Chemistry
Quantum Numbers
Quantum numbers are a set of four values used to describe the unique properties of an electron in an atom. These values include the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms). They provide information about the energy, shape, orientation, and spin of an electron within an atom.
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10 Key excerpts on "Quantum Numbers"
eBook - PDF- Stanley E. Manahan(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Quantum Numbers Summarized Quantum Numbers Summarized The information given by each quantum number is summarized below: The information given by each quantum number is summarized below: The value of the principal quantum number The value of the principal quantum number • • n gives the shell and main gives the shell and main energy level of the electron. energy level of the electron. The value of the azimuthal quantum number The value of the azimuthal quantum number • • l specifies sublevels with specifies sublevels with somewhat different energies within the main energy levels and describes somewhat different energies within the main energy levels and describes the shapes of orbitals. the shapes of orbitals. The magnetic quantum The magnetic quantum • • number m number m l distinguishes the orientations in space distinguishes the orientations in space of orbitals in a subshell and may provide additional distinctions of of orbitals in a subshell and may provide additional distinctions of orbital shapes. orbital shapes. The spin quantum number The spin quantum number • • m s , with possible values of only , with possible values of only ½ ½ and and ½ , , accounts for the fact that each orbital can be occupied by a maximum accounts for the fact that each orbital can be occupied by a maximum number of only 2 electrons with opposing spins. number of only 2 electrons with opposing spins. Specification of the values of four Quantum Numbers describes each electron in an Specification of the values of four Quantum Numbers describes each electron in an atom, as shown in Table 3.3. atom, as shown in Table 3.3. For each electron in an atom, the orbital that it occupies and the direction of its For each electron in an atom, the orbital that it occupies and the direction of its spin are specified by values of spin are specified by values of n , , l , , m l , and , and m s assigned to it.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
(To completely description the terms of the ground and excited states, one must choose a set of Quantum Numbers that describe the state of the entire atom and correspond to quantities that are conserved.) All the electrons with the same principal quantum number n correspond to one electron shell . The maximum number of electrons in the n th shell is equal to 2 n 2 . The shell with n = 1 is called the K -shell, the shell with n = 2 is called the L -shell, and further, by the alphabet. All electrons in a given shell with a fixed number l form an electron subshell with 2 ( 2 l + 1 ) electrons; l = 0 , ... , n – 1 . The electrons of a given shell with a larger value of l have a larger energy due to increasing contribution of the centrifugal energy. The electron state is denoted by the number of the principal quantum number followed by the letter corresponding to its orbital number. If an atom contains several electrons in the same subshell, then the superscript on the right is the number of electrons. We present the electron configurations for several atoms: K -shell: 1 H: 1 s , 2 He: 1 s 2 ; L -shell: 3 Li: 1 s 2 2 s , 4 Be: 1 s 2 2 s 2 , 5 B: 1 s 2 2 s 2 2 p , 6 C: 1 s 2 2 s 2 2 p 2 , 7 N: 1 s 2 2 s 2 2 p 3 , 8 O: 1 s 2 2 s 2 2 p 4 , 9 F: 1 s 2 2 s 2 2 p 5 , 10 Ne: 1 s 2 2 s 2 2 p 6 ; two subshells ( s and p ) in the M -shell: 11 Na: [Ne] 3 s , 12 Mg: [Ne] 3 s 2 , 13 Al: [Ne] 3 s 2 3 p , 14 Si: [Ne] 3 s 2 3 p 2 , 15 P: [Ne 3 s 2 3 p 3 , 16 S: [Ne] 3 s 2 3 p 4 , 17 Cl: [Ne] 3 s 2 3 p 5 , 18 Ar: [Ne] 3 s 2 3 p 6 ; 588 Q UANTUM M ECHANICS . A TOMIC P HYSICS s -subshell in the N -shell: 19 K: [Ne] 3 s 2 3 p 6 4 s , 20 Ca: [Ne] 3 s 2 3 p 6 4 s 2 ; d -subshell in the M -shell: 21 Sc: [Ne] 3 s 2 3 p 6 3 d 4 s 2 , 22 Ti: [Ne] 3 s 2 3 p 6 3 d 2 4 s 2 , 23 V: [Ne] 3 s 2 3 p 6 3 d 3 4 s 2 , 24 Cr: [Ne] 3 s 2 3 p 6 3 d 5 4 s ; and so on. The notation [Ne] stands for the electron configuration of 10 Ne ( 1 s 2 2 s 2 2 p 6 ).
