What does the quantum mechanical view of the atom require?

The newsflash interrupts your favorite TV program. “There has been a hold-up at the First National Bank. The suspect fled in a car and is believed to be somewhere in the downtown district. Everyone is asked to be on the alert.” The robber can be located only within a certain area—the police do not have an exact location, just a general idea as to the whereabouts of the thief.

In 1926, Austrian physicist Erwin Schrödinger (1887–1961) used the wave-particle duality of the electron to develop and solve a complex mathematical equation that accurately described the behavior of the electron in a hydrogen atom. The quantum mechanical model of the atom comes from the solution to Schrödinger’s equation. Quantization of electron energies is a requirement in order to solve the equation. This is unlike the Bohr model, in which quantization was simply assumed with no mathematical basis.

Recall that in the Bohr model, the exact path of the electron was restricted to very well-defined circular orbits around the nucleus. The quantum mechanical model is a radical departure from that. Solutions to the Schrödinger wave equation, called wave functions , give only the probability of finding an electron at a given point around the nucleus. Electrons do not travel around the nucleus in simple circular orbits.

Figure 1. An electron cloud: the darker region nearer the nucleus indicates a high probability of finding the electron, while the lighter region further from the nucleus indicates a lower probability of finding the electron.

The location of the electrons in the quantum mechanical model of the atom is often referred to as an electron cloud. The electron cloud can be thought of in the following way: Imagine placing a square piece of paper on the floor with a dot in the circle representing the nucleus. Now take a marker and drop it onto the paper repeatedly, making small marks at each point the marker hits. If you drop the marker many, many times, the overall pattern of dots will be roughly circular. If you aim toward the center reasonably well, there will be more dots near the nucleus and progressively fewer dots as you move away from it. Each dot represents a location where the electron could be at any given moment. Because of the uncertainty principle, there is no way to know exactly where the electron is. An electron cloud has variable densities: a high density where the electron is most likely to be and a low density where the electron is least likely to be (Figure 1).

In order to specifically define the shape of the cloud, it is customary to refer to the region of space within which there is a 90% probability of finding the electron. This is called an orbital , the three-dimensional region of space that indicates where there is a high probability of finding an electron.

• The Schrödinger wave equation replaced the Bohr ideas about electron location with an uncertainty factor.
• The location of the electron can only be given as a probability that the electron is somewhere in a certain area.

Use the link below to answer the following questions:

//science.howstuffworks.com/atom8.htm

1. What was one problem with the Bohr model of the atom?
2. What did Heisenberg show about electrons?
3. What did Schrödinger derive?

1. What does the quantum mechanical view of the atom require?
2. What is a wave function?
3. What does a high density electron cloud suggest?

Glossary

• electron cloud: The location of the electrons in the quantum mechanical model of the atom.
• orbital: The three-dimensional region of space that indicates where there is a high probability of finding an electron.
• quantum mechanical model: A model of the atom that derives from the Schrödinger wave equation and deals with probabilities.
• wave function: Give only the probability of finding an electron at a given point around the nucleus.

