Postulates of quantum mechanics

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1 Postultes of qutum mechics P1. The stte of qutum mechicl sstem is completel specified b wvefuctio Ψ(q,t) P. For ever mesurble propert of the sstem, there eists correspodig opertor i QM (mesuremet i the lb opertig o the wvefuctio, ÂΨ) P3. Ψ(q,t) is obtied b solvig t-dep Schrodiger equtio, P4. I sigle mesuremet of the observble tht correspods to the opertor A, the ol vlues tht will ever be mesured re the eigevlues of tht opertor P5. If the sstem is i stte described b Ψ(q,t), the verge vlue of ll the mesuremets for the observble A * ψ Âψdτ A * ψψ d τ 1

2 Schrodiger equtio: m 3. Prticle-i-bo models: ˆ HΨ i Ψ, Hˆ ψ Eψ t ˆ + V q,, H ( ), where q:, (, z, ), ( r, θφ, ) or ( rr ) 1 d (,, ) z + + V z 4. Rigid-rotor d gulr mometum eigesttes 1 1 si 1 d,, θ + V ( r θφ) Re siθ θ θ si θ φ 5. moleculr vibrtios d t-dep perturbtio theor 1 Jˆ 1 r d V ( r, θφ, ) kr r r r 1 ˆ J Ze r d V r r r 4πε r 6. the hdroge tom:

3 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) rsiθcosφ rsiθcosφ z rcosθ r + + z cosθ tφ z r 3

4 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) z r r r siθ θ θ si θ φ r siθ ˆ 1 1 J siθ + siθ θ θ si θ φ ˆ 1 J r r r r dτ dddz ( )( si ) ( r si ) dτ dr rdθ r θdφ θ drdθdφ 4

5 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) rsiθcosφ rsiθcosφ z rcosθ r + + z cosθ tφ z r r θ φ + + r θ φ z, z, θφ, z, r, φ z, r, θ r θ φ + + r θ φ z, z, θφ, z, r, φ z, r, θ r θ φ + + z z r z θ z φ,, θφ,, r, φ, r, θ 5

6 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) r + + z r r r siθcosφ z, r r r siθcos φ siθsi φ cosθ z z,, z, r + + z cosθ tφ, z r z z cosθ z + + z r + + z, z ( ) 1 θ 3 siθ z ( + + z ) ( ) rcosθ r z, z,, tφ 1 φ rsiθsiφ ( cosφ ) z, ( r siθcosφ) z,,, 3 ( rsiθcosφ) θ cosθcosφ θ cosθsiφ θ siθ r r z r φ siφ φ cosφ φ rsiθ rsiθ z z 6

7 3.1 Prticle i 1D bo Coductio i metls Opticl properties i polmers Free electro moleculr orbitl theor ˆ d H m d + V V,, otherwise ψ π si h E where 1,,3 8m h E1 8m Zero-poit eerg: turl cosequece of the ucertit priciple 7

8 3.1 Prticle i 1D bo π h si, E where 1,,3 ψ 8m ψ hs wvelegth λ/ Ech ode of the -th stte is locted betwee odes of the (+1)-th stte holds for solutio to the 1D Schrodiger equtio mkig ψ orthogol More odes shorter wvelegth higher KE 8

9 3.1 Prticle i 1D bo π h si, E where 1,,3 ψ 8m 9

10 Outside the bo 3.1 Prticle i 1D bo d ψ ( E ) ψ ψ m d Iside the bo ˆ d H m d + V V,, otherwise d ψ Eψ ψ Asiα+ Bcosα m d ( ) where α BC.. ψ B me π ψ ( ) α, 1,, 3 π ψ si E h π me α 8m π ψ Asi * π ψψd A si d 1 π A 1 cos d A 1 A ˆ d ψ π Hψ ( ψ) Eψ m d m π h E m 8m 1

11 3.1. Epecttio vlues * ψ ψd * ψψd 3 1 π h si, E where 1,,3 ψ 8m π 1 π si d 1 cos d 1 π cos d 4 1 π π si si d 4 π π p p p ψ p ψd d * * ψψ h 4 p p p h h h h > π p π d π si si d i d π π π si cos d i π π π let t si, dt cos d t, t tdt i Prticle is movig with equl probbilit from right to left d from left to right h h p m H m 8m 4 11

12 3.1.3 Orthogolit π h si, E where 1,,3 ψ 8m π mπ d A si si d * ψ ψm * ( + ) π ( ) 1 m m π cos cos d + i) m 1 ( + m) π ( m) π si + si ( + m) π ( m) π ii) m i( A± B) ia ± ib 1 π cos 1 d + 1 ( ) 1 ψ ψ m d δ m e e e ( A± B) + i ( A± B) ( A+ i A)( B± i B) A B A B i( A B A B) A A B B+ A A B B ( A+ B) ( A B) cos si cos si cos si cos cos si si + si cos ± cos si cos cos si si cos cos cos si si cos 1 si Asi B cos A+ B + cos A B ( ( ) ( )) 1

