Particle in a Box
Contents
1. fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl F Eurofysica INSTRUMENTEN Particle in a Box 114430 Februari 2013 13 Exact wat u nodig heeft DATA SECTION Experiment 2 Determining the size of a Quantum Dot 1 What was the peak wavelength for each vial a Red b Orange c Yellow d Green 2 Using the formula from the Theory Section calculate the size of the Quantum Dots in each vial Show any work in the space below or on a separate paper a Red b Orange c Yellow d Green 3 Find the percent error between your measured wavelength and dot radius versus the actual values provided by your instructor a Red b Orange c Yellow d Green 4 Plot the quantum dot radius versus the emitted wavelength what relationship do you see Use graph paper 5 What happens if the radius of the quantum dot gets very large approaching infinity Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica asian 114430 Februari 2013 14 APPENDIX This derivation is intended as supplemental material to provide the full theoretical derivation from first principles to the fin al solution for the radius of the quantum dots The experimental guides provide a more pedagogical approach to leaming about Schr dinger s wave equation Solution to partical in a s2. P Particle in a Box Eurofysica mE X 114430 Exact wat u nodig heeft Februari 2013 1 PRODUCT CONTENTS Quantity Description 1 vial Red Quantum Dot solution 1 vial Orange Quantum Dot solution 1 vial Yellow Quantum Dot solution 1 vial Green Quantum Dot solution 1 Aluminum rack OTHER SUGGESTED MATERIALS Spectrofotometer SpectroVIS 113120 Light source 405 nm Required 114432 SpectroVis Optical fiber 113121 ASSEMBLY OF APPARATUS The apparatus is held securely in place by two hex bolts which can be unscrewed to allow removal of the vials for testing The glass vials containing the quantum dots provided by Nanosys should be left in the rack upright at all times unless being tested The vials are sealed with aluminum caps and are never meant to be opened or punctured If they are opened cracked or leaking please consult the MSDS sheet immediately CARE MAINTENANCE OF DEVICE The rack with vials should always be kept in an upright position and stored in a temperature range of 50 F to 80 F When the vials are removed from the rack take care to not damage the Aluminum PTFE seal Please read the included MSDS before perfoming any demonstrations or experiments If you should experience any difficulty with the apparatus please contact us giving details of the problem To ensure maximum service do not return any apparatus until we have given you authorization COPYRIGHT NOTICE The Quantum Dots operating instructions are copyr
3. a Or 1 1 kr yy Vk 0 Or P D Kr yz 7E r 1 Pr But from above we know k U r 83 3r U r UL Y r U r U r Q amp 8 L0 U r UL y Pr U r Show that d amp dj are raising and lowering operators First show that d di dra dm dea dia 1 k8 r 1 1 1 kr 1 k r 1 D kr 1 kO Or 1 2 k r 1 k6 Or 1 2 kr 1 k 0 dr 1 2 k7G Or l r 1 2 k rG Or 14 2 r k89 18r 142 k r 1 2 k rd Or 1 2 kr Or 14 2 k r f 1 k 0 r P 41 4 1 2 kf r 1 amp 8 r 10 2 1 Er Also d di 1 k Or L kr CV k0 Or L1 k r V 9 Or 1 1 k r 1 r 1 1 k r Or L1 k r V E r 1 1 k r 1 1 k r r 1 41 k2r0 ar L1 Pr 1 k 18 12 0 amp r Therefore d di d a dia Now use along with previous results d U r U r Therefore d r R r r R r 1 k r 1 1 kr r R r R r 1 kr r 1 1 kr r R r R a r 1 k rd r r R r 1 1 kr r R r R r Mkr 1 k0 0r R r Therefore R r Ck r V k amp r 8 Gr R r Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica mE X 114430 Exact wat u nodig heeft Februari 2013 46 Now determine R r amp 10r U r LU
4. 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 Eg Therefore a simple model of the Quantum Dot zero point energy is described by hm hr E 2m R i 2m R E E 2 15 X 107 J m 7 29 X 10 kg m 7 5 47 X 10 kg h B Setup Procedure 1 Using an LED close to 400 nm illuminate each vial so the class can see the luminescence shutting the lights off makes the demonstration more visible Explain to the class that the vials all contain the same solution the only difference is the physical size of the quantum dots contained in each Define quantum dots as small crystals 10 50 times the diameter of an atom 2 After illuminating each vial ask the class to brainstorm about the size of the quantum dots in each vial based on the color the vial radiated Ask for a volunteer to write the colors on the board in order from smallest to largest quantum dots C Questions 1 Why is a violet LED used instead of a red LED Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica 114430 Exact wat u nodig heeft Februari 2013 11 Experiment 2 Determining the size of a Quantum Dot A Theory derivation of formulas Using the theory section from Experiment 1 Classroom Demonstration
