Physics 237, Midterm Exam #3

Tuesday April 24, 2018

8.00 am – 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.

 

You are required to complete the following Honor Pledge for Exams.  Copy and sign the pledge before starting your exam.

 

“I affirm that I will not give or receive any unauthorized help on this exam, and that all work will be my own.”

 

_____________________________________________________________________________

 

_____________________________________________________________________________

 

_____________________________________________________________________________

 

Name:  ______________________________________________________________________

 

Signature:  ____________________________________________________________________


 

One-Electron Atoms – Details

 

The following table lists the n = 1, n = 2, and n = 3 wavefunctions of the one-electron atom.

 

 

In these wavefunctions, the parameter a0 is defined as

 

 

The energy of each wavefunction is equal to

 

 

 


 

 

 

 

The radial probability density for the electron in a one-electron atom for n = 1, 2, 3 and various values of l.


 

 


Problem 1 (35 points)                                                                        ANSWER IN BOOKLET 1

 

Consider the first few low-lying 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 two-electron system, , can be written as combinations of the following four functions

 

 

a)     Write down all possible total wavefunctions for the ground state of Helium (n1 = 1 and n2 = 1).

 

b)    Write down all possible total wavefunctions for the first low-lying excited states of Helium (n1 = 1 and n2 = 2).

 

c)     Make an energy diagram showing the location of the low-lying energy levels of Helium (n1 = 1 and n2 = 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 low-lying 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 low-lying 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.

 

E2

 

2P3/2

 

E1

 

2P1/2

 

 

 

 

 

 

 

E0

 

2S1/2

 

 

a)     What are the Landé 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 E0, E1, and E2, Ķb, 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 one-dimensional 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)     File source: http://commons.wikimedia.org/wiki/File:Pieter_Zeeman.jpgAustralia

 

b)    In which country was Pieter Zeeman born?

a)     The Netherlands

b)    USA

c)     Canada

d)    Brazil

e)     China

f)     Australia