13 3.1.5 Correspodece priciple: serve s useful guide i the developmet of pproimte qutum theories (e, semiclssicl theories) π h si, E 1,, 3 ψ 8m As, Іψ І is the sme for ll As m, h, or, ΔE As h or size become mcroscopic, the predictios of QM become those of CM. 13

14 (Recll) Seprtio of vribles A sstem of two oiterctig prticle, H(q 1, q ) I f Hˆ Hˆ q Hˆ ( ) + ( q ) ( q, q ) ψ ( q ) ψ ( q ) 1 1 Ψ E E 1 ( ) ψ ( ) ψ ( ) ( ) ψ ( ) ψ ( ) where Hˆ q q E q + E Hˆ q q E q Hˆ Hˆ 1( q1) + Hˆ ( q) ( Hˆ 1( q1) + Hˆ ( q) ) Ψ ( q1, q) EΨ( q1, q) let Ψ ( q1, q) ψ1( q1) ψ( q) ( Hˆ 1( q1) + Hˆ ( q) ) ψ1( q1) ψ( q) Eψ1( q1) ψ( q) ψ( q) Hˆ 1( q1) ψ1( q1) + ψ1( q1) Hˆ ( q) ψ( q) Eψ1( q1) ψ( q) Hˆ 1( q1) ψ1( q1) Hˆ ( q) ψ( q) + E E1+ E ψ1( q1) ψ( q) Hˆ 1( q1) ψ1( q1) E1ψ1( q1) Hˆ ( q ) ψ ( q ) Eψ ( q ) A sstem of oiterctig prticle, H(q 1, q, ) If ( ) + ( ) + + ( q ) ( q, q, q ) ψ ( q ) ψ ( q ) ψ ( q ) ˆ ˆ ˆ ˆ H H1 q1 H q H where Hˆ Ψ E E + E + E 1 ( q ) ψ ( q ) Eψ ( q ) i i i i i i i 14

15 3. Prticle i D bo ˆ H, iside + b m 4 π π ψ, (, ) si si b b E 8m b where d 1,,3 h, + + m m Hˆ Hˆ + Hˆ iside ( ) ( ) ( ) ψ ( ) ψ ( ) Hˆ E ( ) ψ E h 8m si π ( ) ψ ( ) ψ ( ) Hˆ E ψ E ( ) h 8mb si π b b π π ψ, (, ) ( ) ( ) si si ψ ψ b b h h E, where d 1,,3 E + E + 8m 8mb 15

16 3. Prticle i D bo 4 π π ψ, (, ) si si b b h E, where d 1,,3 + 8m b ψ1,1 ψ ψ ψ,1 1,, If the bo is squre ( b) h E, ( ) degeerc + 8m M of the sttes re degeerte! ψ ψ 1,,1 π π si si π π si si E(h /8m )

17 3.3 Prticle i 3D bo ˆ H,, iside + + b z c m z 8 π π zπ z ψ,, (,, ) si si si z z bc b c E 8m b c where, d 1,,3 h z,, + + z z b c ( b c) E3. (p47) If prticle is i cubic bo, E,, ( ) h z + + z 8m 3h E111 8m E E E E (tripl degeerte) ψ11 ψ11 ψ11 17

18 3.4 Free-electro moleculr orbitl model E h 8m CC CC A simple model of the π electroic sttes i cojugted orgic molecule: 1. π e s i cojugted molecule c be seprted from the σ e s. the σ e s frmework is froze 3. ll iter electroic iterctios re eglected. 4. the effective potetil ctig o ech π e s is give b prticle-i--bo potetil 5. the epressio for the eerg levels of the π e s is give b E 6. π e s occup E ccordce with the Puli priciple (two e s ech) HOMO: the highest occupied moleculr orbitl LUMO: the lowest uoccupied moleculr orbitl LUMO HOMO If -stte is HOMO, h E E+ 1 E + 8m hc h hν + mi λm 8m ( ( ) ) 1 ( 1) butdiee For lier polees, the bsorptio wvelegth gets loger s the legth of the chi is icresed 18

19 E 3.1 ormliztio coditio : ψ dτ 1 ψ d d wvefuctio ψ is ot ormlizble. ( ψ mes tht the probbilit tht the prticle is where is zero. ) E 3. h 1 D: E 8m D: E 3 D: E, b c h 8m + 8m b c h z,, + + z b h E,, z + + z z 8m lowest stte is o-degeerte. ( ) (,, 1,, 3, ) ( z) ( ) ( ) ( ) The first ecited stte,,,1,1, 1,,1, 1,1, is tripl degeerte. 6h E E E E 8m