5. this theoretical partical 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 Schr dinger equation predicts that for some energy levels there are nodes implying positions at which the particle can never be found Quantum Dots a real world particle in a box While the particle in a box problem serves to explain some of the unique implications of quantum mechanics it is not something that can easily be seen or tested in real life As such these fundamental concepts are often very difficult for students to grasp since there is no good real world example they can use to see these effects Quantum Dots however are a real world particle in a box kk c 5 s La a bz i Pu z E j i pet Quantum Dots Figure 3 are small semiconductor particles that can contain electrons Just like in the semiconductors that are used to make Flash cards and microprocessors these electrons can move freely inside the semiconductor but cannot get out just like a particle in a box A Quantum Dot provides a unique way to teach core concepts associated with quantum mechanics such as energy quantization and wave particle duality as well as the
6. 3 features stated above This Nanosys product uses Quantum Dots to give the student a hands on feel for the implications of quantum mechanics while exposing student to cutting edge nanotechnology MALE T o E F Lal E Li ak i T ri E a t L mL Se T ee ees 1 ia Figure 3 Micrograph of 5 nm Quantum Dot Each bright point represents a single atom This image was taken with a transmission electron Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica mE X 114430 Exact wat u nodig heeft Februari 2013 4 Experiment Guide High School 114430 Quantum Dots Experiment 1 Classroom Demonstration A Theory derivation of formulas This simple product can be used to expand upon elements within the high school curriculum such as waveparticle duality quantization of energy and the Bohr model of the atom By using optical measurements such as florescence the students will be able to use this lesson with tools commonly available within the high school lab Florescence is the emission of light from a substance after it has absorbed energy The absorbed energy can be in the form of electricity light heat or other forms of energy In the case where the absorbed energy is provided in the form of light the emitted wavelength is often of lower energy longer wavelength red end of
7. P U n K U r rU r k U r R r sin k r r Therefore R r Ckry I K rY 0 0r sin kr kr These are the spherical Bessel functions R r 7 sin k r r R r kry I Er 0 Or sin kr kr R r 2 sin k r k r cos k r kr Impose boundary conditions V r 0 forO lt r lt R V r o forr gt R For 120 radia function s orbitals For free particle E n k 2m Wavelength must go to zero atr R Therefore sin k R 0 KR nn k nnu R Therefore E irmn 2mR Actual sizes and Peak Wavelengths Color Peak Wavelength nm Radius nm Gem so ensa Yellow 57 Orange 60 2 718174 292494 Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl
8. a solution for the three dimensional case can be derived This is left as an exercise for either the student or teacher E mR 2mR He Since we will use the wavelength color of the fluorescence from the Quantum Dots to measure the zero point energy we must also convert this wavelength information to energy using E hv and v where E v and A are the energy frequency and wavelength of the emitted light c is the speed of light 3 x 10 mis and h is Plank s Constant 6 62x10 J s B Setup Procedure Each brand of spectrometer will have slightly different operating instructions so please follow your user manual The following is designed for the SpectroVis spectrometer Procedure 1 Carefully remove one of the vials from the aluminum rack by using a hex key loosening both hex bolts 2 Set the vial upright on a flat surface free of other obstructions 3 Set up the spectrometer probe so it is held stationary and pointed at the solution in the vial 4 Turn off the lights or cover the apparatus to improve the quality of the readings 5 Turn on the spectrometer and have the software ready to scan the wavelengths 6 The 400 nm light source needs to be illuminating the vial but perpendicular to the spectrometer probe See Figure S2 on the next page If the light source is directly across from the probe you will not get an accurate reading of the wavelength radiated by the solution 7 Turn on the 400 nm light sourc
