Physics 237, Midterm Exam #3
Tuesday April 24, 2018
8.00 am Ð 9.30 am
Do not turn the
pages of the exam until you are instructed to do so.
Exam rules: You may
use only a writing instrument while
taking this test. You may not consult any calculators, computers,
books, nor each other.
Problems 1 and 2 must be answered in exam booklet 1. Problems 3 and 4 must be answered in exam booklet 2. The answers need to be well motivated and expressed in terms of the variables used in the problem. You will receive partial credit where appropriate, but only when we can read your solution. Answers that are not motivated will not receive any credit, even if correct.
At the
end of the exam, you need to hand in your exam, the blue exam booklets, and the
equation sheet. All items must be
clearly labeled with your name, your student ID number, and the day/time of
your recitation. If any of these items are missing, we will
not grade your exam, and you will receive a score of 0 points.
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ÒI affirm that I will not give or receive any unauthorized help on this exam, and that all work will be my own.Ó
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OneElectron Atoms Ð Details
The following table lists the n = 1, n = 2, and n = 3
wavefunctions of the oneelectron atom.
In these
wavefunctions, the parameter a_{0}
is defined as
The energy of
each wavefunction is equal to
The radial probability density for the
electron in a oneelectron atom for n
= 1, 2, 3 and various values of l.
Problem 1 (35 points) ANSWER
IN BOOKLET 1
Consider the first few lowlying
states of the Helium atom. When the
Helium atom is in its ground state, both electrons are in n = 1 states. When the
Helium atom is in one of the first excited states, one electron is in an n = 1 state and the other electron is in
an n = 2 state. The spatial wavefunction of electron i can be written as
where a contains information about the spatial quantum numbers.
The spin wavefunction of the
twoelectron system, , can be written as combinations of the following four
functions
a)
Write down all
possible total wavefunctions for the ground state of Helium (n_{1} = 1 and n_{2} = 1).
b)
Write down all
possible total wavefunctions for the first lowlying excited states of Helium (n_{1} = 1 and n_{2} = 2).
c)
Make an energy
diagram showing the location of the lowlying energy levels of Helium (n_{1} = 1 and n_{2} = 1, 2), assuming there is
no Coulomb interaction between the electrons. Label each level with the n and l values of each of the two electrons and their total spin. Note: the actual location of these
energy levels is not important, but their relative position is.
d)
Now include the
effect of the Coulomb interaction between the electrons, but ignore the
exchange force. What is the effect
of the Coulomb interaction on the energy of the states shown in the diagram
constructed in c)? In the energy
diagram, indicate the shifts of the energy levels of the lowlying states of
Helium due to the Coulomb interaction. Label each level with the n and l values of each of the two electrons and their total spin. Note: the relative shifts of the levels
are important and need to be correctly motivated and drawn.
e)
Finally, include
the effect of the exchange force.
What is the effect of the exchange force on the energy of the states
shown in the diagram constructed in d)?
In the energy diagram, indicate the shifts of the energy levels of the
lowlying states of Helium due to the exchange force. Label each level with the n and l values of each of the two electrons and their total spin. Note: the relative shifts of the levels
are important and need to be correctly motivated and drawn.
Problem 2 (35 points) ANSWER
IN BOOKLET 1
Consider the three lowest energy
levels in Na, shown in the Figure below.
E_{2} 

2P_{3/2} 


E_{1} 

2P_{1/2} 








E_{0} 

2S_{1/2} 

a)
What are the LandŽ g factors
for these levels?
b)
When the atom is
placed in a weak magnetic field, the energy levels split. Draw an energy level diagram showing the
energy levels and determine the corresponding energies (in terms of E_{0}, E_{1}, and E_{2},
µ_{b}, and B).
c)
Which
transitions between the 2P and the 2S energy levels are possible? Explain why you selected these
transitions?
Problem 3 (30 points) ANSWER
IN BOOKLET 2
In a
onedimensional system the number of energy states per
unit energy is
where L is the length of the system and m is the mass of the electron. There are N electrons in this sample and each state can be occupied by at
most two electrons.
a)
Determine the
Fermi energy at 0 K.
b)
Determine the
average energy per electron at 0 K.
Problem 4 (5 points) ANSWER
IN BOOKLET 2
a)
In which country
was Didi Gregorius born?
a)
The Netherlands
b)
USA
c)
Canada
d)
Brazil
e)
China
f)
Australia
b)
In which country
was Pieter Zeeman born?
a)
The Netherlands
b)
USA
c)
Canada
d)
Brazil
e)
China
f)
Australia