代写PHAS0020作业、代做Python，c/c++，Java程序语言作业、Boundary value作业代写
PHAS0020 Final Assignment 2019
Dr. Christian G¨utschow?
([email protected])
February 2019
Contents
1 Introduction 1
2 Boundary value problems: eigenvalue solutions 2
3 Infinite square well 2
4 Finding the ground-state energy 3
5 Finding the ground-state wavefunction 3
6 Finding the higher energy states 4
7 Beyond the infinite square well 4
7.1 Harmonic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
7.2 Finite square well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
7.3 Coulomb potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
7.3.1 Quantum tunnelling . . . . . . . . . . . . . . . . . . . . . . . . 5
8 General assignment advice 6
9 Assessment 6
10 Submission 6
11 Page limit 7
1 Introduction
In the course so far we have used a wide variety of techniques to give you a flavour
of the sort of problems that can be tackled using computational approaches. In
session 10, we are going to look at one more new aspect of this, including boundary
values in the solving of differential equations. You will then apply this to the problem
for the PHAS0020 final assignment: solving the quantum-mechanical problem of a
particle in a potential well. You will fist solve for an infinite square potential well, and
use the known solutions for this system to check that your results are correct. You will
then be able to extend this to calculate for systems where there is no simple analytical
solution. Before commencing any of the tasks below, make sure you have gone through
and understood the contents of the Jupyter Notebook script for session 10, which will
guide you through the process of finding roots of a function using the secant method,
and hence solving differential equations with boundary value conditions, both of which
you will need to solve this problem.
With thanks to Dr. Louise Dash ([email protected]) who provided most of the material for
this course.
1
2 Boundary value problems: eigenvalue solutions
In this task we are going to be applying our new knowledge of boundary value problems
to the one-dimensional time-independent Schr¨odinger equation,(1)
Initially, we are going to consider one of the most well-known solutions of this—for
a particle in a square potential well of width 2a with infinitely high walls. This is
Figure 1: The infinite square
well potential. Within the well,
the potential is zero, everywhere
else the potential is in-
finite.
shown in Figure 1, and the potential V (x) takes the form
V (x) = (0 if a ≤ x ≤ +a,∞ if |x| > a.
(2)
You will all have studied this in PHAS0022 (or an equivalent course) and should be
familiar with the solutions. These solutions show that the probability of finding the
particle in the region where V (x) = ∞ is zero, and thus the wavefunction must be
subject to the boundary conditions of ψ = 0 at x = a and x = +a. Therefore, this
would seem to be an ideal system to solve using the same approach as we used to
calculate the trajectory of a thrown ball in the Jupyter Notebook for Session 10—we
could transform Equation 1 into two first-order equations, guess some initial values,
and iterate until we find a solution that fulfills the boundary condition at x = +a.
Separating out the Schr¨odinger equation into two first-order equations yields(4)
We know one initial boundary condition, that ψ = 0 at x = a. We would therefore
need to guess an initial condition for φ and then vary this with the secant method (or
another root-finding method) until we find a solution where ψ(x = +a) = 0.
Unfortunately, this approach will not work. The Schr¨odinger equation is a linear
equation, so if we have a solution ψ, any multiple of ψ will also be a solution. Conversely,
a multiple of any candidate wavefunction ψ that does not fulfill the boundary
condition of ψ(x = +a) = 0 will also not fulfill the condition.
This is because the Schr¨odinger equation is an eigenvalue equation—it only has
solutions at particular values of the energy E. For other values, there are no solutions
that fulfill the boundary conditions.
The approach we will adopt, therefore, is not to search for an initial condition on
φ that will give us the required solution, but instead to search on values of E. As the
only effect of φ’s initial condition is to multiply the wavefunction by a constant, we
can choose any starting value. We can then normalize the wavefunction once we have
found a solution if needed.
