Chm 331
Fall 2015, Exercise Set 1
Preliminaries, Waves and Probability
Schrödinger’s Equation, the Parve on a Pole,
Parve in a Well and the Harmonic Oscillator
Mr. Linck
Version: 5.2.
Compiled August 12, 2015 at 10:51:34
Topics of problems, more or less in order:
• Quantum Weirdness
• Mathematica Problems
• Important Equations
• Waves
• Superposition and Operators
• Eigenfunctions, Eigenvalues
• Probability and Expectation Values
• Normalized and Orthogonal Functions
• All of the above in Review
• The Parve on a Pole (POP)
• A Parve on a Ring
• Superposition, again
• Numerical Integration with Boundary Conditions
• Superposition, again
• Finite Well Problems
• Harmonic Oscillator
1.6
2
Figure 1: Device M1. The second magnetic pair simply straighten out the two beams
emerging from the first. The red arrows represent what we think should happen to an up
spin electron and the green ones what should happen to a down spin electron.
1.1. Particles and Waves
Outline factors that are different between a particle and a wave. Use examples from real
life.
1.2. Random Behavior
We usually think that a coin toss is a random event. Can you imagine how, in principle,
you should be able to make it deterministic?
1.3. Stern-Gerlach Experiments
If you block the upper output beam from device M1 (see Figure 1) and put another SternGerlach magnet (another M1 device) to interact with the lower output beam, which way
will that beam be deflected in the second apparatus?
1.4. Stern-Gerlach Experiments
You have a device M1 (see Figure 1) and place two detectors at the output positions, the
upper detector goes “click” when an electron hits it, the lower detector goes “clunk”. Now
you fire five electrons, one at a time, down the input path. What pattern of sounds will
you find? An example answer might be “click, clunk, click, clunk, click.” Please give some
sort of explanation.
1.5. Stern-Gerlach Experiments
We have a two device M1’s (see Figure 1), both up/down. If we put the input of the second
one at the upper output of the first, what is the probability that the parve1 will come out
of the upper output of the second?
1
Our name for a quantum object
Chm 331
Exercise Set 1
1.9
3
Figure 2: Device M2. The second magnetic pair is now twice as long as it was in device
M1, so it not only straightens out the two beams emerging from the first, it bends them
toward each other. The red arrows represent what we think should happen to an up spin
electron and the green what should happen to a down spin electron.
1.6. Stern-Gerlach Experiments
We have a two device M1’s (from lecture), the first up/down and the second in/out. If we
put the input of the second one at the upper output of the first, what is the probability
that the parve will come out of the ”out” output of the second?
1.7. Stern-Gerlach Experiments
We have a two device M1’s (Figure 1), the first up/down and the second upside down. That
is, the first has the south pole of the first magnet up and the second has the south pole
of the first magnet down (in both cases, the north pole is the sharpened pole). If we put
the input of the second one at the upper output of the first, what is the probability that
the parve will come out of the lower (physically lower, the bottom) output of the second?
HINTS: We will analyze this in more detail in a problem below. Take the parve that comes
out of the first M1 device and ask what would happen to it with a second M1 devise in
the same direction. Now you can turn the parve’s spin over 180o , ascertain from which
output the parve will come, then turn both parve and device M1 over 180o again and get
the answer.
1.8. Stern-Gerlach Experiments
Let’s define the output holes of device M1 as the “SN” and the “NS” exit holes, where the
names reflect the polarity of the magnets read from left to right. Now put two M1 devices
together where the second one takes as input the output from the “SN” hole of the first.
(Do you recognize all the first M1 device is doing is to give us a beam of electrons with spin
up?) From the experiments described above can you formulate a function that will predict
the probability that the parve will come out of the the ”SN” exit of the second M1 devise?
You have three data points, so your answer must be a guess; but if you think hard, you will
find a couple more data points that must be true by symmetry and surely you will discover
that the system has to be cyclic.
Chm 331
Exercise Set 1
1.14
4
Figure 3: Device M3. The new magnet in device M3 acts to cause the two beams to combine
and to transversing from right to left again. The red arrows represent what we think should
happen to an up spin electron and the green what should happen to a down spin electron.
1.9. Stern-Gerlach Experiments
The equation for the force on our parve is
Fz = µz
∂Bz
∂z
(1)
where µz is negative for an up spin electron (because the spin is related to the angular
momentum with a proportionality constant that has the charge on the parve). Take the
z axis as pointing up and remember that the magnetic field vector goes from N to S pole.
Here’s how I analyze the force: For a device M1, the field goes from N to S, as does the z
axis, so that is positive. The field is large (intense) at the bottom, and small at the top,
so that is decreasing as z goes up, negative. Finally, µz is negative. Net: positive times
negative times negative equals positive: The force is in the positive z direction. Show that
the result of problem 7 can be predicted using this method.
1.10. Further Stern-Gerlach Experiments
Use the results of the last problem to analyze device M2 (see Figure 2).
1.11. Stern-Gerlach Device M3
Make a guess to determine what will happen if we put an electron of up spin into the left
side of an M3 device and then measure what comes out the right hand side with a simple
M1 device. Also, from where would we get the electron of up spin?
1.12. Learning Mathematica
Work your way through the Mathematica instructions sheet. Not to be collected.
1.13. Mathematica Practice
Use Mathematica to make a table of values of Sin( nπ
12 ) which cover the range of 0 to 2π.
Make a ListPlot of your data. HINT: To make a table of values of 3n from 1 to 531441 (i.e.,
n=12). I would use zz=Table[ xi , {i,1,12}]
Chm 331
Exercise Set 1
1.23
5
1.14. Mathematica Practice
The Mathematica command to pull out a given member of a table is the name of the table,
say zz, with double square brackets. So zz[[3]] would pull out the third member of the table.
Try it on your answer to the last problem.
1.15. Mathematica Practice
We will deal often with two dimensional lists (tables), which normally are called matrices.
Try making the two dimensional table with the command cc = Table[{n + m}, {n,1,3},
{m,1,3}]
1.16. Mathematica Practice
The command to pull out the second row, third column of a two dimensional table (matrix)
is dd[[2,3]] if dd is the name of the table. Extract the third row, first column element from
the matrix in the last problem.
1.17. Mathematica Practice
See if you can figure out how to use Mathematica (in an elegant way) to produce the matrix


1 2 3
 4 5 6 
7 8 9
HINTS: Look at problem ??. I used n + m as my argument and changed the range of m.
The range command {m,1,3} steps m from 1 to 3 in steps of 1; the command {m, 1, 5, 2}
steps m from 1 to 5 in steps of 2, i.e., 1, 3, 5. Take it from there.
1.18. Mathematica Practice
Plot on the same scale the function f = 1 + 2 x + 3 x2 and its derivative with respect to x
over the range of x = 0 to x = 3.
1.19. Mathematica Practice
Integrate the function f of the last problem over the range of x = 0 to x = 3.
1.20. Mathematica Practice
Look at your plot from the problem 18 and estimate the value of x for which the function equals 20.7495. Now use the procedure of guessing an x and evaluating the function for that guess to find the exact value. HINT: You need something like x = 2.2; f
<SHIFT><ENTER> multiple times unless you are really lucky.
1.21. Mathematica Practice
Use the Solve command to find the answer to the last problem. HINTS: Don’t forget the
“double equals”; also, you may have to clear x first with “x =.” or the “Clear” command.
1.22. Using Mathematica
Use Mathematica to enter the function f = 3x4 − 6x3 + 8 and evaluate it at x = 7.
Chm 331
Exercise Set 1
1.32
6
1.23. Using Mathematica
I typed the function in the last problem into Mathematica and struck <SHIFT><ENTER>.
The computer responded with 8. Why? HINT: The M. literature says:
Forgetting about definitions you made earlier is the single most common cause of mistakes when using Mathematica. If you set x=5,
Mathematica assumes that you always want x to have the value 5,
until or unless you explicitly tell it otherwise. To avoid mistakes,
you should remove values you have defined as soon as you have
finished using them.
Recent versions of Mathematica seem to “forget” values faster than older versions, but be
careful.
1.24. Using Mathematica
Use M. to make a table of values of f (from problem 22) over the values of x from 0 to 10
in integer steps.
1.25. Using Mathematica
A function such as that in problem 22 can be evaluated at several points by defining a list
of x values,
x = { 3, 5.4, 7.2 }
and then typing “f <SHIFT ><ENTER >”. This is more convenient than the method in
the last problem if the x are not related in some simple pattern. Try it.
1.26. Using Mathematica
Use M. to find the derivative of f (see problem 22) and evaluate it at x = 4.34.
1.27. Using Mathematica
Plot f (see problem 22) over the range x = -4 to x = 10.
1.28. Using Mathematica
Integrate f (see problem 22) from x = 0 to x = 5.
1.29. Using Mathematica
Do a numerical integration of f (see problem 22) over the limits from x = 0 to x = 5. Compare
your answer to that of the last problem. HINT: Numerical integrations are achieved with
the M. function NIntegrate[func, {var, lower limit, upper limit }] where var is the variable
you are integrating over; for instance, if the integration variable is dx, then var = x.
1.30. Using Mathematica
Make a table of values ranging from 0.2 to 2 in units of 0.2. HINT: Be sure to name your
table.
