Problem 1: 1D Problem (17 pts)
A solid nanowire is subjected to a diffuse source of high energy gamma rays, resulting in a small number of local defects. An electron in the nanowire becomes bound and trapped by one of the local defects, whose potential energy we model as V(x) = −aδ(x) . We would like to know the rough spatial extent of the trapping. To do so, find the value x0 such that the electron has a 50% probability of being found in a range around the defect given by |x| < x0 .
Problem 2: Low Dimensional Spaces, Dirac (17 pts)
Consider a three-dimensional ket space spanned by basis vectors |1⟩ , |2⟩ , and |3⟩ and operatorsA(̂) andB(̂) represented by the matrices below where a and b are real:
a.) WriteA(̂) andB(̂) in terms of |1⟩ , |2⟩ , and |3⟩ and corresponding bras. (6pts)
b.) Is the spectrum of observables A and B degenerate? Recall that the spectra of an operatorQ(̂) is given by solving det(Q(̂) − λ 1(̂)) = 0 for all values of λ . (6pts)
c.) DoA(̂) andB(̂) commute? (5pts)
Problem 3: Hydrogenic Atoms (66 pts)
As discussed in class, you will be solving for the eigenenergies and eigenfunctions of the hydrogen atom. The starting point is qualitatively identical to the case of treating the vibrational and rotational properties of a two chemically bound nuclei with quantum mechanics. The I’m independent Schrodinger equation for the system is:
By allowing ψ = ψtrans ψr ψθφ and switching to the reduced mass reference frame.
(which we did in class for the diatomic molecule, and I won’t replicate here), the above can be rewritten as … .
with
And
But in (2),we don’t care about the translational properties of hydrogen, we care about the internal properties ψr and ψθφ and associated energies. So, we toss out the translational part of (2). Also, if we think for a moment about what potential energy is at play, it is the Coulomb potential between and electron and a proton; or for future flexibility in modeling ions, let’s say between an electron and Z protons … .
But this is a spherically symmetric “central potential,” and we have already solved the angular part for all of those kinds of problems to be ψθφ = ylm !! So we can now just jump straight to the SE for a central potential in spherical coordinates, which is
and with (3) into (4) and some algebra we have … .
So, your task from here amounts to solving equation (5). You will proceed in a similar way as we have done with other non-trivial equations, by first considering a well-known differential equation, called the “associated Laguerre differential equation”:
The equation has two general solutions ifv and β in (6) are non-negative integers.
which are “confluent hypergeometric functions of the first kind” and Laguerre polynomials, respectively. We have come across the 1F1 before, and as you may recall, 1F1 blows up liker2 , so the appropriate probability density function, in the radial coordinate in spherical polar coordinates, would go like … | 1F1 |2r 2 dr ∝ r 6 … .. and not be normalizable in the limit as r → ∞ . So, we toss it! Ok, now your part. [Each part 6 points]
a.) Assume the radial wavefunction can be written as ψr = R(r)r−1 . Now, simplify (5) to aform. like R ′′ + = 0. Write out the entire equation with the blank filled in and label it as equation (8).
b.) Proceed toward a solution by performing a non-dimensionalization procedure by introducing dimensionless energy λ and radius p with the following definitions:
Use these transformations to eliminate all occurrences of E andr in equation (8) that you just derived. Take care to appropriately transform. the derivatives. Eventually arrive at an equation of form. Label it equation (12).
c.) Now continue the process by assuming the additional functional form.
in and label that equation as (14).
d.) Now continue the process by assuming one final functional form. for f:
Simplify (14) to a form. pg′′ + = 0. Derive the entire equation with the blank filled in and label that equation as (16).
e.) Equation (16) should be in the form. of the associated Laguerre differential equation. Based on the information provided thus far in the problem, especially relating to the Laguerre differential equation, its solution and constraints, what are (I) the solutions to (16) with yet unsolved normalization constant and (II) the conditions that must be placed on λ − l − 1 ≡ nr (which we have defined as the “radial quantum number” nr ) and the conditions that must be placed on 2l + 2?
f.) Using the condition just found on λ − l − 1 , must λ be a positive integer or a negative integer? Show your work.
g.) Now relabel λ as n , as it is actually the “principal quantum number” known to you from introductory chemistry. Use logic & inequalities at your disposal to show how the principle & angular momentum quantum numbers are related, specifically, prove l ≤ n − 1, n ∈ {1,2,3,4 … . }
h.) Use the newly found conditions and labeling conventions, along with the transformation equations introduced in part b to determine the eigenenergies of the hydrogen atom. Express your final answer in terms of the modified Bohr radius, aμ , where Additionally, verify that E1 ≈ −13.6 ev.
i.) Use your result from part e.) (I), (15), (13) and our first assumption from part a, to write out ψr,nl (r) and label it equation (20). Use the following notational definition to make things look clean:
j.) Determine the normalization constant, Nnl . For this, we need to utilize an integral relation true for the Laguerre polynomials:
as well as the fact that the Laguerre polynomials are real-valued. NOTICE: I chose a newer normalization convention/approach for this problem, as a result, the normalization constant will be slightly different from the one I presented in class.
k.) Lastly (and finally!) write out the normalized eigenfunction as an explicit function of r, θ, & φ indexed by the appropriate quantum numbers (excluding spin). Note, the final answer should still preserve the aμ and Z notations.