lecture4 quantum chem ks

7/23/2019 Lecture4 Quantum Chem Ks

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K. Sumithra

CHEMISTRY I (CHEM C141)

Lecture 4: 11/8/2010



What Next?

• Light behaves like waves a!" #a$ti%les&• 'a$ti%les %a! behave like waves&• E!e$g is *a!ti+e"&• ,!%e$tai!t #$i!%i#le

How to model the matter wave mathematically ?



Wave -*!%ti.! /0 #si



The wavefunction /0 #si

In quantum mechanics, we abandon the classical concept of

particles moving along trajectories.

The quantum mechanical view is that a particle is spreadthrough space like a wave, being more likely to be found in

some regions than in others.

To a crude approximation, the wavefunction is a blurredversion of the trajectory.



Wave -*!%ti.! Ψ/ψ psi

• Mathe2ati%al t..l i! 3*a!t*2 Me%ha!i%s

• It is a -*!%ti.! t#i%all .- s#a%e .$2.2e!t*2 .$ s#i! a!" #.ssibl .- ti2e

• The laws .- *a!t*2 2e%ha!i%s "es%$ibeh.w the wave -*!%ti.! ev.lves .ve$ ti2e&



Wave -*!%ti.! Ψ psi

%lassi%al wave is %ha$a%te$i+e" b wavele!gth 5

6 a2#lit*"e I!te!sit .- light is #$.#.$ti.!al t. 7

2#lit*"e .- the 3*a!t*2 2e%ha!i%al wave is %alle"

Wave -*!%ti.! /

Matte$ wave 8 3*a!t*2 2e%ha!i%al wave

C.!tai!s all the "!a2i%al i!-.$2ati.! ab.*t the#a$ti%le



Wave -*!%ti.! Ψ psi

2#lit*"e .- the 3*a!t*2 2e%ha!i%al wave is %alle"Wave -*!%ti.! /

Ψ2 dV 9 7":; Ψ may e com!le"

the !roaility that the !article i# located i$ the

i$%i$ite#imal eleme$t o% volume dV aout the

&ive$ !oi$t' at time t.

7

'$.babilit "e!sit



(ua$tum )echa$ical *o#tulate

+out wave %u$ctio$

,he #tate o% a$y (ua$tum mecha$ical #y#tem ca$ e

#!eci%ied a# com!letely a# !o##ile y the State -*!%ti.! or

Wave -*!%ti.! Ψ

(ua$tum mecha$ical wave carrie# the

i$%ormatio$ aout !o#itio$' mome$tum'

Ki$etic e$er&y' *ote$tial e$er&y etc.



-ave%u$ctio$ %or !article with !reci#ely de%i$ed

!o#itio$



+ #har!ly localied wave%u$ctio$ ca$ e &e$erated

y addi$&/#u!er!o#itio$ o% lar&e/i$%i$ite $umer o%

wave%u$ctio$#



• Su!er!o#itio$ o% ma$y wave# corre#!o$d# to

#u!er!o#itio$ o% ma$y di%%ere$t li$ear mome$ta de

ro&lie mome$tum i$%ormatio$ i# lurred

•-e ca$ &et localiatio$ o% !article o$ly at the

e"!e$#e o%/lo## o% !reci#e i$%ormatio$ aout the

mome$ta



Born interpretation of the wavefunction

Ψ i# $ot a #!eci%ic !ath o% electro$.

It is i2#.ssible t. #$e"i%t the #.siti.! .- the#a$ti%le at a! i!sta!t

We %a! .!l #$e"i%t the #$.babilit .--i!"i!g the ele%t$.! i! a %e$tai!