3 Infinite square well
You now have all the information you need to start on the task. The first part of the
task is to calculate the energy of the ground state. You should be able to do this by
the end of Session 10, and you can get help from the demonstrators for this part of
the task during the session. Start by creating a new notebook from scratch. You will
need to define the following physical constants:
reduced Planck constant ~ = h/2π = 1.0545718 × 10?34 Js
electron charge e = 1.60217662 × 1019 C
electron mass m = 9.10938356 × 1031 kg
proton mass mp = 1.6726219 × 1027 kg
2
You can include more digits of precision than this if you wish, but please work in SI
units. When quoting energies however, you will find it more convenient to convert to
electronvolts ( eV) by dividing the value in Joules (J) by e.
Set the half-width of the potential well a, to be 5 × 1011 m, so that the well will
be 2a = 10?10 m wide. Use N = 1000 for the number of Runge–Kutta calculation
points.
For an infinite square well, we know that the particle will never be outside the
well, where V (x) = ∞, and as V (x) = 0 inside the well, we could just ignore this
in the calculations. However, later in the task you will calculate for other forms of
V (x), so write a suitably-named and documented function that will, for the moment,
return a value of V (x) = 0 for all values of x, the function input1
.
1Hint: you just need one line in the
function at the moment: return 0.0.
Make sure you remember the docstring
as well though.
We have already split the Schr¨odinger equation into two first-order equations,
Equations 3 and 4. You will need to write another function, just like the ones you
used in Sessions 7 and 8, to calculate the right-hand side of these equations2
. Now 2Hint: Unlike the previous functions,
this one will need three arguments as
inputs: a vector r of ψ and φ, a value
of x, and a value for the energy E,
which appears in the RHS of Equation
4.
you need a Runge–Kutta function. Use the fourth-order function you already have,
but you will probably want to make some changes. In particular, you can put the
initial conditions within the function. You can use any initial condition you choose
for φ, but phi = 1.0 is probably a good choice. Instead of running over an array of
time points, the loop will now run over an array of N x-points starting at x = a
and finishing at x = +a, with step size h = 2a/N.
3 The Runge–Kutta function will 3Hint: As you will be asked to change
the array range later on in the exercise,
it might be worth passing the
array as an argument to the Runge–
Kutta function.
need to return an array of the wavefunction values representing ψ(x), although note
that we do not actually need to return an array to represent φ(x) = dψ/dx.
4 Finding the ground-state energy
As in the script for Session 10, you will need to set up a code cell to iterate using
the secant method until you find a value of E that gives the required ψ(x) = 0
at x = a. To do this, you will need two initial guesses for the energy (e.g. using
variable names E1 and E2)
4
, then calculate corresponding wavefunction arrays using 4Hint: choose initial guesses that
are close to the initial ground-state
energy, otherwise you may find the
eigenvalue of one of the excited states
instead!
the Runge–Kutta function. Check the final element of these wavefunction arrays–are
they equal to zero? If not, update the guesses E1 and E2 using the secant method
until you find an energy that does fulfill the boundary conditions5
.
5Hint: You will need to set an appropriate
value for the tolerance, e/1000 is If all has gone well, your code will converge on the ground state reasonably quickly. probably a good choice
To check your result, compare it with the known ground-state energy, by recalling
that the eigenvalues of this system are given by:
This is the point that we anticipate most of you will be able to reach by the end of
Session 10. From this point onwards, you have to complete the task entirely on your
own, with no help from demonstrators or from your peers.
5 Finding the ground-state wavefunction
Now calculate and plot, on an appropriately labelled plot, the ground-state wavefunction.
You may wish to represent the walls of the infinite square well potential on the
plot as well6
.
6Hint:
plt.axvline(x=-a,c=’k’,ls=’-’,lw=2)
will plot a thick black vertical line at
x = a. Adapt this as required.
As we noted earlier when discussing the arbitrary choice of initial condition for
φ, our wavefunction as calculated is not normalised. It would be useful to write a
function that will normalize the wavefunction, so that
Z|ψ(x)|2dx = 1. (6)
You can achieve this by calculating the value of the integral on the left-hand side of
Equation 6, and then dividing the wavefunction through by the square root of this
value. The easiest way of calculating the integral is to use the trapezoidal rule7
, where 7Note that our wavefunction is already
in a convenient form to do this,
as it is a numpy array.
