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From One to Many Electrons… Electron Spin Pauli Exclusion Principle Symmetric and Antisymmetric Wave Functions Slides adapted from: http://web.mst.edu/~pringle/Phys107 Spectral lines (absorption or emission) are caused by photons absorbed or emitted when electrons change their energy state. Changes in the principal quantum number n cause the most noticeable changes. However, changes in other quantum numbers also give rise to changes in electron energies. Such changes typically involve less energy, and result in a "splitting" of the primary lines. 1s→2p so selection rules are not violated! The “ordinary” Zeeman effect. Not all splittings can be explained by the quantum theory developed in chapter 6. It turns out we need another quantum number -- spin. Anomalous Zeeman effect: ??? Ordinary Zeeman effect: classical magnetic dipole in B field Let’s think about electrons and magnetism: If you “shoot” an electron through a region of space with no magnetic field, the electron will experience no deflection (assuming no gravitational forces). If you “shoot” an electron through a region of space with a nonzero magnetic field, you know from P222 that the electron will experience a deflection. - Orbital angular momentum of electron gives rise to magnetic dipole moment: !" !" µ = IA I = ev / 2π r = eL / mr 2 Torque, energy of magnetic dipole in an external magnetic field: Normal Zeeman effect: States with same n different l (magnetic dipole moment) will have different energies ! "! "! τ = IA×B or ! "! "! τ =µ×B "! "! U = −µ • B A silver atom has 47 protons and electrons. It has a single outermost 5s electron, and this 5s electron has zero orbital angular momentum. The single electron acts “sort of” like a lone electron (it “sees” a 47 proton nucleus shielded by 46 electrons, so it is “sort of” like hydrogen. The 5s electron has ℓ=0 and so it (the outer electron) should not interact with an external magnetic field. If the silver atom did behave “like” a dipole, it should be deflected by an external magnetic field. If one uses an “oven” to heat silver to “boiling” and makes a beam of silver atoms, the silver atom dipoles should have randomly oriented (in space) dipole moments. A magnetic field should deflect the beam of silver atoms in “all” directions. With these thoughts in mind, let’s consider the Stern-Gerlach experiment, in which silver atoms were “shot” through a magnetic field. The Stern-Gerlach experiment (1924) With field off, atoms go straight through. Also what should happen for µL = 0. Classical expectation if there is some other source of magnetic dipole: With field on, atoms will deflect in “all” directions. (The “funny” shape is due to the magnet geometry.) Let’s see what really happens… Experimental result! Failure of another classical prediction, leads to discovery of new quantum mechanical property: Spin Evidently the silver 5s electron has some binary “property.” It can be this kind of electron: Or it can be this kind of electron: All electrons together do this: But how does the electron obtain a magnetic moment if it has zero angular momentum and therefore is not a "current loop”? This binary (one or the other, but only two choices) property is the electron spin. Spin, as in a spinning top? “A spinning ball of charge is equivalent to a current loop, which would produce a magnetic moment, so the electron would interact with an external magnetic field.” Not exactly…. But before we discard this classical “picture” of electron as spinning ball of charge, let’s think about it for a minute. The picture suggests the electron has an intrinsic angular momentum, associated with the spin and independent of the orbital angular momentum due to its motion about nucleus— this is in fact the case. The picture also “explains” the intrinsic magnetic moment of an electron. So it is OK to keep this picture… …in our heads, and even use it to help explain spin, but just remember that it is ultimately wrong. However, this statement is correct: “The electron spin gives rise to an intrinsic angular momentum, associated with the spin and independent of the orbital angular momentum. It also gives rise to the intrinsic magnetic moment of an electron.” The electron’s orbital angular momentum is quantized, and so is its spin angular momentum. The spin quantum number s which describes the spin angular momentum of an electron has a single value, s=½. All electrons have the same s! Just as is the case with the orbital quantum number ℓ and orbital angular momentum L, the spin angular momentum is given by 3 S = s ( s +1) h = h. 4 capital S lowercase s All electrons have the same spin angular momentum S (magnitude!). S = (3/4)½ ħ is the magnitude of the electron spin angular momentum. Just as the space quantization of L is specified by mℓ, the space quantization of S is described by ms. 1 ms = ± . 2 1 Sz = ms h = ± h . 