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create a presentation to display the results of your research. The report should be a powerpoint or slide. All work should be typed with double-spacing and 11 pt Arial font. You will be graded on the...

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create a presentation to display the results of your research. The report should be a powerpoint or slide.

All work should be typed with double-spacing and 11 pt Arial font. You will be graded on the evidence of work, mathematical detail

and understanding, proper exposition and neatness, time management, and organization of ideas. Your work should also be

supported with properly labeled and embedded plots and equations. Any references used should be cited as well.

Structure:

A suggested outline of the final report include:

  • Title page (including Title, Names, Abstract), Introduction to Problem (Statement and Background)

  • .Formulation of Model and Method of Solution

  • Solution Results (Analytical, Numerical, Plots of Solutions)

  • The presentation includes: Discussion/Results (Summary and Discussion of Solution)

  • References

Answered 4 days AfterMar 09, 2022

Solution

Anirudha answered on Mar 14 2022
69 Votes
Spherically Symmetric Solutions to Schrödinger's Equation for the Hydrogen Atom
Spherically
Symmetric
Solutions to
Schrödinger’s
Equation fo
the Hydrogen
Atom
Tannaz Anvari
Spherically Symmetric Solutions to
Schrödinger’s Equation for the Hydrogen
Atom
Tannaz Anvari
March 10, 2022
1 / 1
Spherically
Symmetric
Solutions to
Schrödinger’s
Equation fo
the Hydrogen
Atom
Tannaz Anvari
Outline
2 / 1
Spherically
Symmetric
Solutions to
Schrödinger’s
Equation fo
the Hydrogen
Atom
Tannaz Anvari
Only one frame
Problem Statement
In quantum mechanics one is interested in determining
the wave function and energy states of an atom. These
are determined from Schrödinger’s equation. In the
case of the hydrogen atom, it is possible to find wave
functions ψ that are functions only of r , the distance
from the proton to the electron. Such functions are
called spherically symmetric and satisfy the simple
equation
3 / 1
Spherically
Symmetric
Solutions to
Schrödinger’s
Equation fo
the Hydrogen
Atom
Tannaz Anvari
Only one frame
1
d
dr2
(rψ) =
−8mπ2
h2
(
E +
e20
)
ψ,
where e20,m, and h are constants and E , also a
constant, represents the energy of the atom, which we
assume here to be negative. With these details in mind,
we set out to answer some questions on the subject.
4 / 1
Spherically
Symmetric
Solutions to
Schrödinger’s
Equation fo
the Hydrogen
Atom
Tannaz Anvari
Only one frame
(a) Show that with the substitutions
=
h2
4π2me20
ρ,E =
2π2me40
h2
ϵ,
where ϵ is an negqtive constant, the equation the
problem statement reduces to
d2(ρψ)
dρ2
= −
(
ϵ+
2
ρ
)
ρψ.
5 / 1
Spherically
Symmetric
Solutions to
Schrödinger’s
Equation fo
the Hydrogen
Atom
Tannaz Anvari
Only one frame
In order to solve this, we simply substitute the ansatzs
for E and r in the differential equation to get
(4π2me20)
2
h4
1
ρ
d
dρ2
(ρψ) =
−8mπ2
h2(
2π2me40
h2
ϵ+
4π2me20
h2
e20
ρ
)
ψ,
=⇒
(4π2me20)
2
h4
1
ρ
d
dρ2
(ρψ) = −
(4π2me20)
2
h4
(
ϵ+
2
ρ
)
ψ
=⇒ d
dρ2
(ρψ) = −
(
ϵ+
2
ρ
)
ρψ
Thus, we’ve obtained the required differential equation.
6 /...
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