{"id":4449,"date":"2024-09-22T18:02:38","date_gmt":"2024-09-22T18:02:38","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4449"},"modified":"2024-09-22T18:02:38","modified_gmt":"2024-09-22T18:02:38","slug":"real-world-particle-in-a-box","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/real-world-particle-in-a-box\/","title":{"rendered":"REAL-WORLD PARTICLE IN A BOX"},"content":{"rendered":"\n

There are very few real-world systems that behave like the simplified mathematical model we used above for a particle in a box. However, recent developments in the nascent field of nanotechnology<\/em> have made it possible to produce nanostructures<\/em> of only 2\u201310 nm in size that exhibit behavior close to the simple particle-in-a-box model we discussed.<\/p>\n\n\n\n

These tiny structures are known as\u00a0quantum dots.<\/em>\u00a0As shown in\u00a0Figure 111, they are small\u00a0semiconductor<\/em>\u00a0crystals of only 10 to 50 times the diameter of an atom. A\u00a0semiconductor<\/em>\u00a0is a material in which an electric current can flow due to the movement of electrons and \u201choles\u201d\u2014states where electrons could sit, but which are empty and thus act as positive charge carriers. You can study about those in most introductory books on electronics. For our purposes, it is sufficient to say that the quantum dots contain free pairs of one electron and one \u201chole\u201d that can move freely inside the semiconductor, but cannot get out\u2014just like in the quantum particle-in-a-box problem. Quantum dots of different dimensions are currently being manufactured using semiconductor materials such as cadmium selenide\/zinc sulfide [CdSe\/ZnS] and indium phosphide\/zinc sulfide [InP\/ZnS].<\/p>\n\n\n\n

Figure 111\u00a0Quantum dots are nanometer-sized crystals in which electrons and \u201choles\u201d get trapped, acting as real-world models of the particle-in-a-box problem. (a<\/strong>) The quantum dots are made of semiconductor materials such as InP\/ZnS, in which the size of the dot can be controlled very well at the time of manufacture. (b<\/strong>) An organized array of atoms is seen in this\u00a0TEM<\/strong>\u00a0image of a 10-nm quantum dot.\"\"<\/p>\n\n\n\n

Do-it-yourself synthesis of quantum dots is possible, and relatively simple procedures using equipment available in undergraduate laboratories have been published,34,35<\/sup> but they involve the use of rather toxic materials which we prefer to avoid. Instead, we purchased a set of four vials, each containing a suspension of quantum dots of a specific radius (2.37, 2.53, 2.72, and 2.92 nm). The quantum dots consist of indium phosphide cores surrounded by thin zinc sulfide shells. They are made by Nanosys for Cenco Physics, and are available from educational scientific supplies stores such as Sargent Welch (catalog number WLS1751-18). Another source of ready-made nanocrystal quantum dots is Evident Technologies. They sell kits containing a number of small (although somewhat expensive) vials of CdSe\/ZnS and lead sulfide (PbS) quantum dots.<\/p>\n\n\n\n

As shown in\u00a0Figure 112, a quantum particle trapped in a box can absorb a photon (or be excited in some other way), causing the system to move to a higher energy state. However, the higher energy state is unstable, and will eventually decay to the ground state by emitting a photon with energy\u00a0E<\/em>photon<\/sub>\u00a0=\u00a0hf<\/em>\u00a0= \u0394E.<\/em>\u00a0Similarly, when a quantum dot is illuminated by a photon of sufficiently high energy, an excited electron-hole pair forms inside the quantum dot. When the electron and hole recombine, the quantum dot returns to its ground state, and a photon is emitted with a wavelength dependent on the radius of the quantum dot. Smaller dots emit photons with shorter wavelengths.<\/p>\n\n\n\n

Figure 112\u00a0A quantum particle in a box can move to a higher energy state when it absorbs a photon of sufficient energy (a<\/strong>), (b<\/strong>) The excited state is not stable, so the system will tend to return to its ground state by emitting a photon with energy equal to the difference in energy between the two states.\"\"<\/p>\n\n\n\n

Before we experiment with real quantum dots, we must modify the equations to accommodate the fact that the quantum dots are not one-dimensional \u201cboxes,\u201d but instead have a spherical geometry. We must also account for the presence of two particles within the box (an electron-hole pair), and consider that the box is not empty, but rather filled with a semiconductor material (which increases the ground state by 2.15 \u00d7 10\u221219<\/sup> J). We won\u2019t go into the derivation of the equation for the InP\/ZnS quantum dot, but instead limit ourselves to stating that the energy of the fluorescence (emitted) photon is:<\/p>\n\n\n\n

\"equation\"\/<\/figure>\n\n\n\n

where<\/em><\/p>\n\n\n\n

R<\/em> = radius of the quantum dot<\/p>\n\n\n\n

m<\/em>e<\/sub> = effective mass of the electron inside the semiconductor = 7.29 \u00d7 10\u221232<\/sup> kg<\/p>\n\n\n\n

m<\/em>h<\/sub> = effective mass of the hole inside the semiconductor = 5.47 \u00d7 10\u221231<\/sup> kg<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

To conduct an experiment, illuminate a quantum-dot suspension with a UV LED (at a wavelength of approximately 405 nm) as shown in\u00a0Figure 113a. Use the spectrometer that you built in chapter 4 (Figure 80) to photograph the spectrum of the light emitted by each quantum-dot suspension. Process the photographed spectra using VisualSpec (as discussed in chapter 4) to obtain spectral plots similar to those of\u00a0Figure 113b. What is the peak wavelength you measure for each quantum-dot size? How well do your measurements match the theoretical peak emission (Table 6)? Plot the quantum-dot radius versus the peak emission. What type of relationship does this graph show?<\/p>\n\n\n\n

Figure 113\u00a0Suspensions of quantum-dot nanocrystals illuminated with UV light behave very similarly to a theoretical quantum particle in a box (a<\/strong>), (b<\/strong>) The fluorescence spectra from various quantum-dot suspensions agree with Schr\u00f6dinger\u2019s equation for the quantum particle in a box if adjustments are made to account for the presence of two particles within the box (an electron-hole pair), and for the fact that the \u201cbox\u201d is filled with a semiconductor material.\"\"<\/p>\n\n\n\n

TABLE 6\u00a0Nanocrystal Radii and Calculated Fluorescence Wavelength for Quantum-Dot Suspensionsa<\/sup><\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n","protected":false},"excerpt":{"rendered":"

There are very few real-world systems that behave like the simplified mathematical model we used above for a particle in a box. However, recent developments in the nascent field of nanotechnology have made it possible to produce nanostructures of only 2\u201310 nm in size that exhibit behavior close to the simple particle-in-a-box model we discussed. These tiny structures are […]<\/p>\n","protected":false},"author":1,"featured_media":4199,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[517],"tags":[],"class_list":["post-4449","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-schrodinger-and-his-zombie-cat"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/3d-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4449","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4449"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4449\/revisions"}],"predecessor-version":[{"id":4450,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4449\/revisions\/4450"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4199"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4449"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4449"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4449"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}