* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter 41: Quantization of Angular Momentum and of Energy Values
Renormalization wikipedia , lookup
Density of states wikipedia , lookup
State of matter wikipedia , lookup
Conservation of energy wikipedia , lookup
History of subatomic physics wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Electromagnetism wikipedia , lookup
Spin (physics) wikipedia , lookup
Quantum vacuum thruster wikipedia , lookup
Condensed matter physics wikipedia , lookup
Nuclear physics wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Photon polarization wikipedia , lookup
Old quantum theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Introduction to quantum mechanics wikipedia , lookup
Chapter 41: Quantization of Angular Momentum and of Energy Values Classical mechanics: Angular momentum and energy are in a continuous range 1 2 p = mv ; E = mv ; L = mr × v 2 v Quantum mechanics: Angular momentum and energy quantized in discrete values. Similar to classical waves confined in a cavity A(x) or Ψ(x) p or E x P ( x) = Ψ 2 1 Classical “picture” of an atom What’s wrong with this picture? + - Quantum mechanical “picture” of an atom P ( x) = Ψ 2 2 Atomic spectral lines http://www.colorado.edu/physics/2000/quantumzone/index.html 3 Quantization of energy (frequency) in standing waves Ψ (x) note : Ψ = 0 at boundaries P(x)= Ψ (x) 2 probability λ for electron? 4 Quantization of energy (frequency) in standing waves 2b λn = n = 1, 2 ,3… n h nh = pn = λn 2b 2 2 2 Energy α n 2 pn nh = En = 2m 8mb 2 0 5 Bohr model of hydrogen since m p >> me we treat nucleus as stationary → m = me v2 p2 p2 e2 F= = me a = me = → = 2 r me r 4πε 0 r 2me 8πε 0 r p2 e2 e2 e2 e2 E = K +U = − = − =− 2m 4πε 0 r 8πε 0 r 4πε 0 r 8πε 0 r L = me vr , v = e 4πε 0 me r → L = e me r 4πε 0 Classical e2 L2 r= me e 2 4πε 0 Angular momentum quantized: Ln = n n2 2 2 = rn = n a0 2 me e 4πε 0 n = 1, 2, 3… 2 , a0 = me e 2 4πε 0 =0.53 ×10-10 m e2 1 e2 13.6 eV =- 2 En = − =8πε 0 rn n 8πε 0 a0 n2 The circumference of the orbit is a integral multiple of λ : 2π rn de Broglie: = n , λn = h pn (de Broglie) λn 2π rn rn pn Ln → = = =n h pn h 2π 6 Bohr model of hydrogen −13.6 eV En = n2 E >0 ? E∞ = ? http://www.colorado.edu/physics/2000/quantumzone/bohr.html Note radiation is emitted ONLY when electron changes energy levels. E1 = ?7 Energy of radiation emitted or absorbed by atoms Population in nth state ∝ e − En kT HW # 38 & 39 (Boltzmann distribution is introduced on p. 560 521, Fishbane et al, Vol. 1) hf = Ei − E f E f < Ei energy released (emission) E f > Ei energy absorbed (absorption) What if T is very small? 8 Ex. 41-1: A certain laser emits light λ=3391 nm, what is ∆E=Ei-Ef in eV? 9 Hydrogen emission/absorption wavelengths hf = h → 1 λ c λ = = Ei − E f Ei − E f hc 1 1 = R∞ 2 − 2 n nf i R∞ = ( me 2hc ) ( e 4πε 0 2 ) 2 = 1.0974 ×107 m -1 Rydberg constant 10 Ex. 41-2: ∆E between lowest energy of hydrogen (n=1) and first excited state (n=2)? At which T is substantial fraction of a gas of H atoms in first excited state? 11 The true spectrum of hydrogenHeisenberg and Schroedinger 1. Classical Orbits → wavefunction Ψ (x) P(x)= Ψ (x) 2 → states → quantum numbers (e.g., n, ) 2. L = , is a vector, so L can have different projections on the z-axis → Lz = m where m = , − 1, − 2…1, 0, −1, − ( − 1) ,→ 2 + 1 orientations z Lz z m=+1 L x L m=0 x m=-1 Example: if = 1 ( L = ), then Lz can have three values corresponding to m = 1, 0, or -1 (Lz = , 0, or - ) 12 Hydrogen orbits n = 0,1,2, … (principal quantum number) l = 0,1,2,…, (n − 1) (orbital momentum) m = 0,±1,±2, …,±l l =1 l =0 l =1 l =1 l=2 l=2 l=2 l=2 l=2 13 Projection of L onto z axis: Lz ( L= L2 = ( + 1) ≈ + 1) 2 Lz = m ≤ n −1 m ≤ n → En principal quantum number 14 Stern-Gerlach experiment Beam of atoms (silver) sent through an inhomogeneous magnetic field. The Ag atoms have a magnetic momentum µ (like a little bar magnet) Classically: µ can point in any direction Reality (and Quantum mechanicly): only certain 15 z-components allowed Potential energy of the atoms in magn. field: U mag = − µ ⋅ B = − µ z B B along zˆ Force on Ag atoms in inhomogeneous magn. Field: Fz = − dU mag dz µ z ∝ Lz dB = µz dz Quantified ⇒only certain values occur ⇒No continuous deflection but separate beams 16 3. Energy values for hydrogen atom 13.6 eV E=− n2 for each n, there are n values , as long as ≤ n − 1 for each , there are 2 + 1 of values m , as long as m ≤ Examples n =1→ = 0 → m = 0 1 state total n=2→ =0→m=0 or = 1 → m = 1, 0, or − 1 4 states total n =3→ =0→m =0 or = 1 → m = 1, 0, or − 1 or = 2 → m = 2,1, 0, −1, or − 2 9 states total n=4→? The total number of states labeled by n is n 2 The energy of the state depends only on n 17 n states total (note that energy only depends