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Although","noIndex":0,"noFollow":0},"content":"<p>Two models of atomic structure are in use today: the Bohr model and the quantum mechanical model. The quantum mechanical model is based on mathematics. Although it is more difficult to understand than the Bohr model, it can be used to explain observations made on complex atoms.</p>\n<p class=\"Remember\">A model is useful because it helps you understand what’s observed in nature. It’s not unusual to have more than one model represent and help people understand a particular topic.</p>\n<p>The quantum mechanical model is based on <i>quantum theory</i>, which says matter also has properties associated with waves. According to quantum theory, it’s impossible to know the exact position and momentum of an electron at the same time. This is known as the <i>Uncertainty Principle</i>.</p>\n<p>The quantum mechanical model of the atom uses complex shapes of <i>orbitals</i> (sometimes called <i>electron clouds</i>), volumes of space in which there is <i>likely</i> to be an electron. So, this model is based on probability rather than certainty.</p>\n<p>Four numbers, called <i>quantum numbers</i>, were introduced to describe the characteristics of electrons and their orbitals:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">Principal quantum number: n</p>\n </li>\n <li><p class=\"first-para\">Angular momentum quantum number: l</p>\n </li>\n <li><p class=\"first-para\">Magnetic quantum number:</p>\n<img src=\"//sg.cdnki.com/what-does-the-quantum-mechanical-view-of-the-atom-require---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NDQzNC5pbWFnZTAucG5n.webp\" width=\"21\" height=\"24\" alt=\"image0.png\"/>\n </li>\n <li><p class=\"first-para\">Spin quantum number:</p>\n<img src=\"//sg.cdnki.com/what-does-the-quantum-mechanical-view-of-the-atom-require---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NDQzNS5pbWFnZTEucG5n.webp\" width=\"21\" height=\"24\" alt=\"image1.png\"/>\n </li>\n</ul>\n<h2 id=\"tab1\" >The principal quantum number</h2>\n<p>The principal quantum number n describes the average distance of the orbital from the nucleus — and the energy of the electron in an atom. It can have positive integer (whole number) values: 1, 2, 3, 4, and so on. The larger the value of n, the higher the energy and the larger the orbital. Chemists sometimes call the orbitals <i>electron shells</i>.</p>\n<h2 id=\"tab2\" >The angular momentum quantum number</h2>\n<p>The angular momentum quantum number <i>l</i> describes the shape of the orbital, and the shape is limited by the principal quantum number n: The angular momentum quantum number <i>l</i> can have positive integer values from 0 to n–1. For example, if the n value is 3, three values are allowed for <i>l</i>: 0, 1, and 2.</p>\n<p class=\"Remember\">The value of <i>l</i> defines the shape of the orbital, and the value of n defines the size.</p>\n<p>Orbitals that have the same value of n but different values of <i>l</i> are called <i>subshells</i>. These subshells are given different letters to help chemists distinguish them from each other. The following table shows the letters corresponding to the different values of <i>l</i>.</p>\n<table>\n<caption>\nLetter Designations of the Subshells\n</caption>\n<tr>\n<th>Value of l (subshell)</th>\n<th>Letter</th>\n</tr>\n<tr>\n<td>0</td>\n<td>s</td>\n</tr>\n<tr>\n<td>1</td>\n<td>p</td>\n</tr>\n<tr>\n<td>2</td>\n<td>d</td>\n</tr>\n<tr>\n<td>3</td>\n<td>f</td>\n</tr>\n<tr>\n<td>4</td>\n<td>g</td>\n</tr>\n</table>\n<p>When chemists describe one particular subshell in an atom, they can use both the n value and the subshell letter — 2p, 3d, and so on. Normally, a subshell value of 4 is the largest needed to describe a particular subshell. If chemists ever need a larger value, they can create subshell numbers and letters. </p>\n<p>The following figure shows the shapes of the s, p, and d orbitals.</p>\n<img src=\"//sg.cdnki.com/what-does-the-quantum-mechanical-view-of-the-atom-require---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NDQzNi5pbWFnZTIuanBn.webp\" width=\"264\" height=\"400\" alt=\"image2.jpg\"/>\n<p>As shown in the top row of the figure (a), there are two s orbitals — one for energy level 1 (1s) and the other for energy level 2 (2s). The s orbitals are spherical with the nucleus at the center. Notice that the 2s orbital is larger in diameter than the 1s orbital. In large atoms, the 1s orbital is nestled inside the 2s, just like the 2p is nestled inside the 3p.</p>\n<p>The second row of the figure (b) shows the shapes of the p orbitals, and the last two rows (c) show the shapes of the d orbitals. Notice that the shapes get progressively more complex.