1.31. Using Mathematica
I named my table from the last problem “tab”. Insert your name into the command
“tab[[5]]” and tell me what it does.
Chm 331
Exercise Set 1
1.36
7
Figure 4: Example Input and Output from Mathematica
1.32. Using Mathematica
Refer to the table in problem 30. Try on your table “Length[tab]”. What does that mean?
Now try “tab[[Length[tab]]]”. What does that do? Does it make sense? HINT: The ability
to select out a member of a table will be useful to you later.
1.33. Using Mathematica
We want to plot the function Exp[-α q2 ] where q varies between -5 and 5 and α are the
values in your table in problem 30. Plot all the functions at once. HINT: If in problem 30
you called your table “tab”, you can simply use that name in a plot command as if it were
the name of the single constant α. Using tab for that makes it into a set of constants from
which M makes a set of plots. Isn’t M. great?
1.34. Using Mathematica
Why might it be best to keep the names you use in M. in lower case letters?
1.35. Using Mathematica
In Figure 4 is some sample input in Mathematica and a result that I did not want to see: a
blank plot. M. does this when it can’t do a plot. Why can’t it do this plot? HINT: If you
don’t see this several times during the semester, then you are far better at M. than I am.
1.36. Using Mathematica
Solve the differential equation
7
Chm 331
∂2f
∂f
−3
= −4f
∂2x
∂x
Exercise Set 1
1.42
8
Figure 5: Input and Output from one of my Mathematica Sessions
HINTS: The M. statement is DSolve[eqn, f, x ]; the equation must be listed with a double
equals, ==, so in this case eqn would be written:
7f 00 [x] − 3f 0 [x] == −4f [x]
Note you must tell M. that the function is a function of x explicitedly, f[x]. What you are
solving for is f and the variable is x. Also note the proper notation for a first and second
derivative is f’[x] and f”[x].
1.37. Checking Mathematica
(Hah, as if M. would be wrong! Although we did find one case last year where it was
questionable, if not wrong.) You can check your answer from the last problem by cutting
it out and pasting it as f into the expression
Simplify[7
∂f
∂2f
−3
+ 4f ]
∂2x
∂x
You should get, of course, zero. Try it. HINT: Without the “Simplify” M. sometimes gives
a complicated expression that is not clearly equal to zero.
1.38. Using Mathematica
When I tried the following problem I got the result shown in Figure 5. Look closely at the
output and tell me what is wrong. HINT: M. tells us, in her own obscure way.
1.39. Using Mathematica and an Important Solution
Use M. to solve the differential equation
∂2f
− k2 f = 0
∂2x
where k is a constant. This is a wave equation we will use quite often in the course.
1.40. Using Mathematica
Use differentiation to show that your solution to the last problem generates the differential
equation given in the last problem.
1.41. Using Mathematica
In this and the next three problems we are, as a Mathematica exercise, going to apply
the Boltzmann equation (of problem 49) to learn about the populations of excited states.
Define a underscore function for the ratio n1 /n0 with temperature and ∆e as variables.
Chm 331
Exercise Set 1
1.47
9
Figure 6: Approximate Results for Two Box Experiment, No Light. Note total area under
curve is assumed to be unity.
1.42. Using Mathematica
Evaluate your function from the last problem for a series of ∆e that range from 1×10−22 J
to 1×10−21 J in steps of 1×10−22 J at a temperature of 298K. Do this all at once. HINT:
“All at once” clearly calls out for a
?
1.43. Using Mathematica
Repeat your evaluation of the last problem, but change the temperature to 100K.
1.44. Using Mathematica
Repeat the evaluation but let the energy changes be from 1×10−20 to 1×10−19 J at 298K.
Compare your result with the last two problems. What do you learn?
1.45. Using Mathematica
Use your data from the last problem to find the fraction of the total number of particles in
the lowest level. HINTS: What is the question in three words or less? If you understand
M., you can do this very slickly.
1.46. Two Box Experiment
In Figure 6 are the results of parves hitting the screen from the two box experiment when
both boxes are opened simultaneously. Clearly there is a interference pattern. Now we
change the experiment slightly. We put a weak light illuminating the holes in the plate.
Sometimes (about 60% of the time) that light strikes a parve as it exits and we then know
which box it came from. The pattern on the screen under these conditions is shown in
Figure 7. Now we turn up the intensity of the light so that it strikes more parves as they
exit one box or the other; we can now see about 90% of them. The pattern is shown in
Figure 8. Comment.
1.47. Classical Black-Body Radiation
A heated black box with a small hole in it acts as an ideal black body emitter. Such an
object achieves equilibrium between the walls of the container and the radiation inside of
it. This radiation can be observed by looking at what escapes from the small hole, whose
Chm 331
Exercise Set 1
1.47
10
Figure 7: Approximate Results for Two Box Experiment, Light Hitting about 60% of Parves.
Note total area under curve is assumed to be unity.
Figure 8: Approximate Results for Two Box Experiment, Light Hitting about 90% of Parves.
Note total area under curve is assumed to be unity.
Chm 331
Exercise Set 1
1.52
11
(small) size allows equilibrium to be maintained inside. Classical theory in which oscillators
in the walls of the container are in equilibrium with the radiation says that the density of
the radiation in a given wavelength interval, ρ, as a function of the wavelength should follow
the equation:
8πkb T
ρ=
(2)
λ4
where kb is Boltzmann’s constant, 1.38×10−23 J K−1 and T is the absolute temperature.
Plot with Mathematica the intensity as a function of λ over the visible range of wavelengths
for a black body of T = 5780K, the temperature of the surface of the sun.
1.48. Classical Black Body-Radiation
What happens to the classical black-body equation (see exercise 47) as the wavelength
becomes smaller, and smaller, and smaller. HINT: Oh, oh.
1.49. Boltzmann’s Population Equation
Boltzmann found an equation that gives the ratio of the number of particles in any two
energy levels given the energy separation between those levels. The equation reads (for the
non-degenerate case):
ni = n0 e−(Ei −E0 )/kb T
(3)
where ni is the number of particles in the ith level, n0 is the number of particles in the
0th level, the E values are the corresponding energies, and kb is Boltzmann’s constant,
1.38×10−23 J K−1 . Find the relative population, ni /n0 , for a energy separation of 1.0×10−20
J (a value typical of a bond vibration in infrared spectroscopy) at a temperature of 150 K.
Do the same at 298 K. Give a qualitative explanation of the difference.
1.50. Planck’s Black Body-Radiation Expression
In 1900, Planck made the assumption that the energy of oscillators in the walls of the blackbody were proportional to their frequency (which was the frequency of light in equilibrium
with them) and therefore that high frequency oscillators might not be able to vibrate because
their energy was not obtainable at moderate temperatures. That energy was quantized,
could occur only in lumps, not continuously, was a bold suggestion but allowed Planck to
derive a new equation for energy flux for a wavelength between λ and λ + dλ per area, ρ,
for black-body radiation:
8πhc2
1
ρ=
(4)
λ5 e λkhcb T − 1
where c is the speed of light, kb is Boltzmann’s constant, and h is also a constant, a new
constant in 1900, called Planck’s constant, which has value of 6.626×10−34 J sec. Show by
plotting the energy flux as a function of λ that this equation removes the difficulty found
in problem 48; use T =5780K. HINT: The (8πhc2 stuff is all fluff (controls the magnitude,
but not the trends) and you could leave it out. All the details in the argument to “e” in
the denominator are important; why?
1.51. Planck’s Black Body-Radiation Expression and Temperature
Show by plotting Planck’s black-body expression, equation 4, which agrees perfectly with
experiment, that the dominant color of a black-body shifts from reddish to bluish as the
temperature of the body goes from 3500 K to 6000 K.
Chm 331
Exercise Set 1
1.61
12
1.52. Planck’s Black Body-Radiation Expression and Temperature
The star Sirius has maximum wavelength for its black-body radiation of about 258 nm.
Make an estimate of the surface temperature of this star. HINT: You might plot your maxima from the last problem and extrapolate, especially if you can find a linear relationship.
1.53. Einstein’s Equation
Which is the more highly energetic end of the visible spectrum, the red end or the blue
end?
1.54. Review of Organic Chemistry, Energy, and an Important Equation
Approximately where in the IR spectrum does a carbonyl bond stretch occur? Please
specify units. Now calculate the wavelength of the light corresponding to the excitation of
this stretch. Finally, find the energy of that wavelength of light. HINT: To do so, you need
to use the equation that Planck first applied to “particles” and Einstein to “photons” in
the period of time immediately prior to the introduction of quantum mechanics. What is
that equation? Please learn it as well as you know how to get to your house.
1.55. Einstein’s Equation
Which is the more highly energetic end of the infrared spectrum, that where C-H stretches,
or that where a carbonyl group stretches?
1.56. Kinetic Energy and another Important Equation
“Kinetic energy is energy of [blank].” Compute the kinetic energy of a softball coming off
Blaire Luna’s hand at 58 mph. Express your answer in Joules.
1.57. Momentum and an Equation
What is momentum? Give a general equation for the kinetic energy in terms of momentum.