$egi.!&



• I! the <.$! i!te$#$etati.!;

•

7 9 #$.babilit "e!sit&

The #$.babilit .- -i!"i!g a #a$ti%le i! a s2all

$egi.! .- s#a%e .- v.l*2e ": 9 7

": (2.$e ge!e$all ==7": si!%e 2a be %.2#lex) ": > :.l*2e ele2e!t

dV d" i$ o$e dime$#io$

d" dy two dime$#io$#

d" dy d three dime$#io$#

Probability ensity



! is physically significant

3ode : the !oi$t where the wave%u$ctio$ !a##e# throu&h ero

$d !oi$t# are $ot $ode#5



"ormali#ation

$ evaluated over the entire space in which the particle exist

should be equal to %, or %&&'

Thus wavefunctions need to be normali#ed

(avefunctions are normali#ed if and only if

) *ntegral limits would be modified to represent the limits of

space a particle inhibits +

1* =

∫

+∞

∞−

dxψ ψ



,s*all the wave-*!%ti.! is 2*lti#lie" with a %.!sta!ts. that !.$2ali+ati.! %.!"iti.! %a! be satis-ie"

*t doesnt effect the shape of the function, it only imposes

a scaling factor on the amplitude



Stati.!a$ States

6$ ma$y #ituatio$# o% i$tere#t i$ chemi#try' o$e i#

co$cer$ed with #tatio$ary #tate#.

the !roaility di#triutio$ i# i$de!e$de$t o% time.

7or #uch #tate#'

2 dV

re!re#e$t# the time9i$de!e$de$t !roailitydi#triutio$.

wave-*!%ti.! (x;;+) is a -*!%ti.! .- the s#atial

%..$"i!ates al.!e&



*ro!ertie# o% acce!tale wave%u$ctio$' Ψ

Si$&le9valued

o$ti$uou#

(uadratically i$te&rale

o$ti$uou# 1#t derivative

ou$dary co$ditio$

-ave %u$ctio$ #hould va$i#h at the ou$darie#



3*a!t*2 The.$

The.$eti%al ##$.a%h 8

1&S%h$"i!ge$ 7& W& Heise!be$g

Wave !at*$e 'a$ti%le !at*$e@i--e$e!tial e*ati.!s Mat$ix Me%ha!i%s

A& Ri%ha$" Be!2a!

'ath I!teg$al ##$.a%h

Sa2e $es*lts

Schr;di$&er <uatio$

,he %u$dame$tal law o% (ua$tum )echa$ic# =



The S%h$"i!ge$ E*ati.!

Schr;di$&er 1> 3oel *rie

-e ca$$ot !rove the Schr;di$&er e<uatio$ .

-e ca$ o$ly veri%y it5

+ !article i$ a @#taleA or time9

i$de!e$de$t #tate ca$ e re!re#e$tedmathematically a# a wave' y a

@wave%u$ctio$A " i$ 19B which

i# a #olutio$ to the di%%ere$tial

e<uatio$




-+V7C3,6D3 /

*S6

,he #tate o% a #y#tem !article i# decried

a# %ully a# !o##ile y it# wave%u$ctio$"'y''t'

,he %u$dame$tal !o#tulate :

(ua$tum )echa$ic# : -ave %u$ctio$# are

#olutio$# o% the Schr;di$&er e<uatio$



• la##ical -ave# : *artial di%%ere$tial

e<uatio$#


• + <ua$tum mecha$ical wave : ca$ al#o e writte$

a# a di%%re$tial e<uatio$

,he #olutio$# : -ave%u$ctio$

,he di%%ere$tial e<uatio$ :

Schr;di$&er e<uatio$

How to model the (ua$tum wave mathematically ?



i&e$value e<uatio$#

D!erator: d/d" acti$& o$ y = ei&e$ %u$ctio$ y

Solutio$: ye"!a"

i&e$value a=( ) ( )axax

eaedx

d =

D!erator o$ %u$ctio$ co$#ta$t E %u$ctio$Dbse$vables 8 #.siti.!; 2.2e!t*2; &E; '&E et%

D#e$at.$s 8 x; '; ; :; F et%&

Eige! > Dw! .$ Cha$a%te$isti%

lecture4 quantum chem ks

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