Description of the EDU testbankscimath.unl.edu/wba/TestBanks/KUiterwaal.pdfDescription of the EDU testbank Quantum Mechanics ... Matrix multiple choice, A ... ($s,"a proton","an electron","a

EDU Testbank Quantum Mechanics 1 / 30 July 2004

Description of the EDU testbank Quantum Mechanics

Kees Uiterwaal, July 2004

The EDU testbank Quantum Mechanics was designed for use in an introductory quantum mechanics course, such as UNL’s PHYS461. The testbank contains 30 questions that are grouped together into six topics, as listed below. In the list, the type of each question is indicated in italics, followed by the letter “A” if the question uses an algorithm. 1. TOPIC 1: Introduction

1.1. Tossing a Coin numeric, A 1.2. Photoelectric Effect numeric, A 1.3. Blackbody Radiation inline 1.4. Wien's Law multiple choice, A 1.5. Compton Scattering numeric, A 1.6. Polarized Light multiple choice, A 1.7. Wavelength of massive particle numeric, A 1.8. Radioactive decay numeric, A 1.9. Diffraction from a slit multiple choice 1.10. Atomic unit of time multiple choice 1.11. Wavefunction Spreading numeric, A 1.12. Founding Fathers matching

2. TOPIC 2: Particle in a box 2.1. Particle in a Box inline, A 2.2. Square Well multiple choice, A 2.3. Number of states in square well multiple choice, A 2.4. Tunneling multiple choice, A

3. TOPIC 3: Harmonic oscillator 3.1. Harmonic Oscillator numeric, A 3.2. A Quantum Mechanical System inline, A 3.3. Parity multiple choice, A

4. TOPIC 4: Spherical harmonics 4.1. Spherical Harmonics matching 4.2. Raising / Lowering Operators multiple choice, A

5. TOPIC 5: Hydrogen-like atoms 5.1. Hydrogen atom line multiple choice, A 5.2. Electron Configuration inline, A

6. TOPIC 6: Mathematical description of QM 6.1. Linear Operators inline 6.2. Matrix multiple choice, A 6.3. Self-Adjoint Operators multiple selection 6.4. Identity Operator multiple choice 6.5. Expectation Value fill in the blanks 6.6. CSCO inline 6.7. Foundations of QM inline


Below, each question is discussed in more detail. We first give a typical screen shot of each question as it displays in an assignment. In some cases, this is followed by a literal copy of the algorithm code used, with red arrows connecting certain variable names to positions at which their values occur in the screen shot. Finally, a short text explains some details. The emphasis is not so much on quantum mechanics but rather on the use of EDU as a testing tool. At various points I have taken the opportunity to point out some bugs or to signal layout problems, hoping that this will contribute to solving some of these issues. On a more general level, I also want to point out here that EDU really is a great tool. I have used it already more than once in my classes and will continue to do so in the future. 1. TOPIC 1: Introduction

1.1. Tossing a Coin numeric, A

The quantities p = probability for “heads”, T = number of tosses, and H = number of heads are all randomly chosen (algorithm). The numerical answer is calculated from

(1 )H T HTans p p

H−

= −

. To make the question more visually attractive, we show T

coins, of which there are H heads and T H− tails. Exploiting the fact that T can never be more than 10 in the algorithm, this is accomplished by always showing 10 pictures, some of which are in fact blank. (As shown here, we have 4 heads, 2 tails, and 4 (invisible!) blank pictures.) There seems to be no for…next like statement in EDU, which could do the same thing more easily using less lines of code.


1.2. Photoelectric Effect numeric, A

A rather standard question using randomized values for both λ and W. The photon

energy is phhcEλ

= . With m being the mass of an electron, the speed is

ph2( )E Wans

m−

= , unless the argument of the square-root turns out to be negative, in

which case the answer is taken to be 0. An acceptable units is m/s. Units have to be converted correctly to get the correct answer.


1.3. Blackbody Radiation inline

An inline question that uses fixed values, so the answers are fixed as well. Note that the use of color and the horizontal bars contribute much to the readability of this question.