2 Aha! There are only two possible values of the z-component of the spin angular momentum. Now we understand the Stern-Gerlach experiment! There are exactly two possible orientations of the electron’s spin angular momentum vector... up down There is nothing special about the z-axis: projection of electron spin along any axis takes on two possible values. Exclusion Principle In 1925 Wolfgang Pauli postulated the (Pauli) exclusion principle, which states that no two electrons in one atom can exist in the same quantum state. Pauli won the 1945 Nobel Prize for discovering the exclusion principle (nominated for the prize by Einstein). Here are a couple of alternate ways to express the exclusion principle: “No two electrons in the same atom can have the same four quantum numbers (n, ℓ, mℓ, ms).” Generalizing: “no two electrons in the same potential can exist in the same quantum state.” (Vital to the understanding of solid state physics.) In 1925, only three quantum numbers were known (n, ℓ, mℓ). Pauli realized there needed to be a fourth! "State" refers to the four quantum numbers n, ℓ, mℓ, ms. Obviously, all electrons have the same s. An even more general statement reads: “No two fermions in the same potential can exist in the same quantum state.” A fermion obeys the Pauli exclusion principle; a boson does not. Examples of fermions: electron, proton, neutron. Examples of bosons: photon, pion. It is found experimentally that all bosons have integer spin. All fermions have half-oddinteger spin. Pauli is perhaps most famous among physicists for the “Pauli Effect*”: catastrophic failure of experiments when he walked into a lab!! *Sources: W. Cropper, Great Physicists, Oxford, 2001, p.256-7; G. Gamow, Thirty Years That Shook Physics, Heinemann, 1966, p.64. Symmetric and Antisymmetric Wave Functions We are about to study many-particle systems (many-electron atoms and many-atom systems). It is important to understand the different kinds of wave functions such systems can have. In this section, the abstract mathematics of quantum mechanics leads us to some interesting results, including the Pauli exclusion principle. For a system of n noninteracting identical particles, the total wave function of the system can be written as a product of individual particle wave functions: ψ(1,2,3,...n) = ψ(1) ψ(2) ψ(3) ... ψ(n) . Electrons, because they satisfy the Pauli exclusion principle, don’t “like” each other and are actually rather good at being “noninteracting.” In a few minutes, we will see that there is a different take on this idea… If the particles are identical, it shouldn't make a difference to our measurements if we exchange any two (or more) of them. (Should it?) Looks the same as before to me! For a two particle system, we express this interchangeability mathematically as ψ(1,2) 2 = ψ(2,1) 2 . Recall that the magnitude of the wave function squared (probability density) is related to what we measure. The equation above implies ψ(2,1) = ψ(1,2) symmetric or ψ(2,1) = - ψ(1,2) . antisymmetric If the wave function does not change sign upon exchange of particles, it is said to be symmetric. If it does change sign, it is said to be antisymmetric. Remember, we can't directly measure the wave function, so we don't know what its sign is, although, as you will see in a minute, we can tell if the wave function changes sign upon exchange of particles. This discussion can be extended to any number of particles. If the total wave function of a many-particle system doesn't change sign upon exchange of particles, it is symmetric. If it does change sign, it is antisymmetric. Now let's take these ideas another step further, and consider two identical particles (1 and 2) which may exist in two different states (a and b). If particle 1 is in state a and particle 2 is in state b then ψI = ψa (1) ψb (2) 1 in state a 2 in state b is the wave function of the system. If particle 2 is in state a and particle 1 is in state b then ψII = ψa (2) ψb (1) 2 in state a is the wave function of the system. 1 in state b But we can’t tell particles 1 and 2 apart (remember, they are identical). So we can’t tell ψI and ψII apart. One is just as “good” as the other. Both ψI and ψII are equally likely to describe our system. ψI = ψa (1) ψb (2) ψII = ψa (2) ψb (1) How do we know which to pick??? Probabilitistic interpretation of the wave function to the rescue: We say that the system spends half of its time in state I and half in state II. ψI = ψa (1) ψb (2) ψII = ψa (2) ψb (1) Our system's wave function should therefore be constructed of equal parts of ψI and ψII. While this approach may seem like nonsense on a macroscopic level; however, it is correct on the quantum level. There are two ways to construct our system's total wave function ψ out of equal parts of ψI and ψII. Symmetric: 1 ψS = ⎡⎣ψa (1) ψb (2) + ψa (2) ψb (1) ⎤⎦ 2 particle 1 in state a particle 2 in state b Antisymmetric: particle 2 in state a particle 1 in state b 1 ψA = ⎡⎣ψa (1) ψb (2) - ψa (2) ψb (1) ⎤⎦ 2 Exchanging particles 1 and 2 changes the sign of ψA but not the sign of ψS. 1 ψS = ⎡⎣ψa (1) ψb (2) + ψa (2) ψb (1) ⎤⎦ 2 1 ψA = ⎡⎣ψa (1) ψb (2) - ψa (2) ψb (1) ⎤⎦ 2 Let’s put both particles (1 and 2) in the same state, say a. 1 2 ψS = ψa (1) ψa (2) ⎡⎣ψa (1) ψa (2) + ψa (2) ψa (1) ⎤⎦ = 2 2 1 ψA = ⎡⎣ψa (1) ψa (2) - ψa (2) ψa (1) ⎤⎦ = 0 2 PS = 2 ψ a (1)* ψ a (2)* ψ a (1) ψ a (2) ≠ 0 PA = 0 What? PS ≠ 0 PA = 0 If individual particle wave functions are antisymmetric, then if we try to put both particles in