</p>\n<h2 id=\"tab3\" >The magnetic quantum number</h2>\n<p>The magnetic quantum number is designated as:</p>\n<img src=\"//sg.cdnki.com/what-does-the-quantum-mechanical-view-of-the-atom-require---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NDQzNy5pbWFnZTMucG5n.webp\" width=\"21\" height=\"24\" alt=\"image3.png\"/>\n<p>This number describes how the various orbitals are oriented in space. The value of this number depends on the value of <i>l</i>. The values allowed are integers from –<i>l</i> to 0 to +<i>l</i>. For example, if the value of <i>l</i> = 1 (p orbital), you can write three values for this number: –1, 0, and +1. This means that there are three different p subshells for a particular orbital. The subshells have the same energy but different orientations in space.</p>\n<p>The second row (b) of the figure shows how the p orbitals are oriented in space. Notice that the three p orbitals correspond to magnetic quantum number values of –1, 0, and +1, oriented along the x, y, and z axes.</p>\n<h2 id=\"tab4\" >The spin quantum number</h2>\n<p>The fourth and final quantum number is the spin quantum number, designated as:</p>\n<img src=\"//sg.cdnki.com/what-does-the-quantum-mechanical-view-of-the-atom-require---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NDQzOC5pbWFnZTQucG5n.webp\" width=\"21\" height=\"24\" alt=\"image4.png\"/>\n<p>This number describes the direction the electron is spinning in a magnetic field — either clockwise or counterclockwise. Only two values are allowed: +1/2 or –1/2. For each subshell, there can be only two electrons, one with a spin of +1/2 and another with a spin of –1/2.</p>","description":"<p>Two models of atomic structure are in use today: the Bohr model and the quantum mechanical model. The quantum mechanical model is based on mathematics. Although it is more difficult to understand than the Bohr model, it can be used to explain observations made on complex atoms.</p>\n<p class=\"Remember\">A model is useful because it helps you understand what’s observed in nature. It’s not unusual to have more than one model represent and help people understand a particular topic.</p>\n<p>The quantum mechanical model is based on <i>quantum theory</i>, which says matter also has properties associated with waves. According to quantum theory, it’s impossible to know the exact position and momentum of an electron at the same time. This is known as the <i>Uncertainty Principle</i>.</p>\n<p>The quantum mechanical model of the atom uses complex shapes of <i>orbitals</i> (sometimes called <i>electron clouds</i>), volumes of space in which there is <i>likely</i> to be an electron. So, this model is based on probability rather than certainty.</p>\n<p>Four numbers, called <i>quantum numbers</i>, were introduced to describe the characteristics of electrons and their orbitals:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">Principal quantum number: n</p>\n </li>\n <li><p class=\"first-para\">Angular momentum quantum number: l</p>\n </li>\n <li><p class=\"first-para\">Magnetic quantum number:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/164434.image0.png\" width=\"21\" height=\"24\" alt=\"image0.png\"/>\n </li>\n <li><p class=\"first-para\">Spin quantum number:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/164435.image1.png\" width=\"21\" height=\"24\" alt=\"image1.png\"/>\n </li>\n</ul>\n<h2 id=\"tab1\" >The principal quantum number</h2>\n<p>The principal quantum number n describes the average distance of the orbital from the nucleus — and the energy of the electron in an atom. It can have positive integer (whole number) values: 1, 2, 3, 4, and so on. The larger the value of n, the higher the energy and the larger the orbital. Chemists sometimes call the orbitals <i>electron shells</i>.</p>\n<h2 id=\"tab2\" >The angular momentum quantum number</h2>\n<p>The angular momentum quantum number <i>l</i> describes the shape of the orbital, and the shape is limited by the principal quantum number n: The angular momentum quantum number <i>l</i> can have positive integer values from 0 to n–1. For example, if the n value is 3, three values are allowed for <i>l</i>: 0, 1, and 2.</p>\n<p class=\"Remember\">The value of <i>l</i> defines the shape of the orbital, and the value of n defines the size.</p>\n<p>Orbitals that have the same value of n but different values of <i>l</i> are called <i>subshells</i>. These subshells are given different letters to help chemists distinguish them from each other. The following table shows the letters corresponding to the different values of <i>l</i>.</p>\n<table>\n<caption>\nLetter Designations of the Subshells\n</caption>\n<tr>\n<th>Value of l (subshell)</th>\n<th>Letter</th>\n</tr>\n<tr>\n<td>0</td>\n<td>s</td>\n</tr>\n<tr>\n<td>1</td>\n<td>p</td>\n</tr>\n<tr>\n<td>2</td>\n<td>d</td>\n</tr>\n<tr>\n<td>3</td>\n<td>f</td>\n</tr>\n<tr>\n<td>4</td>\n<td>g</td>\n</tr>\n</table>\n<p>When chemists describe one particular subshell in an atom, they can use both the n value and the subshell letter — 2p, 3d, and so on. Normally, a subshell value of 4 is the largest needed to describe a particular subshell. If chemists ever need a larger value, they can create subshell numbers and letters. </p>\n<p>The following figure shows the shapes of the s, p, and d orbitals.