1.4. Wien's Law multiple choice, A

This question displays a temperature in either degree celsius (°C) or kelvin. Both cases are shown here. A random temperature T in kelvin (K) is generated for internal use. If T is less than 1000 K it is converted to celsius and displayed as such. The correct answer is

given by 3

9

2.8977685 10(nm) (K) 10

ansT

λ−

−

×= = . Distractors are generated randomly, and it is

checked that all possible answers are different using the EDU condition statement.


1.5. Compton Scattering numeric, A

The algorithm is

$ang = 20+rint(21) $angdisplay = numfmt("00.0",$ang)

$lamscat = 10+rint(21) $lamscatdisplay = numfmt("00.0",$lamscat)

$deltalam = 6.63E-34/9.11E-31/3.00E8*(1-cos($ang/180*3.14159265))$lam = $lamscat*1E-12-$deltalam $freq = 3.00E8/$lam

Note how the randomly generated numbers are displayed using the EDU numfmt command. This gives the programmer full control over the number of significant digits. This is important because we want students to learn that there is a meaning to the number of digits given in an answer, so we must set the tone. The answer is the number $freq, the units should be “Hz” or anything equivalent.


1.6. Polarized Light multiple choice, A

The algorithm is

$ang1 = range(10,30) $ang2 = range(60,90) $ang1f = numfmt("##.00",$ang1) $ang2f = numfmt("##.00",$ang2)

$delang = ($ang1-$ang2)/180.0*3.14159265 $flux = range(1,9)*1000

$t = (cos($delang))^2*$flux $ans = numfmt("#,###",$t)

$wrong1 = numfmt("#,###",abs(cos($delang)*$flux))$wrong2 = numfmt("#,###",abs(sin($delang)*$flux))$wrong3 = numfmt("#,###",(sin($delang)^2)*$flux) $wrong4 = numfmt("#,###",0.0)

For an incoming photon flux Φ ( = $flux ) and and angle θ ( $delang =

$ang1−$ang2 ) the answer is 2cos ( )θΦ ( = (cos($delang))^2*$flux ). Note the use of the numfmt statement to avoid meaningless long sequences of digits in the question. Four distractors are generated.


1.7. Wavelength of massive particle numeric, A

The algorithm is

$s = rint(4) $a_part = switch($s,"a proton","an electron","a deuteron","a muon")

$m = switch($s,1.67E-27,9.11E-31,2*1.67E-27,1.88E-28) $factor = rint(2,9)

$lam = 1.06E-34/(sqrt($factor^2-1)*$m*3.00E8)/1E-10 $lamans = numfmt("0.00E00",$lam)

The random variable $s is used to choose the particle ( $a_part ) to be “a proton”, “an electron”, “a deuteron”, or “a muon” in a switch statement, and sets the chosen

particle’s mass as well. For a wavelength λ and mass m the momentum is hpλ

= , so the

total energy E is given by 2 2

2 2 2 2 20 0 2

h cE E p c Eλ

= + = + (with 20E mc= ). The energy E

must be equal to f ( = $factor) times 0E , so we get 2 2

2 2 2 20 0 2 10 2

(angstrom)10 1

h c hE f E Emc f

λλ −

= = + ⇒ =−

( = $lamans )


1.8. Radioactive decay numeric, A

The algorithm is

$nfmt = numfmt("0.00E00",10^(rand(6,9))) $avgltfmt = numfmt("0.00E00",10^(rand(-9,-3))) $fraction = rand(0.2,0.8)

$time = numfmt("0.00E00",-ln($fraction)*$avgltfmt)$ans = $nfmt*$fraction

We have radioactive decay of particles with an average lifetime τ ( = $avgltfmt ). The activity at t = 0 is 0A ( = $nfmt ). At some time ( = $time ) later, we have an activity of /

0 0tA e fAτ− = ( f = $fraction ) . We have / lnte f t fτ τ− = ⇔ = −

1.9. Diffraction from a slit multiple choice

No algorithm, but the order the answers are presented in is random. Answer: there will be no diffraction “if the opening is very large”.


