Manfra- Ch 6 - Quantum Mechanics II

7/26/2019 Manfra- Ch 6 - Quantum Mechanics II

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6.1 The Schrdinger Wave Equation

6.2 Expectation Values

6.3 Ininite Square!Well "otential

6.# $inite Square!Well "otential

6.% Three!&i'ensional Ininite!"otential Well

6.6 Si'ple (ar'onic )scillator

6.* +arriers and Tunneling

CHAPTER 6

Quantum Mechanics II

Er,in Schrdinger -1*!1/610

A careful analysis of the process of observation in atomic physics has

shown that the subatomic particles have no meaning as isolated

entities, but can only be understood as interconnections between the

preparation of an experiment and the subsequent measurement.

! Er,in Schrdinger


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Opinions on quantum mechanics

I think it is safe to say that no

one understands quantummechanics. Do not keep saying

to yourself, if you can possibly

avoid it, ut how can it be like

that!" because you will get

down the drain" into a blind

alley from which nobody has yetescaped. #obody knows how it

can be like that.

! ichard $en'an

ichard $en'an -1/1!1/0

$hose who are not shocked

when they first come across

quantum mechanics cannot

possibly have understood it.

! iels +ohr


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6.1: The Schr!in"er #a$e Equation

The Schrdinger ,ave equation in its ti'e!dependent or' or aparticle o energE'oving in a potential Vin one di'ension is4

,here iis the square root o !1.

The Schrodinger Equation is T(E unda'ental equation o 5uantu'

echanics.

,here V = V(x,t)


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%enera& So&ution o' the Schr!in"er #a$e

Equation (hen V) *

Tr this solution4

This ,or7s as long as4,hich sas that the total

energ is the 7inetic energ.

( )i kx t i Ae it

= =

22

2 k

x

=

( )( )i i i

t

= =

h h h

2 2 2 2

2

2 2

k

m x m

=

h h

2 2

2

k

m=

hh


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%enera& So&ution o' the Schr!in"er

#a$e Equation (hen V) *

In ree space -,ith V8 90: the general or' o the ,ave unction is

,hich also descri;es a ,ave 'oving in thexdirection. In general the

a'plitude 'a also ;e co'plex.

The ,ave unction is also not restricted to ;eing real. otice that this

unction is co'plex.

)nl the phsicall 'easura;le quantities 'ust ;e real. These

include the pro;a;ilit: 'o'entu' and energ.


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+orma&i,ation an! Pro-a-i&it

The pro;a;ilitP(x) dxo a particle ;eing ;et,eenxandx + dxis

given in the equation

The pro;a;ilit o the particle ;eing ;et,eenx1andx2is given ;

The ,ave unction 'ust also ;e nor'ali


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Properties o' /a&i! #a$e 0unctions

Con!itions on the (a$e 'unction:1. In order to avoid ininite pro;a;ilities: the ,ave unction 'ust ;e

inite ever,here.

2. The ,ave unction 'ust ;e single valued.

3. The ,ave unction 'ust ;e t,ice dierentia;le. This 'eans that itand its derivative 'ust ;e continuous. -=n exception to this rule

occurs ,hen Vis ininite.0

#. In order to nor'ali


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The potential in 'an cases ,ill not depend explicitl on ti'e.

The dependence on ti'e and position can then ;e separated in theSchrdinger ,ave equation. >et4

,hich ields4

o, divide ; the ,ave unction (x)f(t)4

TimeIn!epen!ent Schr!in"er #a$e Equation

The let side depends onl on t: and the right side

depends onl onx. So each side 'ust ;e equal to

a constant. The ti'e dependent side is4


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ultipl ;oth sides ;f(t)/i4

,hich is an eas dierential equation to solve.


+ut recall our solution or the ree particle4

in ,hichf(t) = exp(-it): so4 = B / orB = : ,hich 'eans that4B = E?

So 'ultipling ; (x): the spatial Schrdinger equation ;eco'es4

ignoring the proportionalit

constant: ,hich ,ill co'e ro'

the nor'ali


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This equation is 7no,n as the timein!epen!ent Schr!in"er

(a$e equation: and it is as unda'ental an equation in quantu''echanics as the ti'e!dependent Schrodinger equation.

So oten phsicists ,rite si'pl4

,here4

is an operator.

H E =

2 2

2

2H V

m x

= +

h H


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Stationar States

The ,ave unction can ;e ,ritten as4

The pro;a;ilit densit ;eco'es4

The pro;a;ilit distri;ution is constant in ti'e.

This is a standing ,ave pheno'enon and is called a stationar state.

* *( ) ( )i t i t x e x e =2

( )x=


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6.2: E3pectation /a&ues

In quantu' 'echanics: ,e@ll co'pute expectation values.