1.58. Kinetic Energy of Another Kind
For a particle of mass m circling a point at a distance r from it, the kinetic energy is given
by the expression:
1
T = Iω 2
2
where I is the moment of inertia, the angular equivalent of mass, and ω is the angular
equivalent of velocity, called the angular velocity, and is expressed in radians per second.
Compare this equation to that used in exercise 56. HINTS: If you don’t know the moment
of inertia, now might be a good time to look it up. We use this equation later.
1.59. More Correlation between Linear and Angular
Angular momentum is expressed as ` = Iω. Express kinetic energy of angular motion in
terms of angular momentum such that you have an equation similar to that in exercise 57.
HINT: We use this equation later.
1.60. Another Important Quantum Equation
To deal with quantum mecahnics we have to in some way live with the idea that small things
(in the Dirac sense) have wave-like and particle-like properties. DeBroglie first postulated
an equation that relates the two kinds of behavior. What is that equation?
Chm 331
Exercise Set 1
1.69
13
Figure 9: Two de Broglie Waves
1.61. Wavelength of Massive Objects
Calculate the wavelength associated with the softball in exercise 56.
1.62. Wavelength of Small Objects
Calculate the wavelength associated with a hydrogen nucleus moving at 5×105 m sec−1 .
1.63. Wavelength of Small Objects
Calculate the wavelength associated with an electron moving at 5×105 m sec−1 .
1.64. The Wavelength of Small Objects
Why do the answers from the last two problems differ?
1.65. DeBroglie Waves
In Figure 9 are two deBroglie waves. Which has the highest momentum? Why?
1.66. Looking at a Lady Bug–I
Calculate the energy emitted per second from a 60 watt light bulb. HINT: A watt is a
joule/sec. Assume 0.30 of that energy is visible light, which you can assume (for ease of
calculation) has a λ of 500 nm. That energy is emitted in all directions; the ladybug of
interest to us occupies only a part of all those directions. Assume she is 1.0 meter from
the source. What fraction of the spherical surface does she occupy? HINT: In this and the
following problems you need to estimate certain properties; make reasonable guesses.
1.67. Looking at a Lady Bug–II
How many 500 nm photons strike the ladybug of the last problem in a second, the second
it takes you to observe her. HINT: Assume all the photons of interest are at 500 nm?
1.68. Looking at a Lady Bug–III
We now work with conservation of momentum. The Einstein relationship (from relativity
considerations), p = λh gives the momentum of a photon. So the total momentum incident
on the ladybug in a second is (Number of photons per second)(momentum per photon).
This is the momentum transfered to the ladybug. What is the velocity of the ladybug in a
direction away from the light due to being irradiated with the light?
Chm 331
Exercise Set 1
1.78
14
1.69. Absolute Smallness
Is the ladybug (in the last problem) small in the Dirac sense?
1.70. Wave Theory
Light waves, like all waves, have a wavelength, a frequency, and a speed. Is there a relationship between them? What is it?
1.71. Waves
A typical wave equation is
ψ(x, t) = Asin[kx − ωt]
where A is the wave amplitude, k is the wave number, and ω is the angular frequency. Let
A = 1, t = 0, k = 4, and ω = 1; plot the wave as a function of x from x = 0 to x = 6.
HINTS: 1. Need I ever say it again, use M. 2. If M. objects, then think about these items:
ψ(x,t) is probably not going to make M. happy unless you defined an underscore function.
Also, what would M. do with sin[x]?
1.72. Waves
Plot the wave in problem 71 again for some later time, say t = 0.1. Which way is the wave
moving?
1.73. Waves
What would be the equation for a similar wave (to that in problem 71) moving in the
opposite direction?
1.74. Complex Waves
Show (using M. I would suppose) that the expression e−iφ and the expression Cos[φ] - i
Sin[φ] give the same answer. HINTS: Choose some number for φ and solve with “//N” at
the end of the expression to get a numerical answer, otherwise, M. will just repeat what
you typed in. Also, if you haven’t discovered it yet, M. calls “i” “I”.
1.75. Interference
Use M. to show that the sum of Sin[k π x] for k = 1 to 50 strongly interferes in most of the
region between x = 0 and x = 3. HINT: Use “PlotRange → {{0,3},{-30,30}}” after your
definition of the range of x (and before the end of the square bracket of the Plot command)
to get the full benefit of the plot.
1.76. Using Mathematica and your Brain
As always, look carefully at that “PlotRange” command in the last problem and be sure
you understand what the numbers refer to.
1.77. Linear Combinations and Superposition
Plot the waves Sin[2πx] and Sin[3πx] between x = 0 and x = 2. Now make the addition of
these two waves and plot it:
ψ = Sin[2πx] + Sin[3πx]
In quantum systems, the addition of two states treated as waves to make a summed state
is a superposition. So when we talk about it later, it is no more mysterious than that (at
this mathematical level; it is a different story at the philosophical level).
Chm 331
Exercise Set 1
1.85
15
1.78. Superposition
What is a superposition? HINT: Superpositions will be extraordinarily important to us this
semester.
1.79. Superposition are Mathematically Just Linear Combinations
I claim that the function x(1-x) between x = 0 and x = 1 can be represented by ψ:
8 [Sin[πx] Sin[3πx] Sin[5πx] Sin[7πx] Sin[9πx]
ψ= 3
+
+
+
+
π
13
33
53
73
93
Verify this is true. HINT: You might try plotting the two functions in M. and ask if they
appear the same. REMARK: If ψ represents a quantum state and the various Sin terms
also represent quantum states, then what we have here is a superposition. Before long you
will be able to see exactly where the indicatged expansion came from; it’s easy to make a
superposition.
1.80. Operators
In general, what does an operator do to a function? What does an operator do to a vector?
1.81. Operators
Convince yourself that the operator C4 , clockwise rotation by 90o (or, to reveal the nomenclature, by 360/4 degrees or 2π/4 radians), changes the vector ↑ to the vector →. What
does the operator C2 do to the vector ↑?
1.82. Operators and Vectors
What does the operation C4 C4 do to the vector ↑? Could you express that in some way
that involves only operators? HINT: Think product of operators.
1.83. Operators and Vectors
What does the operation C2 C2 do to the vector ↑? Could you express that in some way
that involves only the operators? HINT: I am asking you to invent an new operator here
which usually has the name “I” (which is not M.’s square root of minus one).
1.84. Eigenfunctions and Eigenvalues
What is an eigenfunction/eigenvalue problem?
1.85. Eigenfunctions and Eigenvalues
Which of the following equations represent an eigenfunction/eigenvalue situation? The
operator is in parenthesis and the function follows it. For each example that is an eigenfunction/eigenvalue equation, give the eigenvalue. The symbols “a” and “k” are constants.
HINTS: 1. If you don’ t know how to multiply matrices, ask; or learn how to get M. to do it
for you. 2. Also note that the multiplication of a matrix by a constant, say “q”, multiplies
all values in the matrix by q.
(x) ex
(d/dx) e[−ax]
(d/dθ]) Sin[k θ]
Chm 331
Exercise Set 1
1.91
16
(d2 /dθ2 ) Sin[k θ]
2
1
2
1
1
1
2
1
1
1
2
−1
1.86. Eigenfunctions and Eigenvalues
If O is an operator and φ is a function, how would you describe the expression
Oφ = 7φ
1.87. Eigenfunctions and Eigenvalues
Is the following matrix equation an eigenvalue/eigenfunction situation? The 3 x 3 matrix
is equivalent to an operator and we call the column matrix a “vector” (and, as we shall
learn soon, is a representation of a ket, |λi). If the equation is an eigenvalue/eigenfunction
situation, what is the eigenvalue?



1
0 1 0
√
1 0 1 − 2
0 1 0
1
1.88. Eigenfunctions and Eigenvalues
Here we have the same matrix operator as in the last problem, but a different vector. Is
this new vector an eigenvector of the matrix? If so, what is the eigenvalue?

 
0 1 0 1
1 0 1 0
0 1 0 1
1.89. Eigenfunctions and Eigenvalues
Here again we have the same matrix operator as in the last two problems, but yet a different
vector. Is this new vector an eigenvector of the matrix? If so, what is the eigenvalue?

 
0 1 0
−1
1 0 1  0 
0 1 0
1
1.90. Operators and Mathematica
You can use the “underscore” functionality in M. to define an operator as well as a function.
For instance, the underscore operator for the second differentiation, ∂ 2 /∂x2 , could be written
pd[f ] := D[ f, {x,2}]
Try taking the second derivative of the function, g = 7x3 , with this “pd” thing. Do you get
the correct answer?
Chm 331
Exercise Set 1
1.100
17
1.91. Operators and Mathematica
Latter we will have the opportunity to work with the operator (q - ∂/∂q). Write this as an
“underscore” operator (see the last problem) and show that it converts the wave function
qe−q
into
2q 2 e−q
2 /2
2 /2
− e−q
2 /2
1.92. Eigenfunctions and Eigenvalues
Is the last problem an eigenfunction/eigenvalue problem?
1.93. Expectation Value
If you had a bunch of dice sitting on a table, how would you compute the average value
showing, a value we will call an “expectation” value in quantum mechanics?
1.94. Probability
What is the probability that a die will show a “1”?
