Physics 3 for Electrical Engineering Ben Gurion University of the Negev Lecturers: Daniel Rohrlich, Ron Folman Teaching Assistants: Daniel Ariad, Barukh Dolgin Week 11. Quantum mechanics – Schrödinger’s equation for multiple electrons identical particles and the Pauli principle atomic ground states periodic table Sources: Feynman Lectures III, Chap. 19 Sect. 6; פרקים בפיסיקה מודרנית, יחידה 8, פרק 3 Tipler and Llewellyn, Chap. 7 Sects. 6-8.

A few quantum principles 1. The classical distinction between particles and waves breaks down [“wave-particle duality”].

A few quantum principles 1. The classical distinction between particles and waves breaks down [“wave-particle duality”]. 2. Physical states are normalized vectors ψ(r), Ψ(r,t),, [ → superposition principle].

A few quantum principles 1. The classical distinction between particles and waves breaks down [“wave-particle duality”]. 2. Physical states are normalized vectors ψ(r), Ψ(r,t),, [ → superposition principle]. 3. Measurable physical quantities – “observables” – correspond to Hermitian or (self-adjoint) operators on the state vectors.

A few quantum principles 1. The classical distinction between particles and waves breaks down [“wave-particle duality”]. 2. Physical states are normalized vectors ψ(r), Ψ(r,t),, [ → superposition principle]. 3. Measurable physical quantities – “observables” – correspond to Hermitian or (self-adjoint) operators on the state vectors. 4. If a system is an eigenstate with eigenvalue a of an observable, then a measurement of on will yield a.

A few quantum principles 1. The classical distinction between particles and waves breaks down [“wave-particle duality”]. 2. Physical states are normalized vectors ψ(r), Ψ(r,t),, [ → superposition principle]. 3. Measurable physical quantities – “observables” – correspond to Hermitian or (self-adjoint) operators on the state vectors. 4. If a system is an eigenstate with eigenvalue a of an observable, then a measurement of on will yield a. Conversely, if a measurement of on any state yields a, the measurement leaves the system in an eigenstate.

A few quantum principles 5. The probability that a system in a normalized state can be found in the state is [Born probability rule]. 6. The time evolution of a quantum state is given by where is the Hamiltonian (kinetic energy + potential energy) of the system in the state [Schrödinger’s equation].

Schrödinger’s equation for helium The Schrödinger equation for helium is

Schrödinger’s equation for helium The Schrödinger equation for helium is

Schrödinger’s equation for helium As usual, we’d like to write Ψ(r 1,r 2 ) as a product ψ 1 (r 1 )ψ 2 (r 2 ), but the electron-electron repulsion gets in the way.

Schrödinger’s equation for helium We can write Ψ(r 1,r 2 ) = ψ 1 (r 1 )ψ 2 (r 2 ) anyway, and treat the electron-electron repulsion as a “perturbation”.

Schrödinger’s equation for helium We can write Ψ(r 1,r 2 ) = ψ 1 (r 1 )ψ 2 (r 2 ) anyway, and treat the electron-electron repulsion as a “perturbation”.

Schrödinger’s equation for helium Let ψ 1 = and ψ 2 = Then and we can take the scalar product of both sides of this equation with to estimate E as an expectation value:

Schrödinger’s equation for helium The expectation value is, explicitly,

Schrödinger’s equation for multiple electrons The Schrödinger equation for an atom with Z protons is

Schrödinger’s equation for multiple electrons The Schrödinger equation for an atom with Z protons is

Identical particles and the Pauli principle A new quantum principle – hard to understand but easy to use – applies only to identical particles. (We did not need this principle for the hydrogen atom, which consists of one proton and one electron, because these particles are not identical.)

Identical particles and the Pauli principle A new quantum principle – hard to understand but easy to use – applies only to identical particles. (We did not need this principle for the hydrogen atom, which consists of one proton and one electron, because these particles are not identical.) Quantum states must be symmetric under exchange of any two identical bosons and antisymmetric under exchange of any two identical fermions. Bosons: photons, pions, …. Fermions: electrons, protons, neutrons,….

Identical particles and the Pauli principle A new quantum principle – hard to understand but easy to use – applies only to identical particles. (We did not need this principle for the hydrogen atom, which consists of one proton and one electron, because these particles are not identical.) Quantum states must be symmetric under exchange of any two identical bosons and antisymmetric under exchange of any two identical fermions. Bosons: photons, pions, … (spin 0, 1,...). Fermions: electrons, protons, neutrons,…(spin,,… ).

Identical particles and the Pauli principle Intuitive connection between spin and symmetry: Consider two ends of a ribbon; exchange the two ends, without rotating either end.

Identical particles and the Pauli principle Intuitive connection between spin and symmetry: Consider two ends of a ribbon; exchange the two ends, without rotating either end. There is now a 2π twist in the ribbon!

Identical particles and the Pauli principle If the “ends of the ribbon” have spin,,… then a 2π rotation yields a – sign because e i2π(1/2) = e i2π(3/2) =…= –1: → antisymmetric wave function. If the “ends of the ribbon ” have spin 0, 1,… then a 2π rotation yields a + sign because e i2π(0) = e i2π(1) =…= 1: → symmetric wave function.