9. al or macroscopic world term and illustrates how these effects become important when dealing with very small length scales Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl F Eurofysica INSTRUMENTEN Particle in a Box 114430 Februari 2013 5 Exact wat u nodig heeft B Setup Procedure Classroom Demonstration 1 Using an LED close to 400 nm illuminate each vial so the class can see the emanations from each shutting the lights off makes the demonstration more visible Explain to the class that the vials all contain the same solution the only difference between them is the physical size of the quantum dots contained in each i e the size of the box Define quantum dots as small crystals 10 50 times the diameter of an atom 2 After illuminating each vial ask the class to brainstorm about the size of the quantum dots in each vial based on the color the vial radiated Ask for a volunteer to write the colors on the board in order from smallest to largest quanturn dots C Data and Calculations Based on the colors that you observed calculate the radius of the quantum dots for each color using equation 1 a Color D Questions 1 Why is a violet LED used instead of a red LED 2 Plot the quantum dot radius versus the emitted wavelength what relationship do you see 3 What happens if the radius of the quan
10. e with it pointed at the solution 8 Initiate the software to record the radiated wavelengths 9 Using the peak wavelength and the formula from Section A Theory Derivation calculate the size of the quantum dots Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl EE Particle in a Box Eurofysica ion 114430 Exact wat u nodig heeft Februari 2013 12 Figure S2 Set up with SpectroVis spectrometer OPTIONAL USES EXPERIMENTS Determining the absorption of the Quantum Dot Solution A Setup Procedure Each brand of spectrophotometer will have slightly different operating instructions but overall the functions are the same The following is design for the SpectroVis spectrophotometer Procedure 1 Since the vials are so small you need to use an adapter to fit the vial into the sample compartment You will also need to place a 15 mm lifter in the bottom of the adapter as well to raise the solution to the level of the light source 2 Starting at the lowest wavelength zero out the unit and then insert the solution 3 Record the data on the absorption ofthe solution increasing the wavelength in increments of 10 nm 4 Repeat steps 2 and 3 until you reach the upper limit of your spectrophotometer 5 Repeat with each of the other three vials Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22
11. ighted and all rights reserved Permission is granted to all non profit educational institutious to make as many copies of these instructions as they like as long as it is for the sole purpose of teaching students Reproduction by anyone for any other reason is prohibited Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica DN 114430 Exact wat u nodig heeft Februari 2013 2 DESCRIPTION OF USE BACKGROUND INFORMATION Along with general relativity quantum mechanics is one of the most significant advances in modern physics While very important and becoming increasingly so with the advent of nanotechnology the concepts of quantum mechaillcs are sometimes difficult to teach owing to their abstract and non intuitive nature Nanosys has developed a product Figure 1 that uses nanotechnology to allow students to readily grasp the underlying principles and implications of quantum mechanics Figure 1 Nanosys Quantum Dots illustrating multiple emissions from a single excitation source 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 characte
12. mple differential equation has the general solution x Asin k x Bcos k x 1i Where keh 2m E 1j This is the general form of the wavefunction and energy of a particle in free space We have not yet however applied the boundary conditions imposed by the infinite potential well the box Since the potential outside the box is infinite the wavefunction must go to zero at the edges of the box Therefore 0 V L 20 1k In order for the wavefunction to go to zero at the left hand wall x20 Y 0 Asin 0 Bcos 0 A 0 B 1 0 1L the boundary condition then implies that the B must equal zero which leaves us with Y x Asin k x 1m Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica DN 114430 Exact wat u nodig heeft Februari 2013 amp In order for the wavefunction to go to zero at the right hand wall x L Y L Asin k L 0 1n the boundary condition then implies that kL f 27 31 lo Therefore nu ke with n 1 2 3 1p and the wave function takes the form V x Asin x where n 1 2 3 1g with m E mi d where n 1 2 3 dr The final step is to determine the value of A The probability of finding a particle with wavefunction VP x at any particular location x is determined by taking the square ofthe abs