1.95. Probability and Expectaton Value
Take our equation for the average value of a bunch of dice
hji =
X jNj
n
NT
where n is an index over the dice, j is the number showing on a given die, Nj is the number
N
of dice with the number j, and NT is the total number of dice. Can you express NTj in terms
P
of probability, Pj ? Do so and rewrite the equation with that substitution to give hji = j
j Pj . HINT: The use of “j” as a subscript and a value is a little confusing. Sort it out.
1.96. Probability
You find a collection of old first class stamps at an antique auction. It contains 4 $0.03
stamps, 8 $0.10, 3 $0.17, 3 $0.24, and 4 $0.29. What is the probability that you will,
reaching randomly into this pile of stamps, pull out an $0.17 stamp?
1.97. Probability
What is the expectation value of the denomination of a stamp randomly withdrawn from
the collection in the last problem?
1.98. Probability and Quantum Mechanics
Here is an interesting issue that will arise over and over again in quantum mechanics: Is the
expectation value of the denomination of a stamp randomly withdraw (see the last problem)
the same as the value of any given stamp? Let me state the conclusion: In this problem,
it is impossible for a single measurement (one stamp withdrawn) to give the average value
you would get after lots of measurements. Mull that over and digest it thoroughly.
1.99. Probability
What is the expectation value of the square of the denomination of a stamp randomly
withdrawn from the collection in problem 96?
Chm 331
Exercise Set 1
1.107
18
Figure 10: Mathematica Commands for Simulation of the Roll of 50 Dice. HINT: To work
right in M. there needs to be a “return” after the “hh” at the end of the second line.
1.100. Mathematica and Probability
In Figure 10 are the M. commands to mimic rolling tn dice (tn is the name of the number
of dice) and to then give as output the number of times a “one”, “two”, etc is rolled as
well as the expectation value of measurement of those tn dice. (HINT: To work right in M.
there needs to be a “return” after the “hh” at the end of the second line.) To be sure you
understand M., see if you can understand the steps. Why is the expectation value near 3.5?
Would it change if the tn were smaller? Larger?
1.101. Counting Occurences
Find the first 25 digits of π and count the number of times that a 3 occurs. HINT: M will do
this with the command N[Pi, 25]. To get a list of those 25 digits you could do the command
rr = RealDigits[N[Pi/10, 25]][[1]]. It might be useful for you to look up “RealDigits” and to
do these steps one at a time to see exactly what each does. HINT for HINT: the [[1]] takes
the first element of a list. If you now use the command Count[rr, 3], you will count the
occurrences of “3” in the list; and if you are clever you can make a table that will list this
information for all the digits between 0 and 9 all in one step. QUESTION for the question:
Upon doing this again, I don’t that we have to divide π by 10. What do you think?
1.102. Probability
What is the probability that you will get a 4 from a random pick of one of the first 25 digits
of π?
1.103. Probability
What is the most probable digit in the first 25 digits of π? What is the average value of
the digits (the expectation value)?
1.104. Probability
What is the standard deviation for the distribution in problem 101?
1.105. Probability
You can make a plot in M. with the command:
Plot[1, {x, 0, 1}, PlotRange → {{0, 1}, {0, 1.2}}, LabelStyle → {FontFamily
→ ”Helvetica”, Bold, FontSize → 12}, Frame → True, FrameLabel → {”x”,
”Probability of obtaining x”}]
Make this plot. Use your intuition to determine the average value of x one would obtain
with this probability function. HINT: You might pay attention to the commands used to
do certain things in this plot; changing them to see what happens is useful.
Chm 331
Exercise Set 1
1.110
19
Figure 11: Probability for problem 107
1.106. Probability
What is the probability you would find a value of x between 0 and 0.5 for the probability
distribution in the last problem?
1.107. Probability and Expectation Value
Normalize the probability shown in Figure 11; determine the expectation value of x, < x >,
and the probability that you would find a value of x between 0 and 0.5.
1.108. Probability and Expectation Value
Normalize the probability shown in Figure 12; determine the expectation value of x, < x >,
and the probability that you would find a value of x between 0 and 5. HINT: Because of
the discontinuity at x = 4, you will have to divide your integral into two portions, 0 to 4
and 4 to 10, and add those portions together.
1.109. Probability and Expectation Values
Let a probability function be
2
P (x) = Ae(−α(x−q) )
where A, α, and q are real constants. Find the normalization parameter A, evaluate < x >
and < x2 >; find the value of the standard deviation, σ. HINTS: (1) The function extends
to infinity in both directions. (2) You might want to add at the end of your Integrate[...]
command, but before the closing bracket, “Assumptions -> Re[α] ≥ 0”, set off with a
comma, but not with the quotation marks, to make your answer more readable.
Chm 331
Exercise Set 1
1.115
20
Figure 12: Probability for problem 108
1.110. Probability and Expectation Values
Imagine you are standing at the midpoint of a marathon and you clock the velocity of the
runners as they pass you. (Some of them are very tired by the time they get to the midpoint
and are moving very slowly!) You find the probability of their velocity can be expressed as:
P (v) = ve(−v/q)
where q is a positive constant and the direction of v is fixed and can range from 0 (because
it is unlikely any runners are going to be going toward the starting line or at a diagonal to
the route) to ∞ (for ease of integration). Find the normalized probability expression for
the velocity of the runners in terms of v and q.
1.111. Probability and Expectation Values
For the probability curve in the last problem, what is the expectation value of the velocity,
< v >? What is the probability that a runner has a velocity of greater than four times the
expectation value of the velocity?
1.112. Schrödinger’s Equation
What are the two fundamental equations that Schrödinger probably used to “derive” his
equation.
1.113. Schrödinger’s Equation
Write the time dependent Schrödinger equation (in one dimension) from memory.
1.114. Schrödinger’s Equation
What is the meaning of ψ in Schrödinger’s equation? This is not a question about ψ 2 .
Chm 331
Exercise Set 1
1.123
21
1.115. Integration by Parts
We use integration by parts quite often in quantum mechanics. This is based on the total
differential
d(uv) = udv + vdu
or
udv = d(uv) − vdu
which, when integrated gives
Z
Z
udv = uv −
vdu
where for most cases each integral is a definite integral and the quantity uv is evaluated as
uv(f inalstate) − uv(initialstate)
By hand, not with M., find the integral of ln(x)dx between x = 1 and x = 2. HINT: Identify
u and v.
1.116. Integration by Parts
Use M. to find the value of the integral from the last problem.
1.117. Normalization
In quantum mechanics, a wave function of x, f, is said to be normalized over the range a to
b if
Z b
f 2 dx = 1.
a
For the function f = A sin[2πx/λ] and a range defined by a = 0 and b = 18, find the
normalization constant A with λ = 6.
1.118. Normalization
Normalize the wave function ψ = x(1-x) over the range from x = 0 to x = 1.
1.119. Probability
What is the probability of finding the parve described by the wave function in the last
problem between x = 0 and x = 0.5?
1.120. Normalization
2
Plot the wave function e−q /2 over the range of q = -5 to q = 5. Normalize the function
over the range of q of -∞ to ∞. Plot the normalized function over the range of q = -5 to q
= 5. Incidentally, why don’t we do the plot in M. over the range of q of -∞ to ∞?
1.121. Probability
What is the probability that a parve will be between q = 1 and q = 2 for the wave function
in the last problem?
1.122. Orthogonality
Two functions of x, f and g, are said to be orthogonal to each other over the range a to b if
Z b
f gdx = 0.
a
For the functions f = sin[2πx/λ] and g = sin[3πx/λ], show they are orthogonal over the
range 0 to 6 with λ = 6.
Chm 331
Exercise Set 1
1.131
22
1.123. Orthogonality
Using the functions from the last problem, plot each over the range 0 to 6. Now plot the
product of the two functions over the range 0 to 6 and examine the area under the curve,
which is, after all, the integral. Does the positive area cancel the negative?
1.124. Orthogonality
Are the two functions in problem 122 orthogonal over the range 0 to 7? Does the range
matter?
1.125. Orthogonality
Are the functions sin[2πx/λ] and cos[4πx/λ] orthogonal over the range of 0 to 8 with λ =
8?
1.126. Mathematical Manipulation
Show that
∂
∂ψ ∗
∗ ∂ψ
ψ
−ψ
∂x
∂x
∂x
is equal to
∂2ψ
∂ 2ψ∗
−ψ
2
∂x
∂x2
HINT: This is a simple differentiation of a product function, followed by addition.
ψ∗
1.127. Superposition
What is superposition? Give an example.
1.128. Approximating a Function
Consider the function
y(x) = x(1 − x)3
Plot the normalized function between x = 0 and x = 1.
1.129. Approximating a Function
I claim that we can approximate the expression for y(x) given in the last problem by a sum
of Sin terms:
y(x) ∼
= u(x) =
6
X
√
an 2Sin[nπx]
(5)
n=1
where the an are numbers. You showed in problem 122 that Sin terms with different n
values are orthogonal to each other (as long as the range limits have both functions going
to zero at those limits). Do all terms of the form Sin[nπx] go to zero at x = 0? at x = 1?