the same state, we get P=0. There is zero probability of finding the system in such a state. The system cannot exist in such a state. Does this remind you of anything we’ve seen recently? In fact, electrons obey the Pauli exclusion principle because their wave functions in a system are antisymmetric. How do we know electron wave functions are antisymmetric? Because electrons obey the Pauli exclusion principle! Chicken and egg …which comes first: wave function antisymmetry or Pauli’s exclusion principle? Pauli “discovered” the exclusion principle in 1925. Heisenberg formulated matrix mechanics in 1925 and Schrödinger “discovered” his equation in 1926. However, all of these discoveries are consequences of the wave nature of matter. Pauli’s exclusion principle is a logical consequence of the wave nature of matter. Giving it a name like “the Pauli exclusion principle” makes it sound like it is something outside the framework of quantum mechanics, but it is not. See the following link for why the Pauli principle is responsible for the fact that matter doesn’t just bunch up and “implode”! http://antwrp.gsfc.nasa.gov/apod/ap030219.html Implications for multi-atomic systems: I) Molecules Consider Hydrogen Molecule with just one electron, in ground state - A single electron sees two potential wells from two protons - Recall: for single well, if ψ is a solution to SE, then so is –ψ. Two unique ways of combining single-well wave functions to get solution to SE for double well: Notes: 1) Talking about 1 electron here, so sym/ anti-sym is NOT w.r.t. exchange of electrons (i.e., not related to Pauli Exclusion Principle). 2) Difference in energies of ψs and ψA is small 3) Protons move closer together, more likely to find electron between protons in ψs à ψs is a “bonding orbital”; electron pulls protons closer together. Band Theory of Solids The essential feature of the band theory is that the allowed energy states for electrons are nearly continuous over certain ranges, called energy bands, with forbidden energy gaps between the bands. • Consider initially the known wave functions of two hydrogen atoms far enough apart so that they do not interact. 31 Band Theory of Solids • Interaction of the wave functions occurs as the atoms get closer: Symmetric Antisymmetric • An atom in the symmetric state has a nonzero probability of being halfway between the two atoms, while an electron in the antisymmetric state has a zero probability of being at that location. 32 Band Theory of Solids • In the symmetric case the binding energy is slightly stronger resulting in a lower energy state. – Thus there is a splitting of all possible energy levels (1s, 2s, and so on). • When more atoms are added (as in a real solid), there is a further splitting of energy levels. With a large number of atoms, the levels are split into nearly continuous energy bands, with each band consisting of a number of closely spaced energy levels. 33 Kronig-Penney Model • An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. • Kronig and Penney (1931) model: an electron experiences an infinite one-dimensional array of finite potential wells. • Each potential well models attraction to an atom in the lattice, so the size of the wells must correspond roughly to the lattice spacing. 34 Implications for multi-atomic systems: Solids and the band structure of electronic states in crystals Electronic characteristics of solids: 1) Insulators 2) Semi-Conductors 3) Conductors Now imagine many single-electron atoms next to each other – Not just two, but many, possible multi-well single-electron wave functions corresponding to a particular single-well state (i.e., 1S), AND all very close in energy, forming a “Band”. Pauli Principle à only two electrons can have the same spatial wave function, so band fills up from lowest to highest energy. For a regular array (crystal) of multi-electron atoms forming a solid, outer shell à band that may or not be filled Implications for multi-atomic systems: Solids and the band structure of electronic states in crystals Example: imagine an array of Sodium atoms (forming a metal): (Sodium has 11 electrons: 1s2 2s2 2p6 3s1 ) Valence and Conduction Bands • The band structures of insulators and semiconductors resemble each other qualitatively. Normally there exists in both insulators and semiconductors a filled energy band (referred to as the valence band) separated from the next higher band (referred to as the conduction band) by an energy gap. • If this gap is at least several electron volts, the material is an insulator. It is too difficult for an applied field to overcome that large an energy gap, and thermal excitations lack the energy to promote sufficient numbers of electrons to the conduction band. 37 Smaller energy gaps create semiconductors • For energy gaps smaller than about 1 electron volt, it is possible for enough electrons to be excited thermally into the conduction band, so that an applied electric field can produce a modest current. The result is a semiconductor. 38 Implications for multi-atomic systems: Solids and the band structure of electronic states in crystals Electronic characteristics of solids: 1) Insulators 2) Semi-Conductors 3) Conductors