13. nductor 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 Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica o 114430 Exact wat u nodig heeft Februari 2013 10 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 see appendix for full derivation 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 E Phere ze where R is the radius of the Quantum Dot m Second there are actually two particles within each quantum dot rather than just one the electron and the hole so Aem hr 2mR 2m R where R is the radius of the Quantum Dot m is the effective mass of the electron inside the semi conductor and Mp is the effective mass of the hole inside the semiconductor Finally unlike the derivation above our Quantum Dot box is not empty but is filled with a
14. olute value of the wavefunction at that location Nro Since we know that the probability of finding the particle somewhere inside the box is 10096 i e the particle cannot exist outside of the box then Ivo 1 Cs so that 4 sin k x d x 1 1t L 4 1 1u a 1v Therefore our final wavefunction and energy levels are Y x E sin YI x 1w 1x where n 1 2 3 Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl xL Particle in a Box Eurofysica a O 114430 Februari 2013 9 This simple example of a particle in a one dimensional box highlights three important implications of quantum mechanics 1 Energy quantization while a classical particle trapped inside a box can have any energy i e it can move back and forth across the box at any speed according to the energy equation above the wave particle duality of quantum mechanics causes the quantum mechanical particle in a box to only have certain discrete quantized energies 2 Zero point energy while a classical particle trapped inside a box can have zero energy i e sit motionless according to the energy equation above the lowest possible energy state for n 1 is not zero 3 Spatial nodes while a classical particle moving back and forth inside a box will spend an equal amount of time at all locations in the box and therefore have an eq
15. phere H h 2MV V r Y R r Y 8 6 H Y EY In spherical co ordinates V Pat amp ay amp oz Vro ered or 1 r sinOe e8 sin0O 00 1 sin 02b A 1 r0 ar r M r L h h 2m r8 10r r M r L 183 V o JY E RGY 2 PS I 2m I r Y lr rR r R r 2mrL Y V r R r Y E R r Y a L Y A I 1 2m I r Y lr rR r R r 2mr e ll 1 V r R Y E R r Y Let U r rR r CRIMEI U r RIC 2mr V r U r E Ur At this point we have not imposed boundary conditions First solve for a free particle in soherical co ordinates and then impose boundaries to get quantization Free particle V r O We also know that for a free particle E h K 2m Therefore h 2m c er U r ht 1 2mr V r U r IPk 2m U r 0 LU r 41 r U r U r Define and raising and lowering operators d amp di d MkO r 1 kr di 2 Mk8l r 1 V kr F d d y y Must be true of raising and lowering operators Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl 3 Particle in a Box Eurofysica INSTRUMENTEN Exact wat u nodig heeft 1 1 4430 Februari 2013 15 1 k Or 1 Vykr 1 K8 Or 1 Vy kr y 1 k0 Or l k r 1 k 0 Or 1 Vy kr 0 Gr 1 Dir L 1 Py I k 80 0r 14 1 k 6 r 1 r 0L D r
16. ristics photons In 1913 Niels Bom 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 the Schrodinger equation is used to calculate the wave like characteris
17. sing the Schr dinger Equation 1c find the wavefunction and energy levels for a particle trapped inside a one dirnensional box where the potential inside the box is zero and outside the box is infinite also known as a particle in a box Qif zxzL V x iih 1d oif 0 gt x or L x 7 N l Ar Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl P Particle in a Box Eurofysica o 114430 Exact wat u nodig heeft Februari 2013 7 In this potential a particle is completely free inside the box but at the two boundaries the infinite potential keeps it from leaving the potential well In other words the probability of the particle being outside the box is zero Please note that the mathematical representation of the well from zero to L is to provide the easiest scenario for solving the wave equation Solution to the particle in a box problem Since the potential inside of the well our region of interest is zero V x 20 we can simplify the Schr dinger Equation to the following form h Ov x S SE EYG 19 2m Ox which can be rewritten as OW x 2mE 10 ex h T0 09 Since the energy of the system cannot be negative E 2 0 we can define a new variable k such that amE 2 k E 1g so that the complete Schr dinger Equation can be simplified to gu RS kY x 1h This si