What terms in the sum would remain if you evaluated the integral
Z 1
√
u(x) 2Sin[2πx]dx
0
Chm 331
Exercise Set 1
1.139
23
1.130. Linear Combinations and Superpositions
What kind of generic expression would you say equation 5 is if the various functions are all
wave functions? HINT: It couldn’t be more obvious with the title of this problem the way
it is.
1.131. Approximating a Function
If the approximation in problem 129 is reasonable, what would you get if you took the
integral?
Z 1
√
y(x) 2Sin[2πx]dx
0
1.132. Approximating a Function
Hopefully you learned in the last problem how to evaluate the an values. Find the six an
values for the expansion in problem 129. HINT: This cries out for a M. table.
1.133. A Check for the Approximate Function
Plot y(x) (problem 128) and your approximation for it (u(x) from the last problem) on the
same plot to see how good the approximation is.
1.134. Review of Important Terms
What is an eigenvalue/eigenfunction problem? What is normalization? What are orthogonal functions? What is an expectation value? What is superposition? How do you evaluate
a probability in quantum mechanics? How do you find an expectation value in quantum
mechanics?
1.135. Normalizing a Wave Function
Find the value of A for the function
2
ψ(x, t) = Ae−ax e−iωt
(where a, and ω are real and positive constants) that normalizes the function. Do so at t =
0. HINTS: (1) Remember that this is a wave function and NOT a probability. How do you
get a probability from a wave function? (2) Also, the function is valid over the range -∞
to ∞. (3) Defining this as an “underscore” or a “colon equals” or a “SetDelayed” function
will be useful below.
1.136. Normalizing a Wave Function
Show that the function in the last problem has a normalization constant that is independent
of time. One way to do this is to set t = 10 and do the last problem again. Is there another
way, a more elegant way?
1.137. Expectation Values
In quantum mechanics, what do you evaluate to find an expectation value for the physical
parameter Q? HINT: I am looking for a generic answer.
1.138. Probability of Position from Wave Function
For the wave function in problem 135, let a = 0.3 and ω = 1; find the probability that
the particle described by the wave function will be found between x = -0.1 and x = 0.1 at
t = 0.0.
Chm 331
Exercise Set 1
1.148
24
1.139. Probability
Imagine you measure the position of the particle described by the wave function in problem 135 and find it to have a negative value of x. Boniface Beebe, the great natural
philosopher from Arkansas, observing this, said “If you make another measurement on an
identically prepared system you will find the particle to have a positive value of x since the
probability that the particle is to be found between x = -∞ and x = 0 is 0.5.” Comment
on his conclusion; express your answer precisely (using probability language).
1.140. Expectation Value of Position
With a = 0.3 and ω = 1, calculate the expectation value of x for the wave function in
problem 135. Is it time dependent?
1.141. Interpreting Expectation Values
How do you make sense of the number from the last problem? HINT: You might plot
something, but it better not have i in it if you are going to do so.
1.142. Expectation Values of Position Squared
Using the a = 0.3 and ω = 1, calculate the expectation value of x2 for the wave function in
problem 135.
1.143. Interpreting Expectation Values
How do you make sense of the number from the last problem?
1.144. Interpreting Expectation Values
With a = 0.3 and ω = 1, calculate the expectation value of -~2 d2 /dx2 for the wave function
in problem 135. HINT: This operator gives us, as we either have or soon shall see, the
expectation value of the momentum squared, which is the kinetic energy times twice the
mass of the parve.
1.145. Eigenfunctions and Eigenvalues
Operate on the function of problem 135 with the operator x. Is this an eigenfunction/eigenvalue
problem? If so, what is the eigenvalue?
1.146. Eigenfunctions and Eigenvalues
Operate on the function
2
g = 2ze−z /2
with the operator -d2 /dz 2 + z2 . Is this an eigenfunction/eigenvalue problem? If so, what
is the eigenvalue? If it is an eigenfunction/eigenvalue problem, what are the implications
for making several measurements of the energy (which it turns out is proportional to the
operator given above)? HINT: If you suspect Oψ = o ψ, you can ask M. to evaluate Oψ
ψ
and you should get the eigenvalue “o”. For this kind of operation, where two functional
relationships are divided by each other, the command “Simplify[ . . . ]” can be useful in M.
1.147. Review of Important Concepts
What is an superposition? What is a complete set of wave functions? What is an expectation value? What is a normalized function? When are two functions orthonormal?
Chm 331
Exercise Set 1
1.155
25
1.148. Review of Important Practical Point
Complete the following sentence (which you should incorporate into your psyche): “Before
you evaluate an expectation value, thou shalt . . . .”
1.149. Review of Important Concept
You have a system for which the wave function is an eigenfunction of the Hamiltonian
operator. What will you get for an answer if you measure the energy on dozens of identically
prepared systems?
1.150. Review of Important Concept
You have a system for which the wave function is not an eigenfunction of the Hamiltonian
operator. What will you get for an answer if you measure the energy on dozens of identically
prepared systems? HINT: Try for a well-phrased answer, even though it will require you to
be long-winded.
1.151. Parve on a Pole Problem
This begins the setup of problems called “parve on a pole.” These are problems in which
the parve is constrained to move only in one dimension. They are not real, but are very
good approximations to all sorts of problems. A thorough understanding of POP problems
will allow you to appreciate all sorts of things in more complex problems.
Schrödinger’s time independent wave equation in one dimension is
−~2 ∂ 2
ψ + V ψ = Eψ
2m ∂x2
Let V = 0 on the pole and infinity outside the ends of the pole. Show that the form of this
equation on the pole becomes
where k =
√
∂2ψ
= −k 2 ψ
∂x2
(6)
2mE/~.
1.152. Parve on a Pole Problem
Show that either ψ = B cos[kx] or ψ = A sin[kx] or ψ = A sin[kx] + B cos[kx] are solutions
to the differential equation in the last problem; A, B, and k are constants, independent of
variable x.
1.153. Parve on a Pole Problem
Let our pole be one unit long, starting a x = 0 and running to x = 1. By making V infinite
outside the pole we are demanding that the parve cannot be at x < 0 or x > 1. What does
that mean the value of ψ 2 is outside those limits? And if ψ 2 = 0, what can you say about
ψ?
1.154. Behavior of Wave Functions
The quantity ψ 2 evaluated at a point in space is a measure of the probability of finding the
parve at that point in space. Does it make sense to argue that the probability of finding
the parve cannot jump from one value to another for adjacent points in space? If that is so,
what can you say about ψ at two adjacent points in space? HINT: Quantum mechanics is
weird, but it is not unreasonable. If a wave function has a value at a position x of 1, then
it cannot have a value of 0 at a position dx from x. This will turn out to be one of our
hypotheses: Wave functions must be continuous.
Chm 331
Exercise Set 1
1.162
26
Figure 13: Attempt to Plot in Mathematica
1.155. Parve on a Pole Problem
In view of the last two problems, why is cos[kx] not a legal wave function for a parve on a
pole starting at x = 0? Why is sin[kx] legal at x = 0?
1.156. Parve on a Pole Problem
Based on the logic used in the last problem what must the value of sin[kx] be at x = 1?
Does that restrict the value of k? Has a boundary condition introduced a quantum number
(usually an integer, but sometimes, as we shall see, a half integer).
1.157. Parve on a Pole Wave Functions
Why does k = nπ
L for a legal wave function, A sin[kx], for the POP?
1.158. Using Mathematica
I am trying to be sure you can trouble-shoot in M. I tried the plot in Figure 13 and got the
indicated result. Why?
1.159. Parve on a Pole
Plot the POP wave function for n = 2 and for n = 4 on a pole of length 1. Exactly why
does this work and my attempt in the last problem did not?
1.160. Parve on a Pole
Use M. to show that the two wave functions that satisfy the parve on a pole problem, ψ2 (x)
and ψ4 (x), are orthogonal to each other.
1.161. Orthogonality of Parve on a Pole Solutions
For the parve on a pole problem, show symbolically that two different wave functions (two
different n values) are orthogonal to each other in the parve on a pole problem. HINT: Use
an expansion for the product sin[a]sin[b]; this is not a M. problem.
Chm 331
Exercise Set 1
1.173
27
1.162. Probability for a Parve on a Pole
Find the probability that a parve (in the parve on a pole system) is between 0.4 L and 0.6
L, where L is the length of the pole, if n = 1; if n = 4; if n = 15. Think carefully and
determine the classical probability for a particle to be in this range?
1.163. Expectation Values for a Parve on a Pole
Evaluate < x>, < x2 >, σx , < p >, < p2 >, σp , and the product σx σp for n = 7 for the
parve on a pole system.
1.164. Schrödinger’s Equation For Parve on Pole
You have Schrödinger’s equation for a POP problem in equation 6. Use M. with one or
another of the known POP wave functions to show this equation works. That is divide the
second derivative of one of the ψ’s by that ψ and show the answer is what k is equal to.
HINT: Use “Simplify[ . . . ]”.
1.165. Parve on a Pole Solutions with Different Starting Coordinates
Solve the parve on a pole problem when the pole is defined to start at x = -L/2 and to end at
x = L/2. HINTS: (1) As is true in many problems, all the ψ’s won’t have the same analytical
form. One way to handle this is to use a summation of possible forms. (2) In contrast to
the pole defined from zero to L, you have to be more careful with boundary conditions.