1.10. Atomic unit of time multiple choice

No algorithm, but the order the answers are presented in is random. Answer: “It equals the time the electron in a Bohr hydrogen atom with n = 1 needs to travel one Bohr radius”


1.11. Wavefunction Spreading numeric, A

The algorithm is

$s = range(0,4)

$particle = switch($s,"an electron","a proton","a gold atom (197Au)","a virus with a mass of 9.00 femtogram","a 1.00-gram particle")

$mass = switch($s,9.11E-31,1.66E-27,3.27E-25,9.00E-15,1.00E-3) $wangstrom = range(3,10)*1E-10

$wvirus = range(1,10)*1E-6 $wheavy = range(1,10)*1E-2

$w0 = if(lt($s,3),$wangstrom,if(eq($s,3),$wvirus,$wheavy)) $w0f = numfmt("0.00E00",$w0)

$time = ($w0)^2*$mass/(8*1.06E-34)

A wavepacket of a particle ( = $particle ) with mass m ( = $mass ) having initial

width 0w ( = $w0f ) spreads to 2 times its initial width in a time 2

0

8w mt

h= . The

random number $s selects “an electron”, “a proton”, “a gold atom (197Au)”, “a virus with a mass of 9.00 femtogram”, or “a 1.00-gram particle”, and also, in the next line of code, its mass. The initial width $w0f is generated in such a way that a realistic value for the particle selected results.


1.12. Founding Fathers matching

Supposedly, the order of the three descriptions and the order of the three graphics should be randomized. Let me report a bug here: The software always gives “This person could …” as the third description, and always shows Max Planck’s picture as the third one.


2. TOPIC 2: Particle in a box

2.1. Particle in a Box inline, A

The algorithm is:

$s = rint(2) $apart = switch($s,"an electron","a proton") $part = switch($s,"electron","proton")

$m = switch($s,9.11E-31,1.66E-27) $L = rand(1.0,9.9) $Lf = numfmt("0.00",$L)

$eneV = (1.06E-34*3.14159265)^2/2/$m/($L*1E-10)^2/1.60E-19


Note how this inline question gains readability through the use of color and horizontal bars. Two unattractive aspects of this question: 1. After each formula there is a sequence of space characters, in HTML code: These get generated by the software one at a time whenever the Java-based detachable editor is opened and then closed in the code development phase. This is a bug. These characters can be eliminated by hand, but that is a tedious procedure. 2. Note that the standard font for displaying questions (probably Arial) and the standard font for diplaying MathML-based formulas are very different; this results in a jagged, nonuniform layout on the screen. This seems hardly avoidable using the standard EDU development tools.


2.2. Square Well multiple choice, A

The algorithm is

$s1 = rint(3) $s2 = rint(3) $s3 = rint(3)

condition:not(eq($s1,$s2)) condition:not(eq($s1,$s3)) condition:not(eq($s2,$s3))

$pic1 = switch($s1,"fig1","fig2","fig3") $pic2 = switch($s2,"fig1","fig2","fig3") $pic3 = switch($s3,"fig1","fig2","fig3")

$path1 = strcat("../classes/phys461/","$pic1",".jpg")$path2 = strcat("../classes/phys461/","$pic2",".jpg")$path3 = strcat("../classes/phys461/","$pic3",".jpg")

$ans = if(eq($s1,2),1,if(eq($s2,2),2,3))

This algorithm randomizes the order in which the three graphics files fig1.jpg, fig2.jpg, and fig3.jpg are displayed.


2.3. Number of states in square well multiple choice, A

Algorithm:

$ashow = rand(0.5,9.5,3) $V0show = rand(1,9,3)

$a = $ashow*1.0E-10 $V0 = $V0show*1.60E-19 $m = 9.11E-31

$hbar = 1.06E-34 $no = sqrt(2*$m*$V0*$a*$a/($hbar*$hbar))/(3.14159265/2)

$ans = int($no+1) $wrong1 = 0 $wrong2 = rint(1,10) $wrong3 = rint(1,10)

condition:not(eq($ans,$wrong2)) condition:not(eq($ans,$wrong3))

condition:not(eq($wrong2,$wrong3)) Three distractors are generated. The answer “0” is always wrong. Using the condition command it is checked that no answers accidentally equal each other.


