Tue. Dec Physics 208, Lecture 261 Last Time… 3-dimensional wave functions Decreasing particle size Quantum tunneling Quantum dots (particle in box) This week’s honors lecture: Prof. Brad Christian, “Positron Emission Tomography”

Tue. Dec Physics 208, Lecture 262 Exam 3 results Exam average = 76% Average is at B/BC boundary Average Course evaluations: Rzchowski: Thu, Dec. 6 Montaruli: Tues, Dec. 11

Tue. Dec Physics 208, Lecture D particle in box: summary Three quantum numbers (n x,n y,n z ) label each state n x,y,z =1, 2, 3 … (integers starting at 1) Each state has different motion in x, y, z Quantum numbers determine Momentum in each direction: e.g. Energy: Some quantum states have same energy

Tue. Dec Physics 208, Lecture 264 Question How many 3-D particle in box spatial quantum states have energy E=18E o ? A. 1 B. 2 C. 3 D. 5 E. 6

Tue. Dec Physics 208, Lecture D Hydrogen atom Bohr model: Restricted to circular orbits Found 1 quantum number n Energy, orbit radius From 3-D particle in box, expect that H atom should have more quantum numbers Expect different types of motion w/ same energy

Tue. Dec Physics 208, Lecture 266 Modified Bohr model Different orbit shapes Big angular momentum Small angular momentum These orbits have same energy, but different angular momenta: Rank the angular momenta from largest to smallest: A B C a)A, B, C b)C, B, A c)B, C, A d)B, A, C e)C, A, B

Tue. Dec Physics 208, Lecture 267 Angular momentum is quantized orbital quantum number ℓ Angular momentum quantized, ℓ is the orbital quantum number For a particular n, ℓ has values 0, 1, 2, … n-1 ℓ=0, most elliptical ℓ=n-1, most circular For hydrogen atom, all have same energy

Tue. Dec Physics 208, Lecture 268 Orbital mag. moment Orbital charge motion produces magnetic dipole Proportional to angular momentum Current electron Orbital magnetic dipole Each orbit has Same energy: Different orbit shape (angular momentum): Different magnetic moment:

Tue. Dec Physics 208, Lecture 269 Orbital mag. quantum number m ℓ Directions of ‘orbital bar magnet’ quantized. Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values Determines z-component of L: This is also angle of L For example: ℓ=1 gives 3 states:

Tue. Dec Physics 208, Lecture 2610 Question For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there? A. 1 B. 2 C. 3 D. 4 E. 5

Tue. Dec Physics 208, Lecture 2611 Summary of quantum numbers n : describes energy of orbit ℓ describes the magnitude of orbital angular momentum m ℓ describes the angle of the orbital angular momentum For hydrogen atom:

Tue. Dec Physics 208, Lecture 2612 Hydrogen wavefunctions Radial probability Angular not shown For given n, probability peaks at ~ same place Idea of “atomic shell” Notation: s: ℓ=0 p:ℓ=1 d:ℓ=2 f:ℓ=3 g:ℓ=4

Tue. Dec Physics 208, Lecture 2613 Full hydrogen wave functions: Surface of constant probability Spherically symmetric. Probability decreases exponentially with radius. Shown here is a surface of constant probability 1s-state

Tue. Dec Physics 208, Lecture 2614 n=2: next highest energy 2s-state 2p-state Same energy, but different probabilities

Tue. Dec Physics 208, Lecture s-state 3p-state n=3: two s-states, six p-states and…

Tue. Dec Physics 208, Lecture 2616 …ten d-states 3d-state

Tue. Dec Physics 208, Lecture 2617 Electron spin New electron property: Electron acts like a bar magnet with N and S pole. Magnetic moment fixed… …but 2 possible orientations of magnet: up and down Spin down N S Described by spin quantum number m s N S Spin up z-component of spin angular momentum

Tue. Dec Physics 208, Lecture 2618 Include spin Quantum state specified by four quantum numbers: Three spatial quantum numbers (3-dimensional) One spin quantum number

Tue. Dec Physics 208, Lecture 2619 Quantum Number Question How many different quantum states exist with n=2? A. 1 B. 2 C. 4 D. 8 ℓ = 0 : m l = 0: m s = 1/2, -1/2 2 states ℓ = 1 : m l = +1: m s = 1/2, -1/2 2 states m l = 0: m s = 1/2, -1/22 states m l = -1: m s = 1/2, -1/2 2 states 2s 2 2p 6 There are a total of 8 states with n=2