eBook - PDF- Gary N. Felder, Kenny M. Felder(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
The term “principal quantum number” refers to the fact that n determines energy: each eigenstate ψ nlm l has energy E n = −(1/n 2 ) Ry, where 1 Ry ≈ 13.6 eV is a unit of energy called a “rydberg.” So the ground state of hydrogen has energy −1 Ry ≈ −13.6 eV, the first excited state has energy −(1/4) Ry ≈ −3.4 eV, and so on. As we discussed in Section 4.1, these energies are negative because the negative potential energy has a higher magnitude than the positive kinetic energy, meaning the electrons are bound to the atom. The energy differences tell you how much energy is required to kick an electron 1 We are using the word “nucleus” rather than “proton” because many of the ideas presented in this section scale up to larger atoms. 2 Some sources call the third quantum number m without the subscript, but we’ll follow the sources that use m l to distinguish it from the electron mass m e and from another quantum number m s that we’ll introduce in Section 7.5. 318 7 The Hydrogen Atom from a low-energy state to a high-energy one, or how much is released if an electron goes the other way. It takes about 10 eV of energy to knock a hydrogen atom from the ground state to the first excited state, but from there it only takes about 2 eV more to kick it up to the second excited state. If you start with a hydrogen atom in the ground state, it takes about 13.6 eV to knock out the electron completely and ionize the atom. Note that one energy eigenstate ψ nlm l of a hydrogen atom is characterized by several different Quantum Numbers, but the actual energy E n associated with that state is just a function of n. Different states that have the same energy – in this case, that means the same n but different values of the other Quantum Numbers – are said to be “degenerate” states. The “Angular Momentum Quantum Number” l The quantum number l can be any integer from 0 to n − 1, inclusive. In the ground state (n = 1) the only option is l = 0; each step up in n allows for one more possible l.
- Albert T. Jr. Fromhold(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
In practice, then, one of the most difficult parts of the problem is to estimate or calculate the potential. The details are quite beyond the scope of our present treatment; however, the important results of such an approach are of interest to us, since they justify using one-electron eigenvalues and eigenfunctions as a semantic framework for describing the multielectron atom. Including spin, there is once again a set of four Quantum Numbers («, /, m h and m s ) required to specify an electronic state. The wave function specified by a given set of Quantum Numbers is called an orbital, or more specifically, an atomic orbital, in analogy with the older Bohr theory in which electrons were considered to travel in planetary orbits in accordance with classical mechanics. The Pauli exclusion principle again requires that no two electrons have the same set of Quantum Numbers [viz., the principal quantum number n characteristic of the total energy of the electron, the angular momentum (azimuthal) quantum number /characteristic of the total orbital angular momentum of the electron, the magnetic quantum number m t characteristic of the orientation of the magnetic moment with respect to the z axis, and the spin quantum number m s characteristic of the orientation of the electron spin magnetic moment]. The orbital angular momentum and magnetic Quantum Numbers / and m x are the same as the Quantum Numbers / and m in the hydrogen atom since the separation of variables with the more general central potential V(f) proceeds in exactly the same way as for the single electron (or hydrogen atom) where U(r) = — Ze 2 /4n£ 0 r, thus yielding the same equations for the Θ(θ) and Φ(φ) factors in the product wave function ^(r, t), φ(τ, t) = ( )©( ) (0) exp[ -( / ) ]. (1.408) The electron spin quantum number m s = ± is likewise the same as in the hydrogen atom.
eBook - ePubCollege Physics Essentials, Eighth Edition
Electricity and Magnetism, Optics, Modern Physics (Volume Two)
- Jerry D. Wilson, Anthony J. Buffa, Bo Lou(Authors)
- 2019(Publication Date)
- CRC Press(Publisher)
∙ . The reasoning is that three Quantum Numbers are needed because the electron can move in three dimensions. Thus the Bohr theory, being essentially a two-dimensional (flat planar) model was deemed incomplete, requiring only one quantum number.The quantum number ∙ is the orbital quantum number . It is associated with the orbital angular momentum of the electron. For each value of n , the ∙ quantum number has integer values from zero up to a maximum value of n − 1. For example, if n = 3, the three possible values of ∙ are 0, 1, and 2. Thus the number of different ∙ values for a given n value is equal to n , and in the hydrogen atom, the energy depends only on n . ▼ Figure 28.4 shows three orbits with different angular momenta, but the same energy (i.e., the same n value). Thus, orbits with the same n value, but different ∙ values, have the same energy and are said to be degenerate .