(3) If you get stuck, use your knowledge of the answer, which must be independent of our
choice of where x = 0 is on the pole to sketch the answers and see if that helps.
1.166. Parve on a Ring
Consider a parve on a ring with no potential energy. Given that kinetic energy is 21 mv2 ,
and that angular momentum, ` = mvr, show that the energy can be written as `2 /(2mr2 )
= `2 /2I, where I = mr2 .
1.167. Parve on a Ring
∂
If the Schrödinger operator for ` is ( ~i ∂φ
), use the expression for the kinetic energy in terms
of ` to find the operator for the kinetic energy.
1.168. Parve on a Ring
Show that Sin[aφ] is a solution to Schrödinger’ s equation for a parve on a ring where a is
a constant.
1.169. Parve on a Ring
Show that Cos[bφ] is also a solution for a parve on a ring where b is a constant.
1.170. Parve on a Ring
As usual, for a solution to be suitable (acceptable), what condition must be satisfied?
1.171. Parve on a Ring
How do you apply a boundary condition for a parve on a ring when there is no apparent
“boundary”? What must be true of ψ(φ)? HINT: Think about ψ(φ+2π).
1.172. Parve on a Ring
Show that Sin[aφ] is a suitable solution for a parve on a ring if a = 1, 2, 3, . . . . HINT: You
will need the relationship that Sin[q + r] = Sin[q] Cos[r] + Sin[r] Cos[q].
Chm 331
Exercise Set 1
1.182
28
1.173. Parve on a Ring
Show that Cos[bφ] is a suitable solution for a parve on a ring if b = 0, 1, 2, 3, . . . . HINT:
You will need the relationship that Cos[q + r] = Cos[q] Cos[r] - Sin[r] Siin[q].
1.174. Parve on a Ring
Normalize your solutions for the last two problems.
1.175. Parve on a Ring
From your solutions to the last problem, find the expectation value of `; of `2 .
1.176. Parve on a Ring
Find some of the eigenvalues of the energy operator for a parve on a ring. Are they also
eigenfunctions of `?
1.177. Parve on a Ring
Make the following linear combination of eigenfunctions of the energy operator for a parve
on a ring:
r
1
1
ψ1n = √
(ıSin[nφ] + Cos[nφ])
(7)
2 π
r
1
1
ψ2n = √
(ıSin[nφ] − Cos[nφ])
(8)
2 π
Show that these linear combinations are eigenfunctions of the energy operator.
1.178. Parve on a Ring
Show that the linear combinations from the last problem are also eigenfunctions of `. Note
carefully what you did. You had eigenfunctions of the energy operator with the same
eigenvalue; you took linear combination of these and found they were still eigenfunctions of
the energy operator. However, the latter are also eigenfunctions of the angular momentum
operator.
1.179. Superposition: The Expansion
A parve on a pole (of length one, running from x = 0 to x = 1) has a potential such that
the wavefunction is
ψ = (1 − x)Sin[4πx]
A bunch of these systems are prepared; all of them have the potential suddenly removed
(to zero). Express ψ in terms of the eigenfunctions of a parve on a pole with zero potential.
HINT: This is the superposition principle in action.
1.180. Superposition and Measurement
What is the energy of the system in the last problem immediately before the potential is
removed? HINT: And after, since energy is conserved.
1.181. Probability of a Given Measurement
What is the probability that an energy of 9~2 π 2 /(2m) will be measured after the potential
of problem 179 is removed?
Chm 331
Exercise Set 1
1.191
29
1.182. Energy in a Superposition
What is the energy you get from your expanded wavefunction in problem 179? How close
is it to the initial energy?
1.183. Allowed Wave Function for a POP
Is the wave function below an acceptable one with respect to the boundary conditions of a
parve on a pole of length 1?
ψ(x) = x(1 − x)2
Plot the functon.
1.184. Expectation Value of the Energy for a Superposition
Find the < E > for the ψ in the last problem. HINT: POP problem has V(x) = 0.
1.185. Superposition
Remember in quantum mechanics, as we shall prove when we get to the postulates, any
arbitrary function can be expanded as a linear combination of the eigenfunctions of an
operator. If φi is the ith eigenfunction of an operator, and ψ is the arbitrary function, we
write:
X
ψ=
ci φi
i
Now because the integral of φi φj is zero unless i = j, the ci are given by the integral of
φi ψ. Write the function of problem 183 as a sum involving the first 10 eigenfunctions of the
energy operator for the parve on a pole.
1.186. Superposition
Plot your expansion of the last problem to see how close it is to the original function of
problem 183.
1.187. Probability of a Given Measurement
Given your expansion in problem 185, what is the probability that a measurement will yield
a value of the energy of ~2 π 2 /(2m)? of 16~2 π 2 /(2m)?
1.188. Expectation Value of Energy
We want to evaluate the expectation value of the energy for a time independent function
that is expanded in terms of the eigenfunctions of the Schrdinger
equation (for a parve on a
P 2
pole) with coefficients ci . Show that < H > is given by n cn En where En are the energies
of the wave functions in the expansion.
1.189. Time Dependent Functions
Imagine the time independent eigenfunction for a system is given by the equation
ψ = Ae−ar
where A and a are constants and r is the variable. Write an equation for the time dependent
function. HINT: Answer this question generally, not specifically.
1.190. The Change in Function with a Change in Position
If you have dψ/dx = -6 x and a condition that ψx=0 is 3, what does the function ψ look
like between x = 0 and x = 1? HINT: This is an old fashion calculus problem, not a M.
problem for which you have the next problem.
Chm 331
Exercise Set 1
1.200
30
1.191. Numerical Integration
M. has a function that solves differential equation problems numerically. It is called NDSolve. It was a topic in our introduction to M. Show that it works on the equations of the
last problem.
1.192. Potential and Kinetic Energy in Quantum Mechanics
Consider a classical case where the total energy is less than the potential energy. What is
the consequence of this? HINTS: Total energy is “potential energy plus . . . ”? Oh, oh!
1.193. Limits on ψ
This problem isn’ t hard; just rewrite SE as instructed! Consider the one dimensional time
independent Schrödinger equation. Rewrite the equation so that the only term on the left
hand side is the second derivative of ψ with respect to x; show that you get
∂2ψ
−2m
=
(E − V (x))ψ
∂x2
~2
(9)
Now expand the second derivative as a derivative of the first derivative, and call the latter
the slope. Show for a small but finite change you can obtain the equation
slopenew = slopeold −
2m
(E − V (x))ψ∆x
~2
(10)
1.194. Limits on ψ
Now examine equation 10 in a region of space where V(x) is greater than E and where ψ
(which remember is a function of x) is positive. What can you say about the new slope
relative to the old slope as ∆x gets larger?
1.195. Limits on ψ
If the slope gets larger as x gets larger in the region defined in the last problem, what
happens to the magnitude of ψ as x gets larger? and larger?
1.196. Limits on ψ
Figure 14 shows part of an (unnormalized) wave function. Look at the region of x from 1.1
to 1.3. Does the function behave as you predicted in the last problems if in this region V(x)
is greater than E? NOTE: This function is, as will become obvious, not an acceptable wave
function for the problem.
1.197. Limits on ψ
Now examine equation 10 in a region of space where V(x) is less than E and where ψ(x) is
positive. What can you say about the new slope as ∆x gets larger? If what you concluded
is true, could you say in the region of space where V(x) is less than E, the wave function
curves toward the ψ = 0 axis?
1.198. Limits on ψ
Examine Figure 14 in the region from x = 0.6 to 0.9. Do you see what you concluded in
the last problem?
Chm 331
Exercise Set 1
1.204
31
Figure 14: Part of an unacceptable wave function for a parve in a well.
1.199. Limits on ψ
From an examination of Figure 14 make a guess at the x value at which V(x) switches from
being less than E to being larger than E.
1.200. Limits on ψ
Look at curve A in Figure 15. Does this agree with your analysis in the last several
problems?
1.201. Limits on ψ
Examine equation 10 in a region where V(x) is greater than E, but ψ is negative. What
should happen to the slope as ∆x gets larger? Is the consequence that ψ gets more and
more negative as x gets larger? That is, that ψ slopes away from the ψ = 0 axis?
1.202. Limits on psi
Examine curve B in Figure 15. Is this in accord with your analysis of the last problem?
1.203. Limits on ψ
What can you say about normalizing ψ in either curves A or B in Figure 15? HINT:
Remember normalization gets the area under the product function equal to unity over all
x values where ψ has magnitude, including (double hint) x greater than 1.5.
1.204. Limits on ψ
You have found in the last several problems that in the region of space where V(x) is greater
than E, ψ always curves away from the ψ = 0 axis as x increases. You should have, provided
you have minimal curiousity, also proved to yourself that ψ curves away from the ψ = 0 axis
as x gets more negative (assuming that V(x) is greater than E. Such functions are therfore
not normalizable, and are thereby useless in quantum mechanics. Is there any situation
where ψ can exist in the region with V(x) greater than E? Consider the positive ψ case
with x getting larger; ψ is curving away from the ψ = 0 axis. However, if ψ starts out
Chm 331
Exercise Set 1
1.208
32
Figure 15: Parts of two more unacceptable wave functions for a parve in a well.
sufficiently positive, it can curve away and yet still be decreasing—see curve A in Figure 15
at x about 1.15. If that curve is such that it reaches the ψ = 0 axis (and hence has ψ = 0)
at the same time as slopeold (in the language of equation 10) is zero, what happens to the
slopenew ? And what happens to the “new new” as x increases some more?