2.4. Tunneling multiple choice, A

Algorithm:

$v = 10.0^rand(5,7) $m = 9.11E-31 $K = 0.5*$m*$v^2

$KeV = $K/1.6E-19 $V0eV = rand(1.02,1.15)*$KeV

$V0 = $V0eV*1.6E-19 $aAng = rand(0.5,3)

$a = $aAng*1E-10 $kappa2 = 2*$m*($V0-$K)/(1.06E-34)^2

$k2 = 2*$m*$K/(1.06E-34)^2 $y = 2*sqrt($kappa2)*$a

$sinh = (e^$y-pi^(-$y))/2 $t = 4*$kappa2*$k2/(($k2+$kappa2)^2*$sinh+4*$kappa2*$k2)

$fluxin = range(1,9)*10000 $fluxout = $t*$fluxin

$wrong1 = sqrt($t)*$fluxin $wrong2 = $K/$V0*$fluxin $wrong3 = rand(0,0.5)*$fluxin $wrong4 = rand(0.6,0.9)*$fluxin

A fairly standard sort of question.

$V0eV


3. TOPIC 3: Harmonic oscillator

3.1. Harmonic Oscillator numeric, A

Algorithm: $hwev = rand(0.1,0.4)

$w = $hwev*1.6E-19/1.06E-34 $wfmt = numfmt("0.00E00",$w)

$n1 = range(0,5) $deltan = range(2,6)

$n2 = $n1+$deltan $deltae = $deltan*$hwev

Here is an example of a code that mixes MathML with randomized variables. For instance, the “ 0 ” in the displayed text is internally represented as <math display='block'>

<mrow><mrow><mo>|</mo><mi>$n1</mi><mo>〉</mo></mrow></mrow>

</math> ,with the randomized variable $n1 embedded in MathML code.


3.2. A Quantum Mechanical System inline, A

This question uses some of the great possibilities of an inline question, such as pull-down lists and formula-type answers. The use of horizontal bars and color enhances the readability. Here, as in other questions, the student is instructed to use three significant digits in numerical answers. In the code, we tolerate up to 5% deviation from the calculated answer; this makes sure that a correct answer using three significant digits is always considered correct. In part 7. there seems to be a problem, with MathML code generating output that does not line up with the body text.


3.3. Parity multiple choice, A

Algorithm:

$n1 = range(0,10) $n2 = range(0,10)

$halfn1 = $n1/2 $halfn2 = $n2/2 $even1 = eq($halfn1,int($halfn1))$even2 = eq($halfn2,int($halfn2))$even = $even1*$even2 $odd = not($even1+$even2) $ans = if($even,1,if($odd,2,3))

condition:not(eq($n1,$n2)) If the random variables $n1 and $n2 are both even, the parity is +1. If they are both odd, the parity is 1− . When one is even and the other one is odd, there is no definite parity. The algorithm detects even-ness by checking if the integer part of $n1/2 equals $n1/2. A nested if(…) statement generates the correct answer. A condition statement makes sure that $n1 and $n2 are not equal, because that would make the question a silly one.


4. TOPIC 4: Spherical harmonics

4.1. Spherical Harmonics matching

Similar in structure to Topic 1, question 12 (and affected by the same software bug).


4.2. Raising / Lowering Operators multiple choice, A

Algorithm: $s1 = rint(2) $op = switch($s1," L+ "," L&minus

") $el = rint(4)

$em = range(-$el,$el) $thph = "(&theta ,&phi

)" $newem = switch($s1,$em+1,$em-1)

$badel = switch($s1,$el+1,$el-1) $ans = if(gt(abs($newem),$el),"0","Y$el,$newem$thph")

$badans = if(le(abs($newem),$el),"0","Y$el,$newem$thph")

Both the operator and the spherical harmonic the operator works on are randomly chosen. A few distractors are generated, too. I have no clue why sometimes a negative sign appears in front of a zero, as in the second answer in the picture above, where it reads: “ 0,1( , )Y θ ϕ− ” instead of the expected “ 0,1( , )Y θ ϕ ” (without a negative sign). [Note that the algorithm (as always in the present text) was copied directly from the Question Bank Editor. Because the character combination ( , )θ ϕ occurs several times in the question text, I decided to define $thph="(θ,φ)"; This contains the HTML codes for the Greek characters θ (θ) and ϕ (φ). When the editor displays code, it apparently displays a line break whenever it encounters a semicolon (;) , which normally means a line break inside an algorithm, but not when HTML character codes are used. This explains the raggedness of the algorithm code shown here.]