Identical particles and the Pauli principle An infinite square well contains two particles. What are the possible states of the particles? Neglect their interactions. The single-particle states are x L 0 outside inside

Identical particles and the Pauli principle Possible states of an electron and a neutron: products of states for any j and k, and any combination of these products. x L 0 outside inside

Identical particles and the Pauli principle Possible states of two neutrons: antisymmetrized products of states for any j ≠ k, and any combination of these antisymmetrized products. x L 0 outside inside

Identical particles and the Pauli principle Possible states of two photons: symmetrized products of states for any j and k, and any combination of these symmetrized products. x L 0 outside inside

Identical particles and the Pauli principle Can two neutrons be in the same state? x L 0 outside inside

Identical particles and the Pauli principle Can two neutrons be in the same state ψ j (x)? No, because x L 0 outside inside

Identical particles and the Pauli principle Can two neutrons be in the same state ψ j (x)? No, because Pauli’s exclusion principle: no two fermions in the same state.

Identical particles and the Pauli principle Can two neutrons be in the same state ψ j (x)? No, because Pauli’s exclusion principle: no two fermions in the same state. “Exchange repulsion”: What is the relative probability that two fermions in the state will be found at the same point x?

Identical particles and the Pauli principle Can two neutrons be in the same state ψ j (x)? No, because Pauli’s exclusion principle: no two fermions in the same state. “Exchange repulsion”: What is the relative probability that two fermions in the state will be found at the same point x? It is

Atomic ground states The ground state of an atom is the lowest-energy state. What is the ground state of helium (Z = 2)?

Atomic ground states The ground state of an atom is the lowest-energy state. What is the ground state of helium (Z = 2)? It is approximately which we can write as or even

Atomic ground states The ground state of an atom is the lowest-energy state. What is the ground state of helium (Z = 2)? It is approximately which we can write as The spin state is a singlet and corresponds to j = 0. (Show that both and annihilate this state.)

Atomic ground states Exercise: What is the ground state energy of helium, neglecting the electron-electron interaction?

Atomic ground states Exercise: What is the ground state energy of helium, neglecting the electron-electron interaction? Solution: Neglecting the electron-electron interaction, each electron has energy with n = 1 and Z = 2. Hence the ground-state energy is 8 × (–13.6 eV) = –108.8 eV. The experimentally measured value of the ground-state energy is –78.95 eV. (We expect it to be higher than our solution because the electron-electron interaction is repulsive.)

Periodic table The ground states of atoms are obtained by filling up the states, one electron to a state, in order of their energy. Although the ground states and their energies are perturbed from the single- electron states and energies that we labeled according to the quantum numbers n,l,m,m s, we still use these quantum numbers to label the perturbed states. A shell contains all states with the same principal quantum number n; an orbital contains all states with the same n and l. A superscript indicates the population of each orbital. How many electrons can an orbital hold? How many electrons can a shell hold?

Periodic table The ground states of atoms are obtained by filling up the states, one electron to a state, in order of their energy. Although the ground states and their energies are perturbed from the single- electron states and energies that we labeled according to the quantum numbers n,l,m,m s, we still use these quantum numbers to label the perturbed states. A shell contains all states with the same principal quantum number n; an orbital contains all states with the same n and l. A superscript indicates the population of each orbital. How many electrons can an orbital hold? 2(2l+1) How many electrons can a shell hold? 2n 2

Periodic table The ground states of atoms are obtained by filling up the states, one electron to a state, in order of their energy. Although the ground states and their energies are perturbed from the single- electron states and energies that we labeled according to the quantum numbers n,l,m,m s, we still use these quantum numbers to label the perturbed states. A shell contains all states with the same principal quantum number n; an orbital contains all states with the same n and l. A superscript indicates the population of each orbital. Examples: H – 1s 1 Li – 1s 2 2s 1 Na – 1s 2 2s 2 2p 6 3s 1 He – 1s 2 Ne – 1s 2 2s 2 2p 6 Cl – 1s 2 2s 2 2p 6 3s 2 3p 5

Periodic table The spin-orbit coupling breaks the degeneracy of the orbitals within a shell. As l increases within a shell, so does the energy. The chemical properties of an atom depend largely on the valence electrons, which populate the highest-energy states and are least attached to the atom. Compare: F – 1s 2 2s 2 2p 5 Cl – 1s 2 2s 2 2p 6 3s 2 3p 5 Ne – 1s 2 2s 2 2p 6 Ar – 1s 2 2s 2 2p 6 3s 2 3p 6 Na – 1s 2 2s 2 2p 6 3s 1 K – 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1

Periodic table

Some shells overlap at small Z. Energy

More about identical particles and the Pauli principle: Consider an interferometer with four half-silvered mirrors. Two particles enter from opposite corners. Assume, at first, that they are not identical. Each reflection induces a phase factor i.

Initial state (if the particles are not identical):

Final state (if the particles are not identical): (Assume that the path lengths are all equal.)

〰 〰〰 〰 Final state – measuring correlations

Final state (if the particles are not identical): But if the particles are identical, we must either symmetrize this state (e.g. for photons) or antisymmetrize it (e.g. for electrons). Symmetrized state:

Final state (if the particles are not identical): But if the particles are identical, we must either symmetrize this state (e.g. for photons) or antisymmetrize it (e.g. for electrons). Symmetrized state: Pairs of photons exit the interferometer in the same state (same port) at the same corner, or in opposite directions (L/R or U/D) at ports on opposite corners.

Final state (if the particles are not identical): But if the particles are identical, we must either symmetrize this state (e.g. for photons) or antisymmetrize it (e.g. for electrons). Antisymmetrized state:

Final state (if the particles are not identical): But if the particles are identical, we must either symmetrize this state (e.g. for photons) or antisymmetrize it (e.g. for electrons). Antisymmetrized state: Pairs of electrons exit the interferometer in different states (different ports) at the same corner, or in orthogonal directions (L/D or U/R) at ports on opposite corners.