▲ Figure 28.4 The orbital quantum number ℓ The orbits of an electron are shown for the second excited state in hydrogen. For the principal quantum number n = 3, there are three different values of angular momentum (corresponding to the three differently shaped orbits and three different values of ℓ ), all of the same total energy. The circular orbit has the maximum angular momentum; the narrowest orbit classically has zero angular momentum.The Quantum Numbers m ∙ is the magnetic quantum number . The name originated from experiments in which an external magnetic field was applied to a sample. They showed that a particular energy level (with given values of n and ∙ ) of a hydrogen atom actually consists of several orbits that differ slightly in energy only when in a magnetic field. Thus, in the absence of a magnetic field, there was additional energy degeneracy. Clearly, there must be more to the description of the orbit than just n and ∙ . The quantum number m ∙ was introduced to enumerate the number of levels that existed for a given orbital quantum number ∙
eBook - PDF- Robert J. Silbey, Robert A. Alberty, Moungi G. Bawendi(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
1 2 1 2 1 2 1 2 ( ) 1 1 ( ) ( ) 10.12 Angular Momentum of Many-Electron Atoms s S s J L S a Table 10.6 in the direction is directly proportional to the spin quantum number for the component for the whole atom. Thus, equation 10.101 becomes (10 102) The value of can range from to , depending on the orientation of the total spin angular momentum for the atom in a magnetic field. For an atom containing two electrons, the spin quantum number is a maximum if the spins are parallel and a minimum if the spins are opposed. If the spin Quantum Numbers of the individual electrons are represented by and , the possible values of are 1 (10 103) For a two-electron atom, 1 0 The total angular momentum of an atom is the vector sum of all the or- bital and spin angular momenta of electrons in it. Like other angular mo- menta is quantized, and its quantum number can take on only integer and half-integer values. In principle these can range up to the sum of the orbital angu- lar momentum Quantum Numbers and spin Quantum Numbers for the individual electrons in the atom. We are interested in the total angular momentum quantum number because states with different angular momenta differ in energy. This is partly because the electrons repel each other electrostatically, and the strengths of the repulsions depend on the distribution of electric charge, which we know is connected with the orbital Quantum Numbers and . This effect is given the shorthand name orbital–orbital, or , interaction. In addition, the atomic states must obey the Pauli exclusion principle, which, because it dictates which orbital states can be associated with which spin states, indirectly brings in a spin–spin, or , interaction determined by and In addition to these effects, there is also a direct spin–orbit, or interaction.
eBook - PDF- Robert J. Silbey, Robert A. Alberty, George A. Papadantonakis, Moungi G. Bawendi(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
5. Electrons have spin of 1 2 , so their spin angular momentum S is added to orbital angular momentum L to yield the total angular momentum of the atomic system J = L + S. 6. In the variational method, a trial wavefunction is used to calculate an upper bound for the energy eigenvalue of a system. 7. The Pauli exclusion principle requires that the wavefunction for any system of electrons must be antisymmetric with respect to the interchange of any two electrons. This explains why the first excited state of helium is a triplet state. 8. The electron configurations of atoms explain the structure of the periodic table. References 389 9. Atomic states can be classified by atomic term symbols that summarize L, S, and J for an atom. 10. Not all possible transitions can occur in spectroscopic transitions because photons have intrinsic angular momen- tum equal to one unit, and angular momentum is conserved. 11. L x , L y , and L z are not compatible observables, do not have simultaneous eigenfunctions and cannot be measured simultaneously. 12. The perturbation-theory approximation method deducts dynamical properties from the “unperturbed” to the “perturbed” case. QUESTIONS ON CONCEPTS AND IDEAS 1. What is the physical meaning of the Quantum Numbers l and m l for the hydrogen atom? 2. Is there any physical reason between the fact that the Coulomb potential bind the electron in the hydrogen atom? How the principal quantum number, n, originates? 3. Order the n = 2 states of H atom in terms of increasing kinetic energy associated with rotation. 4. Correlate the trends in atomic radius and first ionization energy for the rare gases. 5. Which ionization energy (first, second, and so on) should exhibit a large jump for each of the following species: Li, Be, B, and C? 6. Calculate the eighteenth ionization potential of Ar. 7. Which of the first 10 elements of the periodic table have paramagnetic ground states? 8.