on n) 2 Transitions in a H atom s p d f g L photon = → = 1 photon absorption/emission → ∆ = ±1 Angular momentum and energy are conserved in QM Angular momentum of a photon: Lphoton = 18 Ex. 41-4: Balmer series are transitions in a H atom that end at n=2. Longest and shortest λ in series? Paschen Lyman 19 Zeeman Effect: splitting the energies of the different l quantum states with a magnetic field Place an atom into a magnetic field B => the different orientation of the magn. momenta (from the angular momentum) have different pot. Energies in the magnetic field mag z U => Levels with different = −µ ⋅ B = − µ B Lz (or ml ) are split Energy levels in hydrogen For an electron in a circular orbit e L where µ B ≡ Bohr magneton µz = µB = 20 2me (HW 41-52 and next HW 42 − 30) L Zeeman splitting to measure the magnitude of the magnetic fields on the sun’s surface: Strong magnetic fields at “sun spots” split absorbtion lines in two. 21 The Spin of the Electron Energy levels in silver, l =0 is expected to be a single state, but gets split into two states! This means that there is another quantum number for angular momentum that we have not included, that is electronic intrinsic angular momentum or spin: 1 µ=s , s= 2 m =0 1 1 1 → ms ≤ s → ms = , or 2 2 2 2 number of states = 2s + 1 = 2 (2 + 1 = ?) ≡s s= 1 ms = spin up; 2 1 ms = − spin down 2 22 Multi-electron atoms Each electron moves in the attractive Coulomb potential of the nucleus plus a repulsive potential from the other electrons 11 electron 1 electron Note energy now depends on ! 23 For a fixed value of n, the energy increases with l 24 Pauli exclusion principle Due to spin, each ml state is actually two states (a spin up and a spin down state). Therefore the total number of states for the nth level is 2n2. The Pauli exclusion principle states the no two electrons may have the same quantum numbers (unless they are in different atoms, where one can distinguish them!). Therefore each atomic state in a given atom can only accommodate two electrons (a spin up and a spin down). To get the lowest energy of an atom containing more than one electron, start filling from the lowest energy state, allowing no more 25 than two electron per state 26 Science Trek 2000/The Periodic Table Ex. 41-7: Z=37 electrons, what values of n and for the electron that is least tightly bound? 27 Only fermions, which are spin 1/2 particles such as electrons, obey the Pauli exclusion principle. Bosons, which are particles with integer spin (s=0,1,2…), on the other hand are not only allowed to share the same quantum state, but prefer to all be in the same quantum state. Examples of bosons: Photons (s=0) 4He (s=n, where n is an integer) 28 Formation of molecules H2 molecule + + 29 Conditions for formation of molecules 1. Each atom contributes one electron from outside closed shells to form a spin up-spin down pair. Since they have opposite spin, they can move closer to neighboring nuclei and attract the atoms together. Each pair forms a bond, the more pairs the stronger the binding. 2. Only electrons not in closed shells can form bonds with other atoms (electrons in closed shells already have partners). 3. Similarly, electrons that are already paired in a shell cannot form bonds with other atoms. 30 Ionic bond: atom A (alkali metal) has 1 electron outside closed shells, atom B (halogen) is 1 electron short of filling outermost shell, the electron is transferred leaving A+ and B-, ions attract, strong bond (e.g., NaCl, NaF) Valence bond: atoms share electrons to “fill” their outermost shells (e.g., H2, GaAs) 31 Science Trek 2000/The Periodic Table Van der Waals forces Sometimes inert atoms (all shells closed) do form molecules. Charge of one atom induces dipole moment in other, causes attraction. Remember the grass seed in electric field field demo in Physics 108? The van der Waals force is always attractive but falls of as 1/r5 E 32 Molecular spectra H2 (transitions among electronic, vibrational, and rotational states) H (purely electronic transitions) 33 Vibrational motion Evib = n ω n is an integer k ω= is the characteristic frequency M M 1M 2 M= is the reduced mass, M1 + M 2 (M 1 and M 2 are the two nuclear masses) See Chapter 7 (SHO) M hydrogen = ? 34 Rotational states L2 P2 Erot = Etrans = 2I 2M Moment of inertia for two masses connected by a rigid, massless rod of length r0 is: I = Mr02 M= M 1M 2 M1 + M 2 is the reduced mass quantize L → L2 = 2 L = Erot = 2I = ( ( + 1) 2I ( 2 + 1) 2 + 1) ( = 2Mr02 2 2 me e me + 1) 2 4πε 0 4 M 2 8πε 0 2 where we have substituted r0 = 2a0 = me e 2 Erot = ( + 1) E0 me 4M E0 =13.6 eV 35 Molecular energy states 1 eV 0.0001 eV 0.01 eV Eelect − E0 = 2 n Evib = nE0 me Z1Z 2 M E0 = 13.6 eV me −4 ≈ 10 M Erot = ( + 1) E0 me 4M 36