18. the spectrum than the absorbed energy shorter wavelength blue end of the spectrum Since violet has the highest energy shortest wavelength of the visible spectrum it is used in this lesson to excite the quantum dots Additionally even though the same light source is used to illuminate the different solutions of quantum dots their physical size limits the energy each one re emits Small dots will florescence blue while large dots will florescence red Wave particle duality can be readily taught using this teaching aid to introduce students to the particle in a box concept In it s simplest form wave particle duality can be taught be conceptualizing matter waves as skipping ropes or waves in pool With this concept and the constaint that the wave must be pinned at the walls of the box we are able to illustrate the key feature of a wave particle duality For quantum dots mathematically this is expressed as 7 h n p n h n p E 2 2 2m R 2m R h E 1a where R is the radius of the Quantum Dot m is the effective mass of the electron inside the semiconductor m is the effective mass of a hole the absence of an electron inside the semi conductor and E is the bandgap energy of the semiconductor E 22 15 X 10 J Me 7 29 X 10 kg mn 5 47 X 10 kg The first 2 terms of this formula illustrates the size scale at which these wave particle duality effects begin to dominate given that the 3rd term is a bulk materi
19. tics 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 Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl p Particle in a Box Eurofysica mare as 114430 Exact wat u nodig heeft Februari 2013 3 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
20. tum dot gets very large approaching infinity Eurofysica B V Postbus 3435 5203 DK s Hertogenbosch tel 31 0 73 623 26 22 fax 31 0 73 621 97 21 www eurofysica nl info eurofysica nl F Eurofysica INSTRUMENTEN Particle in a Box 114430 Februari 2013 6e Exact wat u nodig heeft Experiment Guide College Experiment 1 Classroom Demonstration A Theory derivation of formulas Introduction to the Time Independent Schr dinger Equation Within the context of wave particle duality the Schr dinger Equation defines the relationship between the wave like characteristic of a particle known as the wavefunction and its energy The Schr dinger equation states HY EW la where Y is the wavefunction of the particle H is the Hamiltonian Operator and E is the energy of the system For a simple one dimensional system such as the one dimensional particle in a box described in this lesson the Hamiltonian Operator is defined as the sum of the kinetic and potential energy which can then be rewritten in the format below h e H V x 1b 2m Ox on where h is the reduced Planck Constant m is the mass of the particle and V x is the potential experience by the particle as a function of location For a particle with a one dimertsional wavefunction W x the complete time independent Schr dinger equation is therefore h x 2m Bx V x EW x lc Problem Particle in a Box U
21. ual probability of being found at any particular locaiton the wavefunction equation above implies that the probability of finding the particle at different locations is not uniform In fact there are locations within the box where the particles can never be found spatial nodes such as at the edges of the box since the wavefunction goes to zero at the edges of the box For energy states with n gt 1 additional spatial nodes can be found in the interior of the box Figure S1 x at left wall of box Figure S1 Wavefunctions for a quantum mechanical particle in a box for various values of n The probability of finding the particle is zero at each point where the wavefunction crosses the dashed blue line Quantum Dots a real world particle in a box While the particle in a box problem above serves to explain some of the unique implications of quantum mechanics it is not something that can easily be seen or tested in real life As such these fundamental concepts are often very difficult for students to grasp since there is no good real world example they can use to see these effects Quantum Dots however are a real world particle in a box 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 semico
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