eBook - PDFChemistry
The Molecular Nature of Matter
- Neil D. Jespersen, Alison Hyslop(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
6d 5f 5d 6s 6p 7s 4f 4d 5s 5p 3d 4s 4p 3s 3p 2s 1s 2p n = 2, = 1, m = 1 n = 2, = 1, m = 0 n = 2, = 1, m = –1 Energy 7p n = 2, = 0, m = 0 n = 1, = 0, m = 0 FIGURE 7.20 Approximate energy level diagram for atoms with two or more electrons. The Quantum Numbers associated with the orbitals in the first two shells are also shown. 7.6 Electron Spin 333 7.6 Electron Spin 333 7.6 Electron Spin Recall from Section 7.3 that an atom is in its most stable state (its ground state) when its elec- trons have the lowest possible energies. This occurs when the electrons “occupy” the lowest energy orbitals that are available. But what determines how the electrons “fill” these orbitals? Fortunately, there are some simple rules that govern both the maximum number of electrons that can be in a particular orbital and how orbitals with the same energy become filled. One important factor that influences the distribution of electrons is the phenomenon known as electron spin. Spin Quantum Number When a beam of atoms with an odd number of electrons is passed through an uneven mag- netic field, the beam is split in two, as shown in Figure 7.21. The splitting occurs because the electrons within the atoms interact with the magnetic field in two different ways. The elec- trons behave like tiny magnets, and they are attracted to one or the other of the poles depend- ing on their orientation. This can be explained by imagining that an electron spins around its axis, like a toy top. A moving charge creates a moving electric field, which in turn creates a magnetic field. The spinning electrical charge of the electron creates its own magnetic field. This electron spin could occur in two possible directions, which accounts for the two beams. The electron can spin in either of two directions in the presence of an external magnetic field. Without a magnetic field the electrons and their spins are usually oriented in random directions.
- Alexandre Zagoskin(Author)
- 2015(Publication Date)
- Teach Yourself(Publisher)
anticommute. You can check that for fermionsTherefore no matter what, the fermion wave functions cannot become usual waves in the classical limit: such a ‘wave’ would have to have zero amplitude. So, e.g. a peace of metal, which consists of fermions, must be solid to exist.6Fact-check1 The principal quantum number determinesa the bound state energy of an electronb the ionization energyc the number of electrons in an atomd the Rydberg constant2 The orbital Quantum Numbers determinea the momentum of an electron in an atomb the shape of the electron wave functionc the number of electrons in an atomd the energy of an electron in an atom3 The Pauli exclusion principle states thata two quantum particles cannot occupy the same quantum stateb two fermions cannot occupy the same quantum statec no more than two bosons can occupy the same quantum stated electrons have spins4 Spin characterizesa an orbital motion of a quantum particleb an orbital angular momentum of a quantum particlec an intrinsic angular moment of a quantum particled the number of electrons in an atom5 If a particle has spin angular momentum , thena the projection of its angular momentum on z-axis may take seven valuesb may take eight valuesc this particle is a bosond this particle is a fermion6 If two particles are exchanged, the wave function of a system of identical bosonsa becomes zerob changes signc acquires a factor γ ≠ 1d stays the same7 Of the following, which are bosons?a α particlesb electronsc helium-4 atomsd helium-3 atoms8 What will be the result of acting on a vacuum state by the following set of creation/annihilation Bose operators (up to a numerical factor)?a |5,3,0,1,0…〉b |5,3,0,0,0…〉c |5,1,0,0,0…〉d |5,2,0,1,0…〉9 The Fock space is a mathematical construction, whicha Is a Hilbert space for a system of infinite number of particlesb replaces the Hilbert space for a system of identical quantum particlesc
eBook - PDFChemistry
The Molecular Nature of Matter
- Neil D. Jespersen, Alison Hyslop(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
N N S S The electron can spin in either of two directions in the pres- ence of an external magnetic field. Without a magnetic field the electrons and their spins are usually oriented in random directions. 7.6 | Electron Spin 323 they are not integers aren’t very important to us, but the fact that there are only two values is very significant. Pauli Exclusion Principle In 1925 an Austrian physicist, Wolfgang Pauli (1900–1958), expressed the importance of electron spin in determining electronic structure. The Pauli exclusion principle states that no two electrons in the same atom can have identical values for all four of their Quantum Numbers. Suppose two electrons were to occupy the 1s orbital of an atom. Each electron would have n = 1, ℓ = 0, and m ℓ = 0. Since these three Quantum Numbers are the same for both electrons, the exclusion principle requires that their fourth Quantum Numbers (the spin Quantum Numbers) be different; one electron must have m s = +½ and the other, m s = -½. No more than two electrons can occupy the 1s orbital of the atom simul- taneously because there are only two possible values of m s . Thus, the Pauli exclusion prin- ciple is really telling us that when the maximum number of electrons, two, are in the same orbital, they must have opposite spins. The limit of two electrons per orbital also limits the maximum electron populations of the shells and subshells. For the subshells we have Subshell Number of Orbitals Maximum Number of Electrons s 1 2 p 3 6 d 5 10 f 7 14 The maximum electron population per shell is 2n 2 : Shell Subshells Maximum Shell Population 1 1s 2 2 2s2p 8 (2 + 6) 3 3s3p3d 18 (2 + 6 + 10) 4 4s 4p4d4f 32 (2 + 6 + 10 +14) Without looking at the tables in the text, how many electrons are there in a d subshell? (Hint: How many orbitals are in the d subshell?) Determine the maximum number of electrons that are in the n = 5 shell.
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