Quantum Dots – A real-world Particle in a Box
Pre-laboratory Questions
Read the experiment and answer the following questions.
1. According to the Quantum Theory of a particle in a one-dimensional box, how are the energy
levels related to the size of the box?
2. What does it mean to say that the energy of the particle in a one-dimensional box is
quantized?
3. What is meant by the “zero-point” energy for a particle in a box.
4. What is the range of radii for the Quantum Dots we are studying in this experiment?
5. What sort of graph should result, if we plot the peak emission wavelength of these Quantum
Dots versus their radius?
Quantum Dots – A real-world Particle in a Box
I. INTRODUCTION AND OBJECTIVES
Figure 1: Quantum Dots Solutions (Cenco)
Quantum dots are particles, with dimensions in the range of 1-30nm, composed of a
semiconductor material whose electron-holes are thus confined to a small volume. The ones
investigated here are spherical, with radius about 3nm, and consist of indium phosphide cores
surrounded by thin zinc sulfide shells. They are suspended in the liquid, tetradecane. When a
photon of high enough frequency (violet light in this case) is incident, it can form an electronhole pair. The electron and hole exist almost independently in the quantum mechanical ground
state determined by the small spherical volume. Upon recombination, a photon is emitted (this is
called florescence) with energy dependent on the radius of the dot. Smaller dots emit higher
energy photons. The elementary quantum mechanical calculation of the ground state for a
particle confined to a sphere yields the energy of the photon as the sum of band gap energy,
kinetic energy of the particle, and kinetic energy of the hole. The kinetic energy parts are
inversely proportional to the square of the radius of the sphere.
This experiment is about verifying the quantitative relationship between radius and emission
wavelength predicted by the elementary quantum mechanical theory. A set of four vials of dots
are obtained from the Cenco company. Emissions are in the red, orange, yellow, and green
regions of the spectrum.
After performing this experiment and analyzing the data, you should be able to:
1. Read the wavelength from Quantum Dot spectra.
2. Determine the width of the spectrum peak through a Gaussian fit.
3. Verify the Quantum Mechanical Predictions of the relationship between the wavelength of
the emitted light and the size of the nanoparticles.
II. EQUIPMENT NEEDED
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Red Tide Emission Spectrometer
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Blue LED Light source 405 nm
Cenco Quantum Dots Solution Set
Vernier Logger Pro Software
III. THEORY
Along with general relativity, quantum mechanics is one of the most significant advances in
modem physics. While very important and becoming increasingly so with the advent of
nanotechnology, the concepts of quantum mechanics are sometimes difficult to understand,
owing to their abstract and non-intuitive nature.
The underpinnings of quantum mechanics came about in the early 1900s with the introduction of
the concept of wave-particle duality, a theory stating that particles can exhibit wave-like
properties and vice versa. In 1905, Albert Einstein explained the photoelectric effect by
proposing that light-waves also have particle-like characteristics ("photons"). In 1913, Niels
Bohr explained the quantized energy states and spectral lines of the hydrogen atom by proposing
that electrons have wave-like characteristics (electron orbitals). Finally, in 1924, Louis de
Broglie put forward his complete theory of matter waves, solidifying the universal concept of
wave-particle duality. Wave-particle duality and the quantized nature of energy in quantum
systems represent two of the central elements of quantum mechanics.
Bohr's model of the hydrogen atom is perhaps the most commonly known quantum theory. Bohr
calculated the energy levels associated with an electron in each orbital based on the wavelength
of the electron orbiting the nucleus. As an electron moves up in energy levels it absorbs a
specific amount of energy, or quanta of energy. When the electrons fall down to a more stable
energy level the same quanta of energy is emitted in the form of light. While the theory was
useful for simple systems such as a hydrogen atom, more complex systems could not be
understood until the development of the Schrodinger equation by Erwin Schrodinger in 1925.
The Schrodinger equation represents a fundamental tool in understanding and using quantum
mechanics even today.
Particle in a Box
One problem that allows a student of chemistry or physics to understand the implications of
quantum mechanics is the "particle in a box" problem, in which the Schrodinger equation is used
to calculate the wave-like characteristics (known as the "wavefunction") and energy levels of a
quantum mechanical particle trapped inside a one-dimensional box.
In this problem, we conceptualize a single particle bouncing around inside of an immovable box,
from which it cannot escape, and which loses no energy when it collides with the walls of the
box. In classical mechanics (otherwise known as Newtonian mechanics), the solution to the
problem is trivial. The particle moves in a straight line, always at the same speed, until it reflects
from a wall (Figure 2a). When it reflects from a wall, it always reflects at an equal but opposite
angle to its angle of approach, and its speed does not change. In quantum mechanics, however,
when the wave-like character of the particle is considered, the results are much more surprising.
