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Identical Particles ¾We Would Like to Move from the Quantum Theory of Hydrogen to That for the Rest of the Periodic Table

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Identical Particles ¾We Would Like to Move from the Quantum Theory of Hydrogen to That for the Rest of the Periodic Table Identical Particles ¾We would like to move from the quantum theory of hydrogen to that for the rest of the periodic table One electron atom to multielectron atoms ¾This is complicated by the interaction of the electrons with each other and by the fact that the electrons are identical The Schrodinger equation for the two electron atom can only be solved by using approximation methods 1 Identical Particles ¾In classical mechanics, identical particles can be identified by their positions ¾In quantum mechanics, because of the uncertainty principle, identical particles are indistinguishable This effect is connected with the Pauli exclusion principle and is of major importance in determining the properties of atoms, nuclei, and bulk matter 2 Identical Particles Consider two particles, then Ψ(x,t) → Ψ(x1,x2,t) The time dependent Schrodinger equation is 2 2 2 2 ∂Ψ()x1,x2,t ⎡ h ∂ h ∂ ⎤ ih = ⎢− 2 − 2 +V ()()x1, x2 ⎥Ψ x1,x2,t ∂t ⎣ 2m ∂x1 2m ∂x2 ⎦ The time independent Schrodinger equation is 2 2 2 2 ⎡ h ∂ h ∂ ⎤ ⎢− 2 − 2 +V ()()x1, x2 ⎥ψ x1,x2 = Eψ ()x1,x2 ⎣ 2m ∂x1 2m ∂x2 ⎦ And note here we have used labels that identify particles 1 and 2 3 Identical Particles ¾An important case is when the particles do not interact with each other This is called the independent particle model It’s the starting point for the He atom e.g. Then V ()x1, x2 = V1(x1 )+V2 (x2 ) 2 2 2 2 ⎡ h ∂ h ∂ ⎤ ⎢− 2 +V1()x1 − 2 +V2 ()x2 ⎥ψ (x1, x2 )= Eψ (x1, x2 ) ⎣ 2m ∂x1 2m ∂x2 ⎦ This suggests we can write ψ ()()()x1, x2 =ψ x1 ψ x2 and E = E1 + E2 Which leads to 2 2 ⎡ h d ⎤ ⎢− 2 +V1()x1 ⎥ψ ()x1 = E1ψ ()x1 ⎣ 2m dx1 ⎦ 2 2 ⎡ h d ⎤ ⎢− 2 +V2 ()x2 ⎥ψ ()x2 = E2ψ ()x2 ⎣ 2m dx2 ⎦ 4 Identical Particles ¾ Let’s say particle 1 is in quantum state m and particle 2 is in quantum state n ψ ()x1, x2 =ψ m (x1 )ψ n (x2 ) ¾ If we interchange the two particles then ψ ()x1, x2 =ψ m (x2 )ψ n (x1 ) ¾ We get different wave functions (and probability densities) so the two particles are distinguishable Two particles are indistinguishable if we can exchange their labels without changing a measurable quantity Thus neither of the solutions above is satisfactory 5 Identical Particles ¾In fact there are two ways to construct indistinguishable wave functions 1 ψ = ()ψ ()()x ψ x +ψ ()()x ψ x s 2 m 1 n 2 m 2 n 1 1 ψ = ()ψ ()()x ψ x −ψ ()()x ψ x a 2 m 1 n 2 m 2 n 1 Ψs is a symmetric wave function under particle interchange of 1↔2 Ψs →Ψs Ψa is a antisymmetric wave function under particle interchange of 1↔2 Ψa → -Ψa The probability density remains unchanged under particle interchange of 1↔2 6 Identical Particles ¾All particles with integer spin are called bosons Spin 0,1,2,… Photon, pions, Z-boson, Higgs ¾All particles with half integer spin are called fermions Spin 1/2, 3/2, … Electron, proton, neutron, quarks, … ¾For the next few lectures we’ll focus on the fermions (electrons) 7 Identical Particles ¾The wave function of a multi-particle system of identical fermions is antisymmetric under interchange of any two fermions of identical bosons is symmetric under interchange of any two bosons ¾The Pauli exclusion principle follows from these principles No two identical electrons (fermions) can occupy the same quantum state 1 ψ = ()ψ ()()x ψ x −ψ ()()x ψ x a 2 m 1 n 2 m 2 n 1 for m = n (same quantum state) 1 ψ = ()ψ ()x ψ ()x −ψ ()x ψ ()x = 0 a 2 m 1 m 2 m 2 m 1 8 Identical Particles ¾What about 3 particles? ¾Symmetric ψ s = ()ψ m ()x1 ψ n (x2 )ψ o (x3 )+ψ o (x1 )ψ m (x2 )()ψ n x3 +ψ n (x1 )ψ o (x2 )ψ m (x3 ) ψ s = ()ψ m ()x1 ψ n ()x2 ψ o ()x3 + permutations ¾Antisymmetric Use the Slater determinant ψ m (1) ψ m (2) ψ m (3) 1 ψ = detψ ()1 ψ (2 )ψ ()3 a 3! n n n ψ o ()1 ψ o (2 )ψ o ()3 1 ψ = (ψ ()1ψ (2 )ψ ()3 −ψ ()1ψ ()3ψ (2 ) a 6 m n o m n o +ψ m ()2 ψ n ()3ψ o ()1 −ψ m ()2 ψ n ()1ψ o ()3 +ψ m ()3ψ n ()1ψ o ()2 −ψ m ()3ψ n ()2 ψ o ()1 ) 9 Identical Particles ¾Just a reminder, we are presently only working with the space wave functions We’ll get to spin in a little bit ¾A consequence of identical particles is called exchange “forces” Symmetric space wave functions behave as if the particles attract one another