Tue. Dec Physics 208, Lecture 2620 Question How many different quantum states are in a 5g (n=5, ℓ =4) sub-shell of an atom? A. 22 B. 20 C. 18 D. 16 E. 14 ℓ =4, so 2(2 ℓ +1)=18. In detail, m l = -4, -3, -2, -1, 0, 1, 2, 3, 4 and m s =+1/2 or -1/2 for each. 18 available quantum states for electrons

Tue. Dec Physics 208, Lecture 2621 Putting electrons on atom Electrons obey Pauli exclusion principle Only one electron per quantum state (n, ℓ, m ℓ, m s ) Hydrogen: 1 electron one quantum state occupied occupied unoccupied n=1 states Helium: 2 electrons two quantum states occupied n=1 states

Tue. Dec Physics 208, Lecture 2622 Atoms with more than one electron Electrons interact with nucleus (like hydrogen) Also with other electrons Causes energy to depend on ℓ

Tue. Dec Physics 208, Lecture 2623 Other elements: Li has 3 electrons n=1 states, 2 total, 2 occupied one spin up, one spin down n=2 states, 8 total, 1 occupied

Tue. Dec Physics 208, Lecture 2624 Atom Configuration H1s 1 He1s 2 Li1s 2 2s 1 Be1s 2 2s 2 B1s 2 2s 2 2p 1 Ne1s 2 2s 2 2p 6 1s shell filled 2s shell filled 2p shell filled etc (n=1 shell filled - noble gas) (n=2 shell filled - noble gas) Electron Configurations

Tue. Dec Physics 208, Lecture 2625 The periodic table Atoms in same column have ‘similar’ chemical properties. Quantum mechanical explanation: similar ‘outer’ electron configurations. Be 2s 2 Li 2s 1 N2p3N2p3 C2p2C2p2 B2p1B2p1 Ne 2p 6 F2p5F2p5 O2p4O2p4 Mg 3s 2 Na 3s 1 P3p3P3p3 Si 3p 2 Al 3p 1 Ar 3p 6 Cl 3p 5 S3p4S3p4 H1s1H1s1 He 1s 2 CaKAs 4p 3 Ge 4p 2 Ga 4p 1 Kr 4p 6 Br 4p 5 Se 4p 4 Sc 3d 1 Y3d2Y3d2 8 more transition metals Ca 4s 2 K4s1K4s1 Na 3s 1

Tue. Dec Physics 208, Lecture 2626 Excited states of Sodium Na level structure 11 electrons Ne core = 1s 2 2s 2 2p 6 (closed shell) 1 electron outside closed shell Na = [Ne]3s 1 Outside (11 th ) electron easily excited to other states.

Tue. Dec Physics 208, Lecture 2627 Emitting and absorbing light Photon is emitted when electron drops from one quantum state to another Zero energy n=1 n=2 n=3 n=4 n=1 n=2 n=3 n=4 Absorbing a photon of correct energy makes electron jump to higher quantum state. Photon absorbed hf=E 2 -E 1 Photon emitted hf=E 2 -E 1

Tue. Dec Physics 208, Lecture 2628 Optical spectrum Optical spectrum of sodium Transitions from high to low energy states Relatively simple 1 electron outside closed shell Na 589 nm, 3p -> 3s

Tue. Dec Physics 208, Lecture 2629 How do atomic transitions occur? How does electron in excited state decide to make a transition? One possibility: spontaneous emission Electron ‘spontaneously’ drops from excited state Photon is emitted ‘lifetime’ characterizes average time for emitting photon.

Tue. Dec Physics 208, Lecture 2630 Another possibility: Stimulated emission Atom in excited state. Photon of energy hf=  E ‘stimulates’ electron to drop. Additional photon is emitted, Same frequency, in-phase with stimulating photon One photon in, two photons out: light has been amplified EE BeforeAfter hf=  E If excited state is ‘metastable’ (long lifetime for spontaneous emission) stimulated emission dominates

Tue. Dec Physics 208, Lecture 2631 LASER :Light Amplification by Stimulated Emission of Radiation Atoms ‘prepared’ in metastable excited states …waiting for stimulated emission Called ‘population inversion’ (atoms normally in ground state) Excited states stimulated to emit photon from a spontaneous emission. Two photons out, these stimulate other atoms to emit.

Tue. Dec Physics 208, Lecture 2632 Ruby Laser Ruby crystal has the atoms which will emit photons Flashtube provides energy to put atoms in excited state. Spontaneous emission creates photon of correct frequency, amplified by stimulated emission of excited atoms.