Nevertheless, it remains a very simple and solvable problem (Figure 2b). In this case, the
wavefunction of the particle can be conceptualized as a vibrating string that is fixed at the walls
of the box. As a result, only certain wavelengths (and therefore energies) can exist for the
particle (Figure 2b).
Figure 2. Particle in Box a) Classical Representation b) Wave Representation
Overall, the wave like property of this theoretical particle in a box, illustrates the following
features of quantum mechanics:
1. Energy quantization -It is not possible for the particle to have any arbitrary energy. Only
specific, discrete energy levels are allowed.
2. Zero-point energy -The lowest possible energy level of the particle, called the zero-point
energy, is nonzero.
3. Spatial nodes -The Schrodinger equation predicts that for some energy levels there are
nodes, implying positions at which the particle can never be found.
The Schrodinger equation for the wave function of a particle is written:
The potential energy function, V(x) for a one dimensional box:
The solution is:
The Quantum Dots we are using are three dimensional spheres of indium phosphide molecules as
shown below in the transmission electron micrograph (TEM):
Quantum dots are small semiconductor particles that can contain one electron and one "hole" (the absence
of an electron). Just like in the semiconductors that are used to make Flash cards and microprocessors,
these electrons and holes act like small particles which can move freely inside the semiconductor, but
cannot get out (just like a particle in a box!). By carefully observing Quantum Dots with different sizes,
we can see the effect of changing the size of the box (L) on the energy levels of the system.
Of course, we must make some small adjustments to the particle in a box equation derived above to take
into account some of the differences between this real-world particle in a box and our ideal model. First,
the box is 3-dimensional and spherical in shape rather than one-dimensional and square. While the calculation is somewhat more complex, the lowest energy state (zero-point energy) for a particle in a sphere
has a very similar form to that derived for the one-dimensional particle in a box. Second, there are
actually two particles within each quantum dot rather than just one (the electron and the hole). Finally,
our Quantum Dot "box" is not empty, but is filled with a semiconductor. As such, there is an additional
base-line energy that must be added to the system that comes from the energy of the semiconductor
bandgap. Therefore, a simple model of the Quantum Dot energy and emission wavelength is described
by:
where is Planck's constant divided by 2 (1.05459 × 10−34 J·s), n is the quantized energy level
(1), me is the effective mass of an electron (7.29 × 10−32 kg), mh is the effective mass of a hole
(5.47 × 10−31 kg), R is the radius of the quantum dot, Eg is the energy gap of the quantum dot
material (2.15 × 10−19 J), h is Planck’s constant (6.63 x 10-34 J), c is the speed of light (3.00 x
108 m/s), and  is the wavelength in meters.
This equation suggests that the Quantum Dot Emission Wavelength is related to the quantum dot
radius through a quadratic relationship. A table of the Quantum Dot radii is shown below:
Table I. TEM data for each quantum dot in solution.
Vial Color
Radius (nm)
Green
2.367
Yellow
2.534
Orange
2.718
Red
2.925
IV. EXPERIMENTAL PROCEDURE
A red tide emission spectrometer will be used to display the Quantum Dot spectra in this
experiment.
Connect the Fiber-Optic Cable to the SMA connector on the Red Tide Spectrometer.
Connect the USB cable from the Spectrometer to the USB port of your Computer. Mount
the other end of the Fiber-Optic Cable in the clamp as shown in the picture below. Point the
fiber at the 1st Vial in the array.
Start up the Logger Pro Program. Load the “Quantum_Dots” file from the “Experiments”
Folder.
Prepare the spectrometer to measure light emission. Click “Collect”. An emission spectrum
will be graphed. Shine the Blue LED on the bottom of the vial producing Green Florescence.
Press the Stop button when a reasonable spectrum is displayed. From the Curve Fit Menu, Select
the Gaussian Function to fit the data.
A typical Quantum Dot spectrum with a Gaussian Fit is shown below:
The Gaussian Fit Constant B is the peak wavelength, , in nanometers (nm) and the
magnitude of the constant C is the standard deviation, , also in nm. Record this data in
the table below. Print the spectrum on the color printer with the fit showing.
Repeat the above procedure for the other Vials on the Apparatus after pointing the fiber
optic cable at that Vial. Record the results in the data table.
Vial
Florescence Color
Quantum-dot
radius
(nm)
Green
Yellow
Orange
Red
2.367
2.534
2.718
2.925
 (nm)
peak wavelength
Standard Deviation
in wavelength
nm)
2. Go to page 2 in Logger Pro to plot the peak wavelength, , versus the Quantum Dot radius, r.
Enter your wavelength and radius data from the data table. Use curve fit with the “quadratic”
function to find the best line that fits the data points.
Print the graph with the Curve fit showing.
Does the data follow the predictions of quantum mechanics? ______________________.
IV. Conclusion
After group discussion, write a one-paragraph conclusion that summarizes the results of this
experiment.