Numerical integration of the Schrödinger

Numerical integration of the Schrödinger

Numerical integration of the Schrödinger equation

***Please read the file under additional files.

6.1 The Problem
Solve for the stationary states of an electron in a ramped infinite square well.

6.2 Introduction
A basic problem in quantum mechanics is to find the stationary states (“energy levels”) of a
bound system, using the time-independent Schrödinger equation (TISE)

(-h^2(bar)/2m)(d^2/dx^2)(psi(x)) + V(x)*psi(x) = E*psi(x)

In this activity, we will look at the numerical
integration of the differential equation.
The problem we face here is not that of the initial value problem of mechanics, that we dealt
with earlier. We can’t just start from some initial value, use numerical integration, and end up
somewhere after some time. Instead, solving the TISE is a boundary value problem. There is
an unknown quantity (E) in the differential equation, and we need to solve for this, subject to
constraints on the boundary conditions of the wave function.
How to go about this? How do we tell if we have a correct value for E (called an eigenvalue)?
We have to look at the properties of the required solution. We find that if E is an energy
eigenvalue, the wave function ψ(x) has the expected behaviour at the boundaries. If we choose
the wrong value for E, then the wave function will not have this property.
Let us consider a specific example. Suppose that we have an electron in a ramped infinite square
well with walls at x = 0 and x = a:

V (x) = {bx for 0 < x < a
{∞ otherwise

Since the potential energy is infinite outside the well, the electron cannot be beyond the well’s
walls. Continuity of the wave function then implies that ψ(0) = ψ(a) = 0, exactly as in the case
of the unramped infinite square well considered in lectures. However, owing to the changing
potential energy in the well, this is a more interesting problem than the one done in class.
A way of finding a solution numerically then suggests itself:

1. Assume a value for E.
2. Start at one of the walls (e.g. the left wall at x = 0), setting ψ to zero there.
3. Integrate numerically to the other wall.
4. If the numerical solution is close (?!) to zero there, we have a correct value for E
5. If we don’t have a correct value, choose another and try again.

Ideally, we would use some sort of root-finding algorithm to drive this process, but we will see
that it can be done by hand.

6.3 Numerical Methods
We can do the numerical integration as we have for Newton’s equation, by splitting the second
order ODE into coupled first order equations. But it’s just as easy to use a numerical
approximation to the second derivative

d^2ψ(x)/dx2 ≈ ψ(x − ∆x) − 2ψ(x) + ψ(x + ∆x)/(∆x)^2

Where does this approximation come from? Substitute the Taylor expansion of ψ(x ± ∆x) into! the RHS of Eq. 6.1 and see.

Note that the integration is over x, using a fixed step size ∆x.
Then we obtain for the TISE:

Then we obtain for the TISE:
ψi+1 = 2ψi − ψi−1 + (∆x)^2g(E, xi) ψi

g(E, xi) = −2m/h^2(bar)(E − V (x)) = −2mc^2/h^2(bar)c^2(E − V (x))

for i = 1 . . . N − 1.

Note that, to start this off, we need two values of the wave function. One is at the left wall:
ψ0 = 0. What about the next, ψ1? We can take any value for this. The wave function is usually
taken to be normalised, but this is not a consequence of the solution, but is imposed by our
interpretation of the wave function. Any multiple of ψ(x) is still a solution. We can take any
value for ψ1 — this simply affects the scaling of the wave function — and normalise afterwards.

6.4 Model
Use the following: a = 3.0 nm, b = 0.5 eV nm^−1,mc^2 = 5.11×105 eV, h(bar)c = 197.3 eV nm, where
c is the speed of light. Note that these values allow you to use “natural scale” units of eV and
nm for an atom-sized problem. (And you don’t have to bother with laborious conversions to SI

6.5 Computer Solution
Given the discussion above, we can outline an algorithm for searching for the energy levels in
our problem.
1. Integrate from x = 0 nm to x = a = 3.0 nm with a step size of, say, 0.01 nm and a suitable
initial value of E (perhaps close to the ground state of the infinite unramped square well
of the same dimensions). For the initial values choose ψ0 = 0 and ψ1 something small,
say 10^−6.Some experimenting might be necessary. Print out the value ψN of the wave function at the right wall.
psi = np . zeros ( N+1,float )
psi [0] = 0. 0 # two initial conditions

psi [1] = 1.0 e−6
# i n t e g r a t e
for i in range ( 1 , N ) :
. . .