20 E 3.3 h E + 8m ( 1) ( ) A 5.6 A 1. the secod highest occupied the lowest empt h 8h E E ( 3 1 ) 8 m 8 m hc λ E 8m hc 19m 8h 3. the highest occupied the secod lowest empt h E4 E ( 4 ) 8 m 8m hc λ 86.3m 1h 1h 8m

21 P 3.1 ( ) ( b) () c h E 8m 1 E m 1

22 P 3. ψ ψd ψψd π π si si d 박스가운데있는입자의 verge vlue 를나타내면 : π π si si d π si si π π si d d

23 P 3.3 ( ) ( ) ψ ( ) BC. ψ d ψ, ψ should be cotiuous d well-behved wvefuctio. ( b) d ψ Eψ ψ Asiα+ Bcosα m d BC. ( ) ( ) ψ ψ B ψ ψ Asiα α π π α ψ Asiα ( ) d Asiα E Asiα m d ( Aα siα) EAsiα m α ( ) ( ) me me α ( ) 3

24 P 3.3 () c ψ ( ) A si α ( ), d m d m α ( A siα ( ) ) ( E Vs ) A siα ( ) ( A α siα ( ) ) ( E Vs ) A siα ( ) ( ) ( ) m E V m E V s α ( d) s Asiα A siα A siα differetil Aαcos α A α cosα A α cosα siα siα 1 1 α cos α α cosα 1 1 tα tα α α 4

25 P 3.4 p h h E p, λ, m λ + 1 π ( + 1) 1 h E m ( + 1) ( + ) 1 π 1 m π m ode 의수가증가하면 λ는감소, E는증가한다. 각각의 eigefuctio 은 orthogolit 를유지하기위해다른 ode 값을가진다. 5

26 P 3.6 ( ) h h E E + E + + 8m b 8m L L L 3L L 3 L, L L E () c h + 8m 9L L ( π ) 1 + 8m 9L π + 18mL ( 9 ) ( 9 ) 6

27 P 3.6 ( b) 7

28 P 3.7 ˆ H + + m z π ψ z,, z bc b c 8m b c cubic bo b c 8 π zπ z h z,, ( z,, ) si si si, E π π zπ z h ψ,, (,, ) si si si, z E z 3,, z 8m 1 ψ ( ψ1,,3 ψ3,1, ) ( + + z ) 1 8 π π 3πz 8 3π π πz si si si si si si ( A B) π π 3πz 3π π πz si si si A, si si si B ˆ 1 8 π 3π π π 3π π Hψ A+ B 3 A+ B A+ B m π 3 m 14π ψ m 7π E m ψ is eigefuctio. Similrl, ψ ψ ψ 1,1,1,, ( A B) is ot eigefuctio. 8

29 P 3.8 E ( z,, ) ( ) ( ) ( z) ψ ψ ψ ψ z z z π π zπ z si si si b b c c h h + + 8m 8mb 8mc b c.6 A z h ( z) ( ) ( z) ( ) ( ) ( ) Whe,, 1,1,1, it hs the lowest eerg. Whe,, 1,1,, 1,,1,,1,1, it hs the lowest ecittio eerg. E h + + 8m ( ) 34 ( J s) 31 1 kg ( m) h E + + 8m ( 1 1 ) 34 ( J s) 31 1 kg ( m) E E E 34 ( J s) 31 1 kg ( m) 18 6 ( 6 3)

30 P 3.9 k 이중결합의수 hc E ELUMO EHOMO λ π (( k+ 1) k ) m π ( k + 1) m π + m 1.4 A ( k+ ) m 1.4 A hc ( k + ) λ π k + 1 ( k 1) 31 1 ( )( ) 34 ( J s) π ( ) kg m 34 8 k + ( J s)( 3 1 m/ s) k m 8 m ( k + ) ( k + ) k + 1 k + 1 3

31 λ가 6 m보다커야하므로, ( k + ) 65m 6m k + 1 ( k+ ) ( k+ ) ( ) 65 k + 4k+ 4 1k+ 6 k k ( 15 + ) ( ) ± k 65 k 14.81,.35 f( k) k 14.81, k.35 k값은양수 k 15 ( ) CH CH CH CH CH CH 13 λ 65m 63m

32 P 3.1 ( ) ( ) ψ ψ ψ, h E + 8m ( ), ( 3 ) ( 3 ) + k k k 6 3k A hc E E+ 1 E λ ( 1 ) h 8m A c λ 8m A 88m 3h 4 si π π si b b 3

Using Quantum Mechanics in Simple Systems Chapter 15

Using Quantum Mechanics in Simple Systems Chapter 15 /16/17 Qutiztio rises whe the loctio of prticle (here electro) is cofied to dimesiolly smll regio of spce qutum cofiemet. Usig Qutum Mechics i Simple Systems Chpter 15 The simplest system tht c be cosidered