1.205. Limits on ψ
Apply the logic of the last problem to the curve in Figure 16. Be sure you see the correct
curvatures. The value of x when V(x) becomes greater than the energy is 1.0; does that
value make sense to you?
1.206. A Scheme to Solve Schrödinger’s Equation
The form of equation 9 is readily solvable for ψ in Mathematica if all the parameters, m, ~,
E, and V(x) are known. We know the first two (or can assign them to arbitrary values) and
usually know how the potential varies with x. So if we guess an E value, we can solve the
differential equation using NDSolve in M. This requires two conditions for the constants of
integration. From all the manipulations we have been doing on the curvature of ψ, what is
the condition that allows us to know if we guessed E correctly?
1.207. Numerical Integration
Use NDSolve to make a graph of the solution to a parve on a pole for the ground state with
the parameters set as follows: ~ = 1, L = 1, and m = 1. HINT: Your NDSolve system should
include Schrödinger’ s equation for the system as well as the two starting conditions, ψ(x =
0) equals something—what would that be for this POP problem? and ∂ψ/∂x(x = 0) equals
something, which is this case can be arbitrary. This problem is very important because we
will use the procedure in lots of cases this semester; it allows solution of problems that have
no analytical solution.
Chm 331
Exercise Set 1
1.212
33
Figure 16: An acceptable (that is normalizable, but not normalized) wave function for a
parve in a well. The edge of the well is at x = 1.0; V(x) is zero between x = 0 and x = 1
and is 10 (arbitrary) units when x is greater than 1. The energy of the parve in those units
is 2.2949.
1.208. Numerical Integration
Try the same procedure as in the last problem for an energy in excess of 100 in the units
with ~ = 1 etc.
1.209. Numerical Integration for an Unusual Potential
We have a “pole” with an infinite potential when x < 0; for x > 0, the potential varies
with x according to the prescription V(x) = 2x. From your knowledge of a POP, guess the
appearance (roughly) of the ground state wave function for this system.
1.210. Numerical Integration for an Unusual Potential
Use the “pole” defined in the last problem. Find the shape and energy of the lowest
level wave function for this situation using numerical methods. LOTS OF HINTS: Take
advantage of the infinite potential at x = 0 to be able to set the boundary conditions. Also,
continue to use the atomic units of ~ = 1, m = 1. Pay attention to the lack of a right
hand boundary; if you don’t make your limit of integration large enough you will miss the
answer; and if you make it too large, numerical instability will give your a wild answer.
Play around and establish both of these latter aspects after you have an answer; to not do
so will make your use of Mathematica much harder later.
1.211. Numerical Integration for an Unusual Potential
For the potential in the last problem, find some excited level wavefunction and its energy.
HINT: I tried something in the range of 25 for energy, which was too high.
Chm 331
Exercise Set 1
1.219
34
1.212. Numerical Integration for an Unusual Potential
You have a potential defined by the equation:
5
+ e−3x − 3.27878
V =
1 + e−x
with V = ∞ at x < 0. Find the lowest energy level using numerical techniques with ~ and
m = 1. NOTE : The term at the end is just to make the lowest potential energy value equal
to zero. HINTS : What must be true of ψ(x) at x = 0? Use an x range of 0 to 7 as the
barrier on the right hand side is rather small. Always plot your V(x) to be sure you have a
reasonable equation for it.
1.213. Expectation Value
Find the expectation value of x for your answer to problem 212.
1.214. M. Practice
Find the value of x for which V is a minimum. HINT: A useful M. command is FindMinimum[ f, {x, x0 }] where x0 is some (guessed) number close to the minimum.
1.215. Interpretation of Quantum Mechanics
Is there anything classically wrong with the probability of finding the parve at x = 4 in the
lowest energy level for your answer to problem 212?
1.216. M. Issue
Try expanding your plot from problem 212 to x = 10. When you observe what happens,
you should do one thing: Ask yourself “Why” do I see what I see? Then you could change
the value of the energy some more and get an answer to more significant figures, or, you
could say “Good enough”. HINT: See problem ??.
1.217. The Harmonic Oscillator Done Numerically
Solve for the ground and a couple of higher energy levels and find the corresponding wave
functions for a harmonic oscillator with a force constant, k, of 4.8; use equation 9. A
harmonic oscillator has a potential energy of the form 21 kx2 in units where ~ and m are
unity.
1.218. An Odd, but Varying Potential, and Interpretation
We have a parve on a pole with length = 1.0 in which the potential energy is zero from x
= 0 to x = 0.8. For 0.8 ≤ x ≤ 1.0 the potential is V. Outside the pole, the potential is
infinite. Find the first three energy levels for V = 0 (trivial), 5, 20, and 80 units (when ~
and m are set equal to unity) – that is, you have to do nine problems: Feel free to work
with others on this problem as there are lots of very similar calculations; but be sure each
one in your group can do any given part of the problem. Be able to give a verbal account of
what your wave functions look like. HINT: Remember to make ”conditional parameters”,
parameters that depend on the range of the value of x, we use the underscore definitions in
M. See original Mathematica hints sheet.
Chm 331
Exercise Set 1
1.224
35
1.219. Probability and an Odd Potential
We want to find the probability that a particle will be between 0.4 and 0.6 in the ground
state and in the second state, n = 2, two separate calculations, for a parve on a modified
pole with the following potential energy situation: V is infinity for x < 0 and x > 1. V is 0
for 0 < x ≤ 0.4 and 0.6 ≤ x < 1.0. For the intermediate area, 0.4 < x > 0.6, V is 250. Let
~ = 1 and m = 1. Before you quit M. after doing this problem, remember that doing the
next problem in the same sitting is useful because it saves some recalculations.
1.220. Superposition, Time Dependency, and Tunneling
Let’s look at the consequences of the barrier we erected between x = 0.4 and x = 0.6 in
the last problem. You have the energies (and therefore the numerical wave functions) for
the first two levels. [HINT: In M., you might want to name these differently, say psi1, and
psi2, although since we can refer to them by their “Out[#]” anyway, it is not absolutely
necessary.] I worked out the energies for the next two: 97.292 and 99.13 in the units of the
last problem. You can therefore find these numerical wave functions; name them differently
from the first two. So you have four wave functions on your machine, located at various
“Out[#]” type locations. [HINT: you have to do this problem in one sitting; closing M.
deletes these numerical values.] Finally, after all that preliminary, here’s the problem: Write
the analytical solution for a parve trapped in the left side of the box in its lowest energy
level, that is, the answer for a POP between x = 0 and x = 0.4. Let’s say we know this at
time zero. You can now expand that wave function in terms of the four numerical functions
at time zero. [HINT: Be careful of your limits on integration; your range of definition of the
parve on the left hand side is between zero and 0.4 L. Anything outside that range is not
defined.] Now look at the probability distributions generated by your superposition wave
function and compare it to that of the localized wave function, all at zero time. Add the
appropriate time dependency to each of the functions in your superposition wave function to
get an approximation to the time dependent wave function (Why just an approximation?)
and look to see what happens to the probability as a function of time. HINT: Take 1, 2,
4, and 5 as your time intervals, remembering that we have defined ~ = 1. When you get
finished you can go “Wow! WOW! DOUBLE WOW! I have just witnessed ‘tunneling’ ”.
1.221. Potential Well Problem
Consider a potential well running from - 1 to 1 of depth V. Solve the system for the right
side (x = 0 to x = 1) for even functions (i.e., ψ is some positive number at x = 0) of this
potential well numerically (using NDSolve). Let m = 1, ~ = 1, and V = -100. HINT: you
know the answers from class. Also, think carefully about the boundary conditions that you
put on the problem.
1.222. Interpretation of the Potential Well Problem
Can the parve in the last problem be in a region of space where it is classically forbidden?
1.223. Non-Classical Behavior
Find the probability that the parve in problem 221 in the ground state will be in a nonclassical region of space where x > 1.
Chm 331
Exercise Set 1
1.232
36
1.224. Superposition and Probability of Obtaining a Certain Energy
Let a particle of mass m start on the left half of a pole of length L with a wave function at
t = 0 in which the probability is constant between 0 and L/2, i.e., a “square wave”. Find ψ
(x, t) = ψ (x, 0), normalize it, and obtain the probability that a measurement of the energy
for the parve on the pole of length L would yield a value of π 2 ~2 /(2mL2 )
1.225. Free Particle Probability
For a free particle determine the relative probability of finding the particle between x = -11
and x = -10 compared to finding it in the region x = 0 to x = 1.
1.226. Free Particle Wave Function
For the step potential (V = 0 when x < 0 and V = V+ for x > 0), and E of the incoming
free particle less than V+ , determine the wave function valid in the region x > 0. How does
this compare to the classical situation?
1.227. Particle in a Finite Well
For the parve in a finite potential well (potential = -V for -a ≤ x ≤ a; potential equal to
zero otherwise; and since the parve is in the well, E < 0) determine the equations that need
to be solved to find the solutions for the odd functions. You don’t have to solve them.