2. You can adjust the value of E in order to home in on the eigenvalue for the ground state.
Find the region where ψN changes sign. This indicates that an eigenvalue has been stepped
over. You will probably need to read in a value of E from the keyboard. This is easy:
. . .
while 1 : # l o o p f o r e v e r w hil e t r yi n g new v al u e s o f E
E = input ( ’ Enter a value for E : ’ )
. . .
# i n t e g r a t e
for i in range ( 1 , N ) :
. . .
# print out last value
3. Normalize and then plot out the wave function. A simple approximation for an integral is
given by the trapezoidal rule

(Please look at the file under “Additional files”
to see the equation)

Can you interpret this equation?
Using numpy this can be written: …

***Please look at the file under “Additional files” to see the rest of the assignment

Violence in Rap Music

Violence in Rap Music

use these sources
“Preface to ‘Does Rap Culture Provide a Positive Outlet for Young People?’.” Rap Music and Culture, edited by Kate Burns, Greenhaven Press, 2008. Current Controversies. Opposing Viewpoints in Context, Accessed 9 Oct. 2018
Robinson, Matthew. “There Is a Link Between Violent Lyrics and Violent Behavior.” Violence in the Media, edited by Dedria Bryfonski, Greenhaven Press, 2014. Current Controversies. Opposing Viewpoints in Context, Accessed 9 Oct. 2018. Originally published as “A Look at Violence in Music,”, 20 Apr. 2011.
Washington, Ahmad Rashad. “Integrating Hip-Hop Culture and Rap Music Into Social Justice
Counseling With Black Males.” Journal of Counseling and Development, vol. 96, no. 1, 2018, p. 97+. Student Resources In Context,
d=271bfe10. Accessed 9 Oct. 2018
my thesis is also
A law should be put in place restricting the ability to monetize music because that discourages a artist to spread his creativity with the world, prevents people from the ability to relate to the music if they need something to seperate them from the outside world, and to help others create a good diversity between themselves.



To support the budgetary process, research how can you reduce your tax liability and/or avoid paying taxes when you file this year. Work with classmates to develop a tip sheet for students on tax avoidance—be sure your suggestions are legal!

Informative Speech Power Point

In the Week 2 forum, you chose your informative topic. In the Week 3 Forum, you wrote your speech outline. In a well-organized speech, inform your audience about your topic.

Read the Week 3 Overview to learn more detail on how to write a speech. Choose one topic from these 4 kinds of Informative speech topics:

4 Kinds of Informative Speech topics (Choose one topic only)

Explanatory Informative speech topic–explain the state of something: Explain the state of Bilingual Education in middle school.

Descriptive Informative speech topic–give the reader a mental picture of your topic: What moon craters look like

Demonstrative Informative speech topic–show someone how to do something: How to lose weight, How to become rich

Definition Informative speech topic–define a concept or idea.: What is an atom?

For the Informative Speech assignment, you can either write a Power Point Presentation speech, or create a voice over Power Point speech. Just upload the Power Point Presentation to the assignments portal like you would a paper. If you want to do a voiced power point, here are the Voiced Powerpoint instructions.

When you write your speech, you need to have a main idea, a body, and a conclusion in your Power Point speech presentation. Here is the structure of your speech when you use Power Point. Take a look at this Power Point speech template

Slide 0: Title Slide/Write the title of your speech and your name

Slide 1: English Only is not better than Bilingual Education because Reason1, Reason 2 and Reason 3. (Write your thesis and write all 3 reasons on your first slide.)

Slide 2: Reason 1 Use no more than three bullet points on your slide. Use 5 to 7 sentences in your speaker notes.

Slide 3: Reason 2 Use no more than three bullet points on your slide. Use 5 to 7 sentences in your speaker notes.

Slide 4: Reason 3 Use no more than three bullet points on your slide.Use 5 to 7 sentences in your speaker notes.

Slide 5: Conclusion

Slide 6: Sources–Cite the sources you used for this speech.

Write whatever you want to say in your speech, in the speaker notes.

PURPOSE: to inform

LENGTH: 450-500 words
AUDIENCE: author, instructor, others interested in the field, peers in the classroom
SOURCE MINIMUM: Total of 3: 1-2 from the Library and 1-2 from discipline specific websites
FORMAT STYLE: the citation style that is appropriate for your discipline and is appropriate for a speech

chapter 7-8

chapter 7-8

The discussion assignment will require you to read and respond to one question from each chapter. Your response should be in a essay format, grammatically correct, logical and well thought out. The response should also be posted as a attachment to the post. Your response should include at least two references outside of your book to support your response.

Discussion Questions: Post in the discussion board: Respond to two of your classmates posting

There were three factors that chapter 7 identified that may affect the extent of a vocational handicap associated with a disability. The physical capacity, the acquired vocational skills and skill acquisition potential and the psychological functioning. Identify a disability and tell us how these three factors may impact that person’s ability to achieve vocational success?
Although work is an activity central to the life of the great majority of adults of working age, persons, why have so many persons who have low vision and or blind been so unrepresented in the competitive labor market?
Discussion Questions: Post in the discussion board: Respond to one of your classmates posting

Discuss in detail the concept of the two-hat theory in relation to the role of the Rehabilitation Counselor?
What should the relationship consist of between a Rehabilitation counselor and the client he/she serves?
Due Date
Oct 21, 2018 11:59 PM