Antisymmetric wave functions behave as if the particles repel one another 10 Infinite Square Well ¾Quick review. For an infinite well at x=0 and x=L 2 nπx Solutions are ψ (x) = sin n L L π 2 2 E = n2 h 2mL2 11 Identical Particles ¾Let’s return to the infinite well problem only now with two particles Case 1 distinguishable particles ψ ()x1, x2 =ψ m (x1 )ψ n (x2 ) Case 2 identical particles (symmetric) 1 ψ ()x , x = ()ψ ()()x ψ x +ψ ()()x ψ x 1 2 2 m 1 n 2 m 2 n 1 Case 3 identical particles (antisymmetric) 1 ψ ()x , x = ()ψ ()x ψ ()x −ψ ()x ψ ()x 1 2 2 m 1 n 2 m 2 n 1 12 Identical Particles ¾What is the probability that both particles will be on the left side of the well? LL/ 2 / 2 2 P = ψ x , x dx dx ∫∫ ()1 2 1 2 0 0 ¾Case 1 L / 2 2 L / 2 2 P = ψ x dx ψ x dx ∫ m ()1 1 ∫ n ()2 2 0 0 2 2 L / 2 sin 2 mπx L / 2 sin 2 nπx P = ∫ dx ∫ dx L L 0 L 0 L 4 L L 1 P = 2 = L 4 4 4 13 Identical Particles ¾ Case 2 and 3 1 L / 2 P = ψ 2 x ψ 2 x +ψ 2 x ψ 2 x ± 2ψ x ψ x ψ x ψ x dx dx ∫ []m ()1 n ()2 m ()2 n ()1 m ()()1 n 2 m ()()2 n 1 1 2 2 0 1 ⎡1 1 LL/ 2 / 2 ⎤ P = + ± ψ x ψ x dx ψ x ψ x dx ⎢ ∫∫m ()1 n ()1 1 m ()2 n ()2 2 ⎥ 2 ⎣4 4 0 0 ⎦ 2 1 ⎛ L / 2 ⎞ P = ± ⎜ ψ x ψ x dx ⎟ ⎜ ∫ m ()()2 n 2 2 ⎟ 4 ⎝ 0 ⎠ 1 P = ± K where K > 0 4 ¾ For a symmetric wave function (+) the particles are more likely on the same side (attracted) ¾ For the antisymmetric wave function (-) the particles are more likely on opposite sides (repel) 14 Identical Particles ¾The wave function of a multi-particle system of identical fermions is antisymmetric under interchange of any two fermions of identical bosons is symmetric under interchange of any two bosons ¾The Pauli exclusion principle follows from these principles No two identical electrons (fermions) can occupy the same quantum state 1 ψ = ()ψ ()()x ψ x −ψ ()()x ψ x a 2 m 1 n 2 m 2 n 1 for m = n (same quantum state) 1 ψ = ()ψ ()x ψ ()x −ψ ()x ψ ()x = 0 a 2 m 1 m 2 m 2 m 1 15 Periodic Table ¾The Pauli exclusion principle is the basis of the periodic table This is why all the electron’s don’t simply fall into the ground state – because they are fermions Recall there were 4 quantum numbers that specified the complete hydrogen wave function: n, l, ml, and ms No two electrons in the same atom can have these same quantum numbers 16 Periodic Table ¾Building the periodic table Principle quantum number n forms shells Orbital angular momentum quantum number l forms subshells l=0,1,2,3,… are called s,p,d,f,… Each ml can hold two electrons, one spin up (ms=1/2) and one spin down (ms=-1/2) The electrons tend to occupy the lowest energy level possible Electrons obey the Pauli exclusion principle 17 Periodic Table ¾Energy levels Note for multielectron atoms, states of the same n and different l are no longer degenerate This is because of screening effects Referring back to the radial probability distributions, because the s states have non-zero probability of being close to the nucleus, their Coulomb potential energy is lower 18 Ionization Potential 19 Periodic Table ¾Hydrogen (H) 1 1s ¾Helium (He) 2 1s – closed shell and chemically inert ¾Lithium (Li) 2 1s 2s – valence +1, low I, partially screened, chemically very active ¾Beryllium (Be) 2 2 1s 2s – Closed subshell but the 2s electrons can extend far from the nucleus 20 Periodic Table ¾Boron (B) 2 2 1 1s 2s 2p – Smaller I than Be because of screening ¾Carbon (C) 2 2 2 1s 2s 2p – I actually increases because the electrons can spread out in 2/3 l state lobes The valence is +4 since an energetically favorable configuration is 1s22s12p3 ¾Nitrogen (N) 2 2 3 1s 2s 2p – See comments for C. Electrons spread out in 3/3 l state lobes ¾Oxygen (O) 2 2 4 1s 2s 2p – Two of the l state electrons are “paired” Electron-electron repulsion lowers I 21 Periodic Table ¾Fluorine (F) 2 2 5 1s 2s 2p – Very chemically active because it can accept an electron to become a closed shell ¾Neon (Ne) 2 2 6 1s 2s 2p –Like He 22 Periodic Table 23 Periodic Table ¾Periodic table is arranged into groups and periods ¾Groups Have similar shell structure hence have similar chemical and physical properties Examples are alkalis, alkali earths, halogens, inert gases ¾Periods Correspond to filling d and f subshells Examples are transition metals (3d,4d,5d), lanthanide (4f), and actinide (5f) series Because there are many unpaired electrons, spin is important for these elements and there are large magnetic effects 24.
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