3
an integral with limits a and b can be evaluated as(7)
Having done this, compare your calculated wavefunction with the known ground-state
wavefunction, remembering that the normalized wavefunctions of the system are given
(8)
If all has gone well, you should find a perfect match between your numerical results
and the analytical solutions we already knew.
6 Finding the higher energy states
Once you are satisfied that your ground-state solutions are correct, try finding the
first few excited states (n = 2, 3, 4, as a minimum), and verify that these are also
correct. Comment on how you chose the initial guesses for the energies for these
states.8 Once more, plot these solutions and compare with the known solutions. 8Hint: As analytical solutions for the
energies are not always known, you
will want to find a general approach
to guessing the energies that does not
rely on prior knowledge of the textbook
results.
Does your method also work to find eigenstates for large n (for example, n ～ 20)?
Show this explicitly.
7 Beyond the infinite square well
Hopefully now you have code that works well to solve for the eigenstates of the infinite
square well. However, you probably found it a fair amount of work to calculate
answers that we already knew from a relatively simple analytical solution. In fact,
the main point of doing this was to confirm that your numerical version does indeed
work as expected, before going on to calculate for systems that are not easily solvable
analytically. This is what you will do now, and then compare your results to the
systems you have already met.
We are going to keep the infinite square walls of the potential well, which guarantee
that the particle cannot be found in the V (x) = ∞ region. This means we can keep
the boundary conditions of the original problem.
However, we are going to change the base of the potential well so that it is no
longer square—in effect we are going to embed another potential within the square
well. An example of this is shown in Figure 2.
Figure 2: Potentials can be
embedded within our infinite
square well, as shown by the
blue line. We can calculate for
any arbitrary potential defined
within the surrounding infinite
square well.
This opens up the possibility of solving for a wide range of systems for which there
is no simple analytical solution. By starting with embedded potentials with familiar
forms, you will be able to explore the similarities and differences from the analytical
solutions and get a feel for the physics of the new, embedded system. You will then
be able to extend this to any (arbitrary) embedded potential. You can implement this
simply by changing the form of the potential defined in the Python function earlier.
In each case, calculate at least the lowest four eigenstates (include some of the
higher eigenstates if you wish), and plot the wavefunctions. Also plot the potential
itself, with the energy eigenvalues as horizontal lines.
Compare these results qualitatively to the form of the wavefunctions you would
expect for the standard form of these potentials—explain the similarities and differences
in your text cell commentary. You may also want to attempt a quantitative
comparison using the known eigenvalues and wavefunctions if possible.
4
7.1 Harmonic potential
Set the potential to have the form
This will give you a harmonic potential embedded within the infinite square well, as
shown in Figure 3. You will need to pick an appropriate value for V0. 800e is a good
Figure 3: A harmonic potential
embedded in the infinite
well. Note that the particle
is still constrained within the
infinite walls indicated by the
black lines.
starting point, as it gives a fairly deep harmonic potential, but experiment to find a
value you think is most suitable. In what way do your results differ from the known
results for a simple harmonic oscillator potential? Why?
7.2 Finite square well
Set the potential to have the form
V (x) = (0 if a/2 ≤ x ≤ +a/2,V0 if |x| > a/2.
(10)
Again, you will need to adjust V0, and you may also want to experiment with the
width of the finite square well. How do your results differ from the known results for
a normal finite square well, and why?
7.3 Coulomb potential
Finally, we will consider the Coulomb potential that we have already met in Session 5.
In this scenario, an electron experiences a repulsion due to a potential that falls off
as ～ 1/r, except for very small ranges below a = 1 × 10?11 m where the potential is
zero. In this case, one of the infinite potential walls is placed at r = 0 and the other
one at r = 3a, resulting in
V (r) =0 if 0 < r < a,
V0/r if a ≤ r ≤ 3a,
∞ otherwise.
(11)
This will give you the Coulomb potential embedded within the infinite square well,
as shown in Figure 4. You may want to experiment again with the value for V0, but
Figure 4: A Coulomb potential
embedded in the infinite
well. Note that the particle
is still constrained within the
infinite walls indicated by the
black lines.
adjusting it such that Vmax is about 10 keV is probably a good starting point. It
might be worth converting all energies to keV for this part of the exercise. Make sure
you include a full interpretation of the physics of your results in your notebook.
