Topic 4 Extras Flashcards
What is the charge of the He nucleus?
+2e
What does r represent in the Hamiltonian equation?
the cartesian coordinates of each particle
What does the upside down delta represent?
kinetic energy operator
What does the Hamiltonian operator express for a Hydrogen atom?
Total energy (kinetic and potential)
Give the Hamiltonian for a Hydrogen atom and define the potential and kinetic energy term
H ̂= -ℏ^2/2m ∇^2-e^2/(4πϵ_0 r)
Kinetic is the first term and potential is the second
Why is the potential energy term negative? How can it be made positive?
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)=+
What does it mean when a wave function is normalized?
when∫0^π∫_0^2π∫_0^∞〖r^2 sinθ ψ* ψdrdφdθ〗=1
Normalization refers to calculating probability everywhere (which has to be 1)
For the normalization expression, which integrals correspond to which variable?
∫0^π∫_0^2π∫_0^∞〖r^2 sinθ ψ* ψdrdφdθ〗=1
The first integral corresponds to r, the second to phi, and the third to theta
Why do we not have a 1p orbital?
L must be less than n and cannot be 1 unless n is 2 already
Explain what a helium-like system is.
A quantum mechanical system consisting of one nucleus with a charge of Z and just 2 electrons.
Why is the distance between r1-nucleus and r2-nucleus shorter than r12?
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
What is a Hamiltonian and what does it tell us?
An operator that tells us about the motion or time evolution of a system through its energy
What does KE tell us?
How quickly the electron changes positions.
Explain the Hydrogen Hamiltonian Equation and its parts.
-ћ^2/2m ∇^2 : KE of electron
-e^2/(4πε_0 ) 1/r_1 : PE of electron
Explain the Helium Hamiltonian and its parts.
-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
Why is the KE term negative?
The KE term is a product of the positive and negative charges from the nucleus/electron.
What does the term 1/r mean?
- Why is it -2e^2 for helium?
The charge of the nucleus for helium is +2, so +2e times –e.
What makes the Schrodinger equation unsolvable for helium?
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 )
How does the Coulombic repulsion between electrons affect the behavior of the He atom?
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.
Compare and contrast the two methods for estimating the SE.
— 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
What are the limitations of the variational method?
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.
How do you know if your estimated energy is accurate?
a. If compared to other estimated energies, the lower the result= the more accurate the estimate is.
How can you optimize the trial function?
By taking the derivative and setting it to 0: dEφ/dα=0
What is the purpose of Z in the helium Hamiltonian?
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.
What happens if z=2?
a. If z= 2 then the ___ term would go to 0 and
Why is there not 3 s/ p orbital. Because N