5. TOPIC 5: Hydrogen-like atoms

5.1. Hydrogen atom line multiple choice, A

Algorithm:

$nf = rint(1,5) $deln = rint(1,5)

$ni = $nf+$deln $oneoverlam = 1.095E5*(1/$nf^2-1/$ni^2)

$lam = int(1/$oneoverlam/(1E-7)+0.5)$nf1 = rint(1,5)

$deln1 = rint(1,5) $ni1 = $nf1+$deln1

condition:not(eq(ni,ni1)) condition:not(eq(nf,nf1)) A standard formula for the Bohr model of the hydrogen atom is tested. A few distractors are generated, with checks to make sure no two answers are accidentally the same.


5.2. Electron Configuration inline, A

Algorithm: $z = range(1,36) $code = switch($z-

1,"H","He","Li","Be","B","C","N","O","F","Ne","Na","Mg","Al","Si","P","S","Cl","Ar","K","Ca","Sc","Ti","V","Cr","Mn","Fe","Co","Ni","Cu","Zn","Ga","Ge","As","Se","Br","Kr")

$name = switch($z-1,"hydrogen","helium","lithium","beryllium","boron","carbon","nitrogen","oxygen","fluorine","neon","sodium","magnesium","aluminium","silicon","phosphorus","sulphur","chlorine","argon","potassium","calcium","scandium","titanium","vanadium","chromium","manganese","iron","cobalt","nickel","copper","zinc","gallium","germanium","arsenic","selenium","bromine","krypton")

$outer = switch($z-1, "1s", "1s2", "2s", "2s2", "2s22p", "2s22p2", "2s22p3", "2s22p4", "2s22p5", "2s22p6", "3s", "3s2", "3s23p", "3s23p2", "3s23p3", "3s23p4", "3s23p5", "3s23p6", "4s", "4s2", "4s23d", "4s23d2", "4s23d3", "4s3d5", "4s23d5", "4s23d6", "4s23d7", "4s23d8", "4s3d10", "4s23d10", "4s23d104p", "4s23d104p2", "4s23d104p3", "4s23d104p4", "4s23d104p5", "4s23d104p6")

$He = "1s2" $K1 = "2s22p6" $K2 = "3s23p6" $Ne = strcat("$He","$K1") $Ar = strcat("$Ne","$K2") $whole = if(ge($z,19),strcat("$Ar","$outer"),if(ge($z,11),strcat("$Ne","$outer"),if(ge($z,3),strcat("$He","$outer"),"$outer")

))

This was a challenging question to write. Using the random variable $z we pick one of the first 36 atoms in the Periodic Table of Elements and generate its electron configuration. The result is stored in the variable $whole. Because the first atom in the Periodic Table has atomic number one, but variables used in switch(…) statements start from the value zero, we have used $z-1 in these switch(…) statements.


6. TOPIC 6: Mathematical description of QM

6.1. Linear Operators inline


The question type “inline” allows us to ask about 10 operators at the same time. I found the software “MathType” very convenient for generating formulas of the sort shown here. Once the MathType “translator” has been set up correctly, the user can copy & paste MathType-generated formulas directly into the testbank code as MathML.

6.2. Matrix multiple choice, A

Complex numbers? Yes, complex numbers, even though as far as I know EDU algorithms cannot handle them. The real and imaginary parts of each matrix element were separately and randomly generated in the algorithm and then combined in a MathML code. There are switch(…) statements embedded in the MathML code to make sure the desired sort of matrix results.


6.3. Self-Adjoint Operators multiple selection

No algorithm used. Note that operators are displayed using a bold font (HTML code …). The dagger symbol was generated using &dagger;, and the “does not equal” sign using ≠. Using HTML codes like this a more visually smooth layout can be achieved than using MathML (compare Topic 6, question 6) Of course, however, MathML can do a lot more than standard HTML.

6.4. Identity Operator multiple choice

A very simple question, the first one I wrote in this EDU course.


6.5. Expectation Value fill in the blanks

Finding good distractors is half the work for this sort of question…


6.6. CSCO inline


No algorithm. Most of the work for this inline question went into preparing the formulas and making everything display the way it should.

6.7. Foundations of QM inline

Inline questions like this one are an ideal way to test if the terminology of a course is getting across.

Description of the EDU testbankscimath.unl.edu/wba/TestBanks/KUiterwaal.pdfDescription of the EDU testbank Quantum Mechanics ... Matrix multiple choice, A ... ($s,"a proton","an electron","a

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