The e3pectation $a&ue4 4is the ,eighted average o agiven quantit. In general: the expected value oxis4

I there are an ininite nu';er o possi;ilities: andxis continuous4

5uantu'!'echanicall4

=nd the expectation o so'e unction ox:g(x)4

1 1 2 2 N N i i

i

x P x P x P x P x= + + + = L

x

( )x P x x dx=

* *

( ) ( ) ( ) ( )x x x x dx x x x dx= = *

( ) ( ) ( ) ( )g x x g x x dx=


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5raet +otation

This expression is so i'portant that phsicists have a special

notation or it.

The entire expression is thought to ;e a A;rac7et.B

=nd is called the -ra,ith the 7et.

The nor'ali


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To ind the expectation value op: ,e irst need to representpin ter's

oxand t. Consider the derivative o the ,ave unction o a ree particle,ith respect tox4

With k=p/ ,e have

This ields

This suggests ,e deine the 'o'entu' operator as .

The expectation value o the 'o'entu' is

Momentum Operator


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The positionxis its o,n operator. &one.

Energ operator4 The ti'e derivative o the ree!particle ,ave

unction is4

Su;stituting = / ields

The energ operator is4

The expectation value o the energ is4

Position an! Ener" Operators


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Su;stituting operators4

E4

K+V4

8eri$in" the Schro!in"er Equation

usin" operators

The energ is4

Su;stituting4

2

2

pE K V Vm

= + = +

E i t

=

h

221

2 2

pV i V

m m x

+ = + h

2

2

pE Vm

= +

2 2

22V

m x = +

h

2 2

22i V

t m x

= +

hh


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Some !i''erentia& equation so&utions

Consider this dierential equation4

+ecause the constant k2is positive: the solution is4

o, suppose the dierential equation is4

+ecause the constant !k2is negative: the solution is4

kis real

2

2

2

d kdx

=

( ) kxx Ae =

2

2

2

dk

dx

=

( ) sin( ) cos( )ikxx Ae A kx B kx = +or


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6.9: In'inite Square#e&& Potentia&

The si'plest such sste' is that o a particle

trapped in a ;ox ,ith ininitel hard ,alls thatthe particle cannot penetrate. This potential is

called an ininite square ,ell and is given ;4

Clearl the ,ave unction 'ust ;e


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+oundar conditions o the potential dictate

that the ,ave unction 'ust ;e


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Quanti,e! Ener"

The quanti


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6.: 0inite Square

#e&& Potentia&

The inite square!,ell potential is4

Considering that the ,ave unction

'ust ;e


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Inside the square ,ell: ,here the potential Vis


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The partic&e penetrates the (a&&s;

The penetration depth is

the distance outside the

potential ,ell ,here the

pro;a;ilit signiicantl

decreases. It is given ;

The penetration distanceis proportional to "lanc7@s

constant.

This violates classical

phsics?


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6.6: Simp&e Harmonic Osci&&ator

Si'ple har'onicoscillators descri;e

'an phsical

situations4 springs:

diato'ic 'olecules

and ato'ic lattices.

Consider the Talor expansion o a

potential unction4


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Consider the second!order ter'

o the Talor expansion o a

potential unction4

Simp&e Harmonic

Osci&&ator

>ettingx0= 0

>et and ,hich ields4

Su;stituting this into

Schrdinger@s equation4

2102

( ) ( )V x x x=


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The Para-o&ic

Potentia& #e&&

The ,ave unction solutions

are ,hereHn(x)

are Hermite po&nomia&so

order n.


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The Para-o&ic

Potentia& #e&&

Classicall: the pro;a;ilit

o inding the 'ass is

greatest at the ends o

'otion and s'allest atthe center.

Contrar to the classical

one: the largest

pro;a;ilit or this lo,estenerg state is or the

particle to ;e at the

center.


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Ana&sis o' the Para-o&ic Potentia& #e&&

=s the quantu' nu';er increases: ho,ever: the solution

approaches the classical result.


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The Para-o&ic Potentia& #e&&

The energ levels are given ;4

The


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6.


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Re'&ection an! Transmission

The ,ave unction ,ill consist o an incident ,ave: a relected ,ave: and

a trans'itted ,ave.

The potentials and the Schrdinger ,ave equation or the three regions4

=ll three

constants

are

negative.

Sines and

cosines in all

three regions

Since the ,ave 'oves ro' let to right: ,e can reDect so'e solutions4

The corresponding solutions are4


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Pro-a-i&it o' Re'&ection an! Transmission

The pro;a;ilit o the particle ;eing relectedRor trans'itted Tis4

+ecause the particle 'ust ;e either relected or trans'itted4

R+ T= 1.

+ appling the ;oundar conditions

x ,x= 0: andx=L: ,e arrive at

the trans'ission pro;a;ilit4

ote that the trans'ission pro;a;ilit can ;e 1.

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Manfra- Ch 6 - Quantum Mechanics II