triple integral of an arbitrary continuous function
f(x, y, z) in spherical coordinates over the solid shown. (Assume a
= 4 and b = 8. ) f(x, y, z) dV E = 0 π/2 f , , dρ dθ dφ 4
The Schrodinger equation is effectively the classical wave
equation recast to account for matter waves by using the deBroglie
relation. The simplest possible application of the Schrodinger
equation is the so-called free particle system where potential
energy, V(x)=0.
a) Show that plane waves, such as , are solutions (k is the
wavevector).
b) Use your solution above to find the energy of a free
particle.
c) What values are possible for position, x, and momentum,
p.
Write the heat conduction equation (without flow in and out) in
cartesian coordinates for the following case:
1. Steady-state, 1-D, without heat generation (2 points)
2.Transient, 1-D, without heat generation (2 points)
3.Transient, 3-D, with heat generation (3 points)
3. Write the three types of boundary conditions. (3 points)