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 ψ 2 2 2 2 ⎡ h ∂ h ∂ ⎤ ⎢− 2 − 2 +V ()()x1, x2 ⎥ 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 ∂ ψ ⎢− 2 +V1()x1 − 2 +V2 ()x2 ⎥ x1 (, x2 = E )x1, x2 ( ) ⎣ 2m ∂x1 ψ2m ∂x2 ⎦ ()()()ψ This suggests we can write x1ψ, x2 = x1 x2 and E = E1 + E2 ψ Which leads to ψ 2 2 ⎡ h d ⎤ ψ ⎢− 2 +V1()x1 ⎥ x1 ()= E1 x1 () ⎣ 2m dx1 ⎦ 2 2 ψ ⎡ h d ⎤ ⎢− 2 +V2 ()x2 ⎥ 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 thereψ are two ways to construct indistinguishableψ wave functions 1 ψ ψ = ()ψ ()()x x + x ψ ()()x s 2 m 1 ψ n 2 m 2 n 1 1 ψ ψ = ()()()x x − 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

Š Ψs →Ψs

„ Ψa is a antisymmetric wave function under particle interchange of 1↔2

Š 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 identicalψ electrons (fermions) can occupy ψ the same quantum state 1 ψ ψ a = ()m ()()x1 n x2 − m x2 ψ ()()n x1 2 ψ for m = n (same quantumψ state) 1 ψ ψ = ()()x x ()− x ψ ()x = 0 () a 2 m 1 m 2 m 2 m 1 8 Identical Particles ¾What about 3 particles? ¾Symmetric ()ψ = ψ ()x ψ (x )ψ (x )+ψ (x )ψ (x )()ψ x +ψ (x )ψ (x )ψ (x ) s m 1 ψ n 2 o 3 o 1 m 2 n 3 n 1 o 2 m 3 ()() () () ψ s = m x1 n x2 ψ o x3 + permutations

¾Antisymmetricψ

„ Use the Slater determinantψ ψ ψ m (1) ψ m (2) m (3) ψ1 ψ ψ = det()ψ 1 ( )2 ψ ()3 a 3! ()n ψ (ψ n )n () ψ o 1 o 2ψ ψ o 3 ψ ψ 1 ψ ψ = (ψ ()1 2 (ψ )3 − ()1 3 ()2 () ( ) a 6 m ψn o ψ m n o ()ψ () ()ψ () () () + 2 3 1 −ψ 2 1ψ 3 m ()n ()o ()m ψ n ()o () () + m 3 n 1 o 2 − m 3 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 n x Solutions are (x) = sin ψ n L L π 2 2 π 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 ()x , x =ψ (x )ψ (x ) 1 2 m ψ1 n 2 ψ „ Case 2 identical particles (symmetric) 1 ψ ψ ()x , x = ()()()x x + x ψ ()()x 1 2 2 ψm 1 n 2 m 2 n 1 „ Case 3 identical particles (antisymmetric)ψ 1 ψ ψ ()x , x = ()()x x ()− 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 ψ 0 0 ¾Case 1 ψ L / 2 2 L / 2 2 P = ()x dx ()x dx ∫ m 1 1 π∫ n 2 2 0 0 2 2 L / 2 sin 2 m x 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

ψ¾ Case 2 and 3 L / 2 ψ 1 P = []2 ()x 2 x ()+ 2 x ()2 x ± 2ψ ()x ψ x ψ ()()x ψ x dx dx ()() ∫ m 1 n ψ 2 m 2 n 1 m 1 n 2 m 2 n 1 1 2 2 0 ψ 1 ⎡1 1 LL/ 2 / 2 ⎤ P = + ±ψ ()x x ()dx x ()x dx () ⎢ ∫∫m ψ1 n 1 1 m 2 n 2 2 ⎥ 2 ⎣4 4 0 0 ⎦ 2 ψ 1 ⎛ L / 2ψ ⎞ ψ P = ± ⎜ ()()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 identicalψ electrons (fermions) can occupy ψ the same quantum state 1 ψ ψ a = ()m ()()x1 n x2 − m x2 ψ ()()n x1 2 ψ for m = n (same quantumψ state) 1 ψ ψ = ()()x x ()− 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 Š 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