Tue. Dec Physics 208, Lecture 2633 PUMP Ruby laser operation Metastable state Relaxation to metastable state (no photon emission) Transition by stimulated emission of photon Ground state 1 eV 2 eV 3 eV

Tue. Dec Physics 208, Lecture 2634 The wavefunction Wavefunction =  = |moving to right> + |moving to left> The wavefunction is an equal ‘superposition’ of the two states of precise momentum. When we measure the momentum (speed), we find one of these two possibilities. Because they are equally weighted, we measure them with equal probability.

Tue. Dec Physics 208, Lecture 2635 Silicon 7x7 surface reconstruction These 10 nm scans show the individual atomic positions

Tue. Dec Physics 208, Lecture 2636 Particle in box wavefunction x=0x=Lx=L Prob. Of finding particle in region dx about x Particle is never here

Tue. Dec Physics 208, Lecture 2637 Making a measurement Suppose you measure the speed (hence, momentum) of the quantum particle in a tube. How likely are you to measure the particle moving to the left? A. 0% (never) B. 33% (1/3 of the time) C. 50% (1/2 of the time)

Tue. Dec Physics 208, Lecture 2638 Interaction with applied B-field Like a compass needle, it interacts with an external magnetic field depending on its direction. Low energy when aligned with field, high energy when anti-aligned Total energy is then This means that spectral lines will split in a magnetic field

Tue. Dec Physics 208, Lecture 2639

Tue. Dec Physics 208, Lecture 2640 Orbital magnetic dipole moment Dipole moment µ=IA Current = Area = Can calculate dipole moment for circular orbit magnitude of orb. mag. dipole moment In quantum mechanics,

Tue. Dec Physics 208, Lecture 2641

Tue. Dec Physics 208, Lecture 2642 Electron magnetic moment Why does it have a magnetic moment? It is a property of the electron in the same way that charge is a property. But there are some differences Magnetic moment has a size and a direction It’s size is intrinsic to the electron, but the direction is variable. The ‘bar magnet’ can point in different directions.

Tue. Dec Physics 208, Lecture 2643 Additional electron properties Free electron, by itself in space, not only has a charge, but also acts like a bar magnet with a N and S pole. Since electron has charge, could explain this if the electron is spinning. Then resulting current loops would produce magnetic field just like a bar magnet. But… Electron in NOT spinning. As far as we know, electron is a point particle. N S

Tue. Dec Physics 208, Lecture 2644 Spin: another quantum number There is a quantum # associated with this property of the electron. Even though the electron is not spinning, the magnitude of this property is the spin. The quantum numbers for the two states are +1/2 for the up-spin state -1/2 for the down-spin state The proton is also a spin 1/2 particle. The photon is a spin 1 particle.

Tue. Dec Physics 208, Lecture 2645 Orbital mag. moment Since Electron has an electric charge, And is moving in an orbit around nucleus… produces a loop of current, and a magnetic dipole moment, Proportional to angular momentum Current electron Orbital magnetic moment magnitude of orb. mag. dipole moment Make a question out of this

Tue. Dec Physics 208, Lecture 2646 Orbital mag. quantum number m ℓ Possible directions of the ‘orbital bar magnet’ are quantized just like everything else! Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps Number of different directions = 2ℓ+1 N S m ℓ = +1 S N m ℓ = -1 For example: ℓ=1 gives 3 states: m ℓ = 0 SN

Tue. Dec Physics 208, Lecture 2647 Particle in box quantum states n=1 n=2 n=3 WavefunctionProbability L npE

Tue. Dec Physics 208, Lecture 2648 Particle in box energy levels Quantized momentum Energy = kinetic Or Quantized Energy Energy n=1 n=2 n=3 n=4 n=5 n=quantum number

Tue. Dec Physics 208, Lecture 2649 Hydrogen atom energies Zero energy n=1 n=2 n=3 n=4 Energy Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each described by a quantum number n

Tue. Dec Physics 208, Lecture 2650

Tue. Dec Physics 208, Lecture 2651 Quantum numbers Two quantum numbers

Tue. Dec Physics 208, Lecture 2652 Pauli Exclusion Principle Where do the electrons go? In an atom with many electrons, only one electron is allowed in each quantum state (n, ℓ, m ℓ, m s ). Atoms with many electrons have many atomic orbitals filled. Chemical properties are determined by the configuration of the ‘outer’ electrons.

Tue. Dec Physics 208, Lecture 2653 Atomic sub-shells Each