Topic 4 Extras

Question 1

What is the charge of the He nucleus?

Answer 1

+2e

Question 2

What does r represent in the Hamiltonian equation?

Answer 2

the cartesian coordinates of each particle

Question 3

What does the upside down delta represent?

Answer 3

kinetic energy operator

Question 4

What does the Hamiltonian operator express for a Hydrogen atom?

Answer 4

Total energy (kinetic and potential)

Question 5

Give the Hamiltonian for a Hydrogen atom and define the potential and kinetic energy term

Answer 5

H ̂= -ℏ^2/2m ∇^2-e^2/(4πϵ_0 r)
Kinetic is the first term and potential is the second

Question 6

Why is the potential energy term negative? How can it be made positive?

Answer 6

The potential energy term (based on Coulomb’s law of attraction) is a product of the positive charge of the nucleus and the negative charge of the electron. It can be made positive if we are considering interactions between two electrons (-e)(-e)=+

Question 7

What does it mean when a wave function is normalized?

Answer 7

when∫0^π∫_0^2π∫_0^∞〖r^2 sin⁡θ ψ* ψdrdφdθ〗=1
Normalization refers to calculating probability everywhere (which has to be 1)

Question 8

For the normalization expression, which integrals correspond to which variable?

Answer 8

∫0^π∫_0^2π∫_0^∞〖r^2 sin⁡θ ψ* ψdrdφdθ〗=1
The first integral corresponds to r, the second to phi, and the third to theta

Question 9

Why do we not have a 1p orbital?

Answer 9

L must be less than n and cannot be 1 unless n is 2 already

Question 10

Explain what a helium-like system is.

Answer 10

A quantum mechanical system consisting of one nucleus with a charge of Z and just 2 electrons.

Question 11

Why is the distance between r1-nucleus and r2-nucleus shorter than r12?

Answer 11

The attraction between each electron and the nucleus is stronger because of the nucleus’s +2 charge, which makes the bond shorter. Therefore, their attraction to each other is less, making their bond longer

Question 12

What is a Hamiltonian and what does it tell us?

Answer 12

An operator that tells us about the motion or time evolution of a system through its energy

Question 13

What does KE tell us?

Answer 13

How quickly the electron changes positions.

Question 14

Explain the Hydrogen Hamiltonian Equation and its parts.

Answer 14

-ћ^2/2m ∇^2 : KE of electron
-e^2/(4πε_0 ) 1/r_1 : PE of electron

Question 15

Explain the Helium Hamiltonian and its parts.

Answer 15

-ћ^2/2m ∇_1^2 -ћ^2/2m ∇_2^2: KE of 1st and 2nd electron respectively
-e^2/(4πε_0 ) 1/r_1 -e^2/(4πε_0 ) 1/r_2 : attraction term
(e^2/(4πε_0 ) 1/r_12 ) coulumbic repulsion tern

Question 16

Why is the KE term negative?

Answer 16

The KE term is a product of the positive and negative charges from the nucleus/electron.

Question 17

What does the term 1/r mean?

Question 18

10. Why is it -2e^2 for helium?

Answer 18

The charge of the nucleus for helium is +2, so +2e times –e.

Question 19

What makes the Schrodinger equation unsolvable for helium?

Answer 19

The coulumbic repulsion term makes it impossible because it cares about both distances of the electrons at the same time. (e^2/(4πε_0 ) 1/r_12 )

Question 20

How does the Coulombic repulsion between electrons affect the behavior of the He atom?

Answer 20

The Coulombic repulsion between electrons causes them to repel each other, which leads to a complex interaction within the atom. This repulsion term makes the SE unsolvable for the He atom using exact methods, necessitating the use of approximation methods to describe its behavior.

Question 21

Compare and contrast the two methods for estimating the SE.

Answer 21

— Trial wave Function (Variational Method): Since the true wave function (Ψ) is unknown a trial wave function (φ) is used to approximate the wave function of the system. A trial function is a proposed mathematical function used to approximate the wave function of a quantum system. It typically contains one or more variational parameters that can be adjusted to optimize the function and minimize the system’s energy.
— Perturbation Theory: Used to handle systems where the Hamiltonian can be separated into a solvable part and a perturbation. It expresses an unsolvable SE in terms of a related solvable SE

Question 22

What are the limitations of the variational method?

Answer 22

It provides approximations to the ‘true’ wavefunction and might not yield exact solutions. It also depends on how well the trial function approximates the true ground state wave function.

Question 23

How do you know if your estimated energy is accurate?

Answer 23

a. If compared to other estimated energies, the lower the result= the more accurate the estimate is.

Question 24

How can you optimize the trial function?

Answer 24

By taking the derivative and setting it to 0: dEφ/dα=0

Question 25

What is the purpose of Z in the helium Hamiltonian?

Answer 25

It is used in the variational method to make the helium/other elements Hamiltonians to be more general and allow the answer to be more accurate.

Question 26

What happens if z=2?

Answer 26

a. If z= 2 then the ___ term would go to 0 and

Question 27

Why is there not 3 s/ p orbital. Because N