7.3.1 Quantum tunnelling
In classical physics, the radius of closest approach rc for a particle of energy E is
where V (r = rc) = E, as indicated in Figure 4. In quantum physics, however, the
particle is allowed to tunnel through the potential barrier where V (r) > E, with some
transmission probability T, which is approximately given by(12)
assuming the exponential is the dominant term of the full expression. Write a function
that solves the integral in the exponent of Equation 12, e.g. using the trapezoidal rule
from Equation 7, and use it to estimate the transmission probability for the lowest
eigenstate energy for which E > V (r) near the infinite potential walls.
The shape of the Coulomb potential can also be used to model other phenomena,
such as radioactive decay or stellar nucleosynthesis. For instance, in order for
two hydrogen nuclei to form a heavier nucleus by fusion, one proton would have to
5
tunnel through the repulsive Coulomb potential of another proton. The range of the
attractive strong force is only around a ≈ 1 fm and so
Vmax =Z1Z2e24π0a≈ 1.4 MeV (13)
where Z1 = Z2 = 1 for the two protons. For a diproton system with reduced mass
mr = m1m2/(m1+m2), use your function to give a rough estimate of the transmission
probability assuming that the average thermal proton energy is roughly kBT ≈ 1 keV
at the Sun’s core where the temperature is around 15 million degrees Celsius.9 9You can ignore the infinite potential
walls for this back-on-the-envelope
calculation.
8 General assignment advice
Submit your work as a fully self-contained Jupyter Notebook. Your notebook should
have a similar writing style to a formal report, i.e. use complete and grammatically
correct sentences, avoid bullet lists, and use formal scientific language.
Pay particular attention to your text cell commentary. Because this task is fairly
open-ended, we will only be able to tell what you have done, and why you have chosen
to do it in that way, if you use the text cells to tell us. Likewise, make sure your code
is fully commented, as you may not get full credit if we cannot figure out what it
does.
Your notebook should not include any interactive code requiring user input.
9 Assessment
In this assignment, most of the available marks (65 %) are for successfully calculating
the eigenvalues and eigenfunctions, and analysing and discussing these results in the
text commentary.
The remaining marks will be allocated for the overall quality of your plots (10 %),
the overall quality of your coding style and code comments (10 %), and the overall
quality of your discussion in the text cells (including grammar and spelling, 15 %).
As a guide: if you have successfully completed only the first part of the task (i.e.
calculation of the ground-state energy), including a reasonable text cell commentary
etc., you would receive a maximum grade of around 40 %. A complete, fully correct
solution of the low-lying eigenstates of the infinite square well would expect to receive
a grade of no more than 55 %. Higher grades than this will only be achieved if you
have extended your code for other potentials, and given a sensible analysis of your
results.
10 Submission
This is an individual assignment. You must work on this on your own, with
no help or input from staff or other students. Everything in your submission must
be either completely your own work, or have the source explicitly acknowledged and
suitably referenced. We will be using several methods to check for plagiarism, copying,
and collusion. This assignment will be graded anonymously. Please make sure that:
your name does not appear anywhere in your submission;
your submission title starts with your student number; and
you include your student number at the start of your submitted Jupyter Notebook.
If you have an e-sticker from Student Disability Services please copy
and paste the wording (but not your name!) into a clearly labelled text
cell at the top of your Jupyter Notebook.
6
You can only submit one file on Moodle, so it will not be possible to upload
additional figure files. You may include images as linked URLs (as, for example,
in the script for session 10), and make sure that the original source of the image is
made clear. If you want to include figures you have created yourself, you will need
to upload these to an external hosting service (e.g. Imgur, Flickr, etc.) and ensure
that the images are public. In this case please make it clear in the accompanying
text that you have created this image yourself. You are advised to include a suitably
formatted bibliography or reference section if you refer to any textbooks, lecture notes
(e.g. PHAS0022) or other sources of information.
11 Page limit
This year we have introduced a page limit of the equivalent of 60 sides of A4 paper
for the final assignment.
You can find out how to determine the page count for your completed notebook
by following the procedure laid out at the bottom of the Moodle page for Session 10.

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