1.228. Simple M. Problem for Ulterior Purposes
Show that a polynominal can be wavy. Try f = -2 + 12q - 16q3 + 3.2q5 over a range of -2
to 2.1.
1.229. Simple M. Problem for Ulterior Purposes
2
Plot the function ψ = e−x /2 .
1.230. Basis of Harmonic Oscillator Potential Energy
In class we said the force on an object obeying harmonic oscillation is given by F = −kx(t),
where the time dependency of x has been made obvious. Let’s establish k in other terms.
Harmonic oscillation is governed by the following relationship between position and time
x(t) = x0 cos[ωt]
2
Find the second derivative of this expression, ∂ ∂tx(t)
2 , which is the acceleration. Then use
Newton’s law, F = ma, and evaluate F. Compare the two expressions for the force and
prove that
k = mω 2
1.231.
Define
The Dummy Variable in the Harmonic Oscillator Problem
r
q≡
mω
x
~
and use the chain rule to show that
∂2
mω ∂ 2
=
∂x2
~ ∂q 2
Chm 331
Exercise Set 1
1.242
37
1.232. Schrödinger’s Equation for the Harmonic Oscillator in “q” Language
Use your result of the last problem to show that Schrödinger’s equation in “q” language is
∂2ψ
= (q 2 − K)ψ
∂q 2
where K is
2E
~ω .
1.233. Harmonic Oscillator Wave Function by Numerical Techniques
Use numerical techniques to plot the wavefunction for a harmonic oscillator in the first
excited level. HINTS : Use the S.E. in “q” form,
∂2ψ
= (q 2 − K)ψ
∂q 2
where K is 2E/(~ω) and solve for K; also remember you are looking for the first excited
level: make sure your function “waves” appropriately.
1.234. Part of Harmonic Oscillator Wave Function Solution
For the function f=a0 + a1 q + a2 q2 + a3 q3 + . . . , show that
q
X
df
=
nan q n
dq
(11)
n=1
1.235. Part of Harmonic Oscillator Wave Function Solution
Show that you could write equation ?? in the following form:
q
X
df
=
nan q n
dq
n=0
1.236. Part of Harmonic Oscillator Wave Function Solution
Given the function f in the last problem, show that
X
d2 f
=
n(n − 1)q n−2
2
dq
n=2
1.237. Part of Harmonic Oscillator Wave Function Solution
Find the relative coefficients of the q, q3 , and q5 terms for K = 11 in the Hermite polynomials
given the relationship derived in class for an+2 in terms of an .
1.238. Harmonic Oscillator Wave Function Solution
Write an appropriate (non - normalized if you wish) wave function for the harmonic oscillator
for E = (11/2) ~ω.
1.239. Harmonic Oscillator Wave Function Solution
Plot the function from the last problem.
Chm 331
Exercise Set 1
1.252
38
1.240. Harmonic Oscillator Wave Function Solution
Normalize the function for the harmonic oscillator in the third excited state (with n = 3).
1.241. Harmonic Oscillator Wave Function Solution
Plot the normalized ψ from the last problem versus q.
1.242. Probability Distribution for a Harmonic Oscillator Wave Function
Plot the probability distribution for the wave function in problem 240.
1.243. Expectation Value of Position for a Harmonic Oscillator
What is the expectation value that the oscillator in problem 240 will be found with a value
of q between 0.5 and 1.?
1.244. Expectation Value of Position for a Harmonic Oscillator
What is the expectation value that the oscillator in problem 240 will be found with a value
of q between 2.0 and 2.5?
1.245. Expectation Value of Position for a Harmonic Oscillator
Do your answers to the last two problems make sense? Explain.
1.246. Classical Limit of Harmonic Oscillator
A classical harmonic oscillator has total energy equal to kinetic (1/2 mv2 ) plus potential
(1/2 kx2 , where x is the displacement). When the oscillator reaches the limit of its stretch,
the point we could call the turning point, how much of each kind of energy does it have?
Write an expression for the maximum displacement, xt , (t for turning point) in terms of
total energy.
1.247. Classical Limit of Quantum Oscillator
Using your relationship from the last problem, write xt in terms the total energy (quantized);
substitute in for k the equivalent value given in problem 230. Then use the definition of q
to obtain an expression for qt .
1.248. ω for a Real Harmonic Oscillator
The harmonic oscillator system we have been working with has the mass of the object in
it; one object. When dealing with two atoms, the appropriate mass is called the reduced
mass, defined by
1/µ = 1/m1 + 1/m2
such that k = µω 2 . If the force constant for the C-O bond in CO is 1.85×103 N/m, evaluate
ω. HINT: Don’t forget 12 g/mole is not 12 g/molecule.
1.249. Turning Point for a Harmonic Oscillator
Find the value of the classical turning point for the n = 2 level for CO.
1.250. Energy of Transition for a Harmonic Oscillator
Calculate the wave number for the transition from n = 0 to n = 1 for CO.
1.251. Calculation of “Impossible Probability” for a Harmonic Oscillator
Find the probability of the CO molecule being outside the classical turning point region;
that is, stretched or compressed past the classical turning points when n=0.
Chm 331
Exercise Set 1
1.259
39
1.252. Numerical Solution for a Harmonic Oscillator
You have an analytically derived solution for ψ with n = 4 for the harmonic oscillator.
Find ψ4 numerically. Normalize both and show that the integral of their product is unity,
establishing that they are the same function.
1.253. Expectation Values for a Harmonic Oscillator
Use the wave function for the harmonic oscillator, n = 3, derived with the dummy variable
q and find < x >, < x2 >, < p >, < p2 >, σx , and σp for CO. Use data from problem 248.
Also determine the product σx σp and verify the uncertainty principle.
1.254. Expectation Values for a Harmonic Oscillator
Find < p2 > for a harmonic oscillator in the n = 0 and n = 2 (two separate calculations,
which, if I were you, I would use M. to do). Comment on any differences that you find.
1.255. Morse Function for a Harmonic Oscillator
A Morse potential is a more realistic potential curve than that of the harmonic oscillator.
The Morse curve is given by
V = p(1 − e−a(x−x0 ) )2
Plot this function for a value of p = 30,000 cm−1 , x0 = 1.45 Å, and a = 4.5×10−1 Å. HINT:
Plot over the range of x = 0.1 to x = 6.
1.256. Differences between Morse Function and Harmonic Oscillator Function
Qualitatively, how would you expect the energies and the wave functions for the system
described by the Morse potential in the last problem to differ from those of the harmonic
oscillator?
1.257. Harmonic Oscillator Potential Energy
We have two harmonic oscillators of the same reduced mass. One with a k of 350 N/m,
the other with a k of 720 N/m. How do the potential energy curves differ? What can you
say about the energy levels of the two oscillators? Solve this by thought, not with M. or,
indeed, with mathematics.
1.258. Numerical Solution of an Anharmonic Oscillator
An anharmonic oscillator might have a potential (in q langauge) such that the q2 term in
(q2 - K) is replaced by
q 2 − 0.04q 3 − 0.004q 4
Plot this function (between q = -8 and q = 8 and see that it does give a curve similar to
the Morse curve in problem 255.
1.259. Numerical Solution of an Anharmonic Oscillator
Set up a numerical integration of Schrödinger’s equation in q language
∂2ψ
= (q 2 − K)ψ
∂x2
with the potential given in the last problem instead of just the value q2 . Find the first
three energy levels. HINTS: The trick in this problem is establishing reasonable constants
of integration. Since the potential is not symmetrical in q, you can’t use such stuff as y[0]
Chm 331
Exercise Set 1
1.262
40
== 1 and y’[0] == 0 for a symmetrical function. I had success with the following logic. I set
y[0] == 1 (actually, of course, in non M. language, ψ = 1 at q = 0) and then let y’[qend ] ==
0 where I chose qend dependent upon which function I was integrating. For lower n values,
maybe qend = 3.5; for higher, qend = 5. Other choices would surely work but just be aware
of the ambiguity, and arbitrary nature, of these choices. ALSO, remember than the answers
for K, which is the surrogate for the energy (in q language) in the harmonic oscillator are
1, 3, 5. That helps with your initial guesses as this potential is close to “harmonic”.
1.260. Orthogonality of Strange Wave Functions
Show that the wave functions with n = 0 and n = 2 in the last problem are orthogonal.
1.261. Talking As Spectroscopists Do
The energy levels in a harmonic oscillator are given by the equation:
E=
~ω
~ω
K=
(2n + 1)
2
2
Spectroscopists usually write these energies in units of wave numbers, cm−1 by dividing the
E (in Joules) by hc, where c is in cm/sec, so the above equation becomes
1
e = ω̄e n +
2
For nitric oxide, NO, the harmonic oscillator has an energy gap of 3.79×10−20 J. Find the
value of ω̄e .
1.262. Anharmonic Parameters in Spectroscopic Language
Spectroscopists write the energy levels of an anharmonic oscillator in terms of a power series
in (n + 21 ),
1
1 2
e = ω̄e n +
− x̄e ω̄e n +
2
2
Fit your data from problem 259 to this power series and evaluate x̄e .
Chm 331
Exercise Set 1