Advance Science Institute Chapter 1 Mathematic Derivation Chemistry Worsheet

I need help with my HW. I also provide you the book that my instructor are teaching, so if you need anything, please use the book. You can cite information from the book.

We are already cover chapter 1, 2, 3, 4 and 7.1, 7.4

https://drive.google.com/file/d/1VzmPcv-UvhcNAvXfz…

1) A particle resides in a three-dimensional box where
and
𝑉 = ∞ 𝑒𝑙𝑠𝑒𝑤h𝑒𝑟𝑒
a) Write an expression for the total energy.
b) Write an expression for Ψ(𝑥, 𝑦, 𝑧).
c) Show using mathematic derivation that the uncertainty principle obeyed
for this case.
2) Given the following wavefunctions (You are NOT allow to use a
computer program to solve):
a) Show that Ψ1 (𝑥) is normalized.
b) Show that Ψ0 (𝑥) and Ψ1 (𝑥) are orthogonal to each other.
c) Evaluate < 𝑝x > for Ψ1 (𝑥).
(Hint: The wavefunctions are Hermite polynomials.)
3) The Hamiltonian can be written in a variety of ways to address different
coordinate systems. If a particle were confined to the shape given to the
right, what two coordinate systems would you recommend for a fellow
student to use? Write the two expressions for the Hamiltonian operator.
Which one would be best given the shape of confinement for the particle?
Explain your reasoning.
4) Find [X̂,
2
] (Hint: Use an arbitrary function f(x)).
Recall that X̂ = x (multiply by x) and
=−𝑖ħ
𝑑
𝑑𝑥
Figure 1: This is an example for question 3a in the 2nd HW I sent you. I attach this
just to show what my instructor expectations/solving ways are.
5) Explain
a) Why the nodal structure of eigenfunctions is important
b) The meaning of wave-particle duality and its importance in quantum
mechanics.
6) The complex conjugate of a function is given by
What is original function?
7) Given the 2s and 2pz orbitals for a hydrogen atom listed in the table
below, determine the following:
a) Calculate the radial functions for both states.
b) Calculate the expectation value of r in terms of the Bohr radius for the 2s
orbital.
c) Is the 2pz orbital normalized? Explain. (Hint: Show by mathematically
solving.)
d) Are these orbitals orthogonal to each other? (Hint: Show by
mathematically solving.)
8) This is a two-part question:
a) A porphyrin is a conjugated macrocycle that forms the structural basis of
the heme group, chlorophylls, and is often used form molecular transport
with the addition of a transition metal. If the radius of the ring is estimated
at 450 pm, what is the wavelength of radiation for the lowest transition?
What range of the electromagnetic spectrum does this correspond to?
b) Suppose you wanted to synthesize a polyene that has its lowest
excitation energy at a wavelength of 600nm. What would be the structure of
this molecule?
She left a question mark here
She ask “what is the final
answer?”
And I got 6/10
4) a)
Trust me, this is exactly what she
wrote:
“Please write IN English the steps you
take. I do Not Understand how the
answers just appear.”
8/10