</p>\n<img src=\"//www.dummies.com/wp-content/uploads/164436.image2.jpg\" width=\"264\" height=\"400\" alt=\"image2.jpg\"/>\n<p>As shown in the top row of the figure (a), there are two s orbitals — one for energy level 1 (1s) and the other for energy level 2 (2s). The s orbitals are spherical with the nucleus at the center. Notice that the 2s orbital is larger in diameter than the 1s orbital. In large atoms, the 1s orbital is nestled inside the 2s, just like the 2p is nestled inside the 3p.</p>\n<p>The second row of the figure (b) shows the shapes of the p orbitals, and the last two rows (c) show the shapes of the d orbitals. Notice that the shapes get progressively more complex.</p>\n<h2 id=\"tab3\" >The magnetic quantum number</h2>\n<p>The magnetic quantum number is designated as:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/164437.image3.png\" width=\"21\" height=\"24\" alt=\"image3.png\"/>\n<p>This number describes how the various orbitals are oriented in space. The value of this number depends on the value of <i>l</i>. The values allowed are integers from –<i>l</i> to 0 to +<i>l</i>. For example, if the value of <i>l</i> = 1 (p orbital), you can write three values for this number: –1, 0, and +1. This means that there are three different p subshells for a particular orbital. The subshells have the same energy but different orientations in space.</p>\n<p>The second row (b) of the figure shows how the p orbitals are oriented in space. Notice that the three p orbitals correspond to magnetic quantum number values of –1, 0, and +1, oriented along the x, y, and z axes.</p>\n<h2 id=\"tab4\" >The spin quantum number</h2>\n<p>The fourth and final quantum number is the spin quantum number, designated as:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/164438.image4.png\" width=\"21\" height=\"24\" alt=\"image4.png\"/>\n<p>This number describes the direction the electron is spinning in a magnetic field — either clockwise or counterclockwise. Only two values are allowed: +1/2 or –1/2. For each subshell, there can be only two electrons, one with a spin of +1/2 and another with a spin of –1/2.</p>","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33762,"title":"Chemistry","slug":"chemistry","_links":{"self":"//dummies-api.dummies.com/v2/categories/33762"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"The principal quantum number","target":"#tab1"},{"label":"The angular momentum quantum number","target":"#tab2"},{"label":"The magnetic quantum number","target":"#tab3"},{"label":"The spin quantum number","target":"#tab4"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":253707,"title":"How to Make Unit Conversions","slug":"make-unit-conversions","categoryList":["academics-the-arts","science","chemistry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/253707"}},{"articleId":251836,"title":"How to Convert between Units Using Conversion Factors","slug":"convert-units-using-conversion-factors","categoryList":["academics-the-arts","science","chemistry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/251836"}},{"articleId":251010,"title":"How to Build Derived Units from Base Units","slug":"build-derived-units-base-units","categoryList":["academics-the-arts","science","chemistry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/251010"}},{"articleId":251005,"title":"How to Do Arithmetic with Significant Figures","slug":"arithmetic-significant-figures","categoryList":["academics-the-arts","science","chemistry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/251005"}},{"articleId":250992,"title":"How to Add and Subtract with Exponential Notation","slug":"add-subtract-exponential-notation","categoryList":["academics-the-arts","science","chemistry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/250992"}}]},"hasRelatedBookFromSearch":true,"relatedBook":{"bookId":282434,"slug":"organic-chemistry-i-for-dummies-2nd-edition","isbn":"9781119293378","categoryList":["academics-the-arts","science","chemistry"],"amazon":{"default":"//www.amazon.com/gp/product/1119293375/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119293375/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119293375-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119293375/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119293375/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//catalogimages.wiley.com/images/db/jimages/9781119293378.jpg","width":250,"height":350},"title":"Organic Chemistry I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><p><B><b data-author-id=\"9321\">Arthur Winter</b>, PhD, </b>is the author of the popular <i>Organic Chemistry Help!</i> website chemhelper.com and <i>Organic Chemistry I For Dummies</i>. 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Two models of atomic structure are in use today: the Bohr model and the quantum mechanical model. The quantum mechanical model is based on mathematics. Although it is more difficult to understand than the Bohr model, it can be used to explain observations made on complex atoms.