4 Dinner guests

4 Dinner guests

This paper must have an introduction-thesis statement and a conclusion. It is a paper on 4 dinner guests of my choice. It must contain biographical information on each guest and a conversation between you and the guests. Each biography must have at least two scholarly sources. The four guests are Fred Smith, Justin Timberlake, Elvis Presley, and Morgan Freeman. Each of these guests are from Memphis, TN. Be creative and make a story out of it.

exam questions

exam questions

What will be the main challenges facing the UK as a trading nation in the period post-Brexit?

for example
1-factories like cars
2- the value of the pound exchange rate
3- skills worker and immigration

this is an exam question so should cover the main challenges facing UK economy after Brexit.

thank you.

intro to contract law

intro to contract law

PART A: coursework of a maximum of 3,500 words (75% of the module grade)
Seaside Events is an events business run by Rahul in Brighton. Nuptials Nibbles is a local catering company run by Sarah.
Sarah receives a telephone call from Rahul on Monday, when he says, “I am interested in Nuptials Nibbles catering for a
wedding for the Mayor’s daughter we are putting on in two weeks’ time at our Seaside Towers Hotel. How much would you
charge per head?” He is anticipating 100 guests.
Sarah responds that she is already contracted to do a wedding that day with a local builder named Jenny South, but that if
the price is right she may be prepared to abandon that contract and do the Mayor’s daughter’s wedding instead. “But it
would cost you at least £50 per head,” she says to Rahul, knowing that Jenny was only going to pay £25 per head for 100
guests. Rahul responds that this is more than he was thinking of paying, but that, “If you promise to make all of the food
yourself, make sure that there are ample vegetarian canapés as well as meat dishes, and serve each course promptly within
a two-hour window” he will meet this price. “You’re on,” Sarah says, continuing – “And don’t worry darling. It’ll all have my
personal touch with quick service and plenty of choices for the ‘fid fads’”. Sarah has a habit of using this term to refer to
anybody with particular dietary requirements. “It’ll be the best wedding food any of them have ever tasted!” she concludes
before hanging up the phone.
Sarah does cancel Jenny’s contract and caters for Rahul’s event instead, leaving Jenny very angry and upset when the
alternative caterer she is forced to engage serves almost inedible food and ruins her wedding.
The event is, in Rahul’s words, “a total disaster”. First, he complains that there were hardly any vegetarian canapés, with
“Over 90% of them meat”. So much so that he says he was forced to go out to Waitrose during the first hour of the event to
buy some vegetarian canapés to supplement what Sarah had provided: this cost him £500. Secondly, he is furious that,
although the first course was served within ten minutes of the guests sitting down, the next course took over an hour and a
half to arrive, with dessert then served almost immediately after the second course had been consumed. He also suspects
that Sarah did not make all of the food herself, since, “With your reputation, you can’t have been responsible for that awful
Sarah responds angrily to all of these allegations, pointing out that she oversaw all food preparation, the food was served
within two-hours and that in her opinion, “Ten percent of that veggie muck is more than ample”.
Rahul feels that he has been misled and that Sarah has failed to deliver her side of the bargain. The Mayor has also refused
to pay Rahul £10,000 of the £20,000 agreed price, citing the delay, the preponderance of meat in the canapés and the fact
that it felt like, “mass-produced garbage”.
Rahul is also furious because his venue Seaside Towers burnt down the day after the Mayor’s daughter’s wedding so he
could not host the local Rotary Club’s annual dinner dance. The Rotary Club is threatening legal action for breach of
contract. Rahul thinks this is unfair. “How could I hold the dance in a hall that wasn’t there?” he complained in a subsequent
email to the Rotary Club secretary, Neville Rayner. Neville in turn wrote back to point out that they would have been quite
happy to have relocated to Seaside Events’ sister hotel, Pier Oh, but that Rahul refused to accommodate this. Rahul bitterly
resents this, and pointed out to Neville in correspondence that Pier Oh was already booked out to the local bingo club,
which could not be moved.
(1) Advise all parties on any potential claims and liabilities which could arise from the above facts.
(2) Draw up a brief contract which would have encapsulated the terms orally agreed by Rahul and Sarah for the
catering of the Mayor’s daughter’s wedding.

epistemological positions Superior

epistemological positions Superior

Pick one of the following epistemological positions and argue for its superiority over the other option: (1.) Skepticism about the external world, (2.) The rejection of skepticism about the external world.

Media’s influence on beauty

Media’s influence on beauty

My final paper is a research paper that has to include some original or thoughtful ideas about media and its influence on a subject, which I chose to be beauty. I have to have at least 5 sources but because the paper is supposed to be 9-12 pages long, I think that having 6-7 sources would be better. I think that 10-11 pages is suitable for this paper, there’s no real requirement other than 9 being the minimal and 12 being the maximum, so you have free range between those limits to write this paper in those length. These sources, my professor said, doesn’t all have to be scholarly sources but they just need to be relevant to the topic at hand and correctly cited in the works cited (MLA form).