A model is useful because it helps you understand what’s observed in nature. It’s not unusual to have more than one model represent and help people understand a particular topic.

The quantum mechanical model is based on quantum theory, which says matter also has properties associated with waves. According to quantum theory, it’s impossible to know the exact position and momentum of an electron at the same time. This is known as the Uncertainty Principle.

The quantum mechanical model of the atom uses complex shapes of orbitals (sometimes called electron clouds), volumes of space in which there is likely to be an electron. So, this model is based on probability rather than certainty.

Four numbers, called quantum numbers, were introduced to describe the characteristics of electrons and their orbitals:

• Principal quantum number: n

• Angular momentum quantum number: l

• Magnetic quantum number:

• Spin quantum number:

The principal quantum number

The principal quantum number n describes the average distance of the orbital from the nucleus — and the energy of the electron in an atom. It can have positive integer (whole number) values: 1, 2, 3, 4, and so on. The larger the value of n, the higher the energy and the larger the orbital. Chemists sometimes call the orbitals electron shells.

The angular momentum quantum number

The angular momentum quantum number l describes the shape of the orbital, and the shape is limited by the principal quantum number n: The angular momentum quantum number l can have positive integer values from 0 to n–1. For example, if the n value is 3, three values are allowed for l: 0, 1, and 2.

The value of l defines the shape of the orbital, and the value of n defines the size.

Orbitals that have the same value of n but different values of l are called subshells. These subshells are given different letters to help chemists distinguish them from each other. The following table shows the letters corresponding to the different values of l.

Letter Designations of the Subshells Value of l (subshell) Letter

0	s
1	p
2	d
3	f
4	g

When chemists describe one particular subshell in an atom, they can use both the n value and the subshell letter — 2p, 3d, and so on. Normally, a subshell value of 4 is the largest needed to describe a particular subshell. If chemists ever need a larger value, they can create subshell numbers and letters.

The following figure shows the shapes of the s, p, and d orbitals.

As shown in the top row of the figure (a), there are two s orbitals — one for energy level 1 (1s) and the other for energy level 2 (2s). The s orbitals are spherical with the nucleus at the center. Notice that the 2s orbital is larger in diameter than the 1s orbital. In large atoms, the 1s orbital is nestled inside the 2s, just like the 2p is nestled inside the 3p.

The second row of the figure (b) shows the shapes of the p orbitals, and the last two rows (c) show the shapes of the d orbitals. Notice that the shapes get progressively more complex.

The magnetic quantum number

The magnetic quantum number is designated as:

This number describes how the various orbitals are oriented in space. The value of this number depends on the value of l. The values allowed are integers from –l to 0 to +l. For example, if the value of l = 1 (p orbital), you can write three values for this number: –1, 0, and +1. This means that there are three different p subshells for a particular orbital. The subshells have the same energy but different orientations in space.

The second row (b) of the figure shows how the p orbitals are oriented in space. Notice that the three p orbitals correspond to magnetic quantum number values of –1, 0, and +1, oriented along the x, y, and z axes.

The spin quantum number

The fourth and final quantum number is the spin quantum number, designated as:

This number describes the direction the electron is spinning in a magnetic field — either clockwise or counterclockwise. Only two values are allowed: +1/2 or –1/2. For each subshell, there can be only two electrons, one with a spin of +1/2 and another with a spin of –1/2.