modern physics 7

7/24/2019 Modern Physics 7

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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

,(-"TE 6

Quantum Mechanics IIQuantum Mechanics II

I think it is safe to say that no one understands quantum mechanics. Do

not keep saying to yourself, if you can possibly avoid it, But how can it

be like that? because you will get down the drain into a blind alley from

which nobody has yet escaped. !obody knows how it can be like that.

! ichard $e/n'an


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6.10 The Schrdinger Wave Equation

The Schrdinger ave equation in its ti'e!dependent or' or a

particle o energ/ "'oving in a potential #in one di'ension is

The extension into three di'ensions is

here is an i'aginar/ nu'er.

c

Fdt

dP

=-nalogous to0


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eneral Solution o the Schrdinger

Wave Equation

The general or' o the ave unction is

hich also descries a ave 'oving in the$direction.

In general the a'plitude 'a/ also e co'plex.

The ave unction is also not restricted to eing real.

4otice that the sine ter' has an i'aginar/ nu'er. )nl/

the ph/sicall/ 'easurale quantities 'ust e real.These include the proailit/5 'o'entu' and energ/.


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4or'aliation and "roailit/

The proailit/ %7$8 d$o a particle eing eteen$and x9 d$as given in the equation

The proailit/ o the particle eing eteen$1and$2is given

/

The ave unction 'ust also e nor'alied so that theproailit/ o the particle eing so'ehere on the$axis is 1.


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"roperties o Valid Wave $unctions

Boundary conditions18 In order to avoid ininite proailities5 the ave unction 'ust e initeever/here.

28 In order to avoid 'ultiple values o the proailit/5 the ave unction

'ust e single valued.

38 $or inite potentials5 the ave unction and its derivative 'ust econtinuous. This is required ecause the second!order derivative ter'

in the ave equation 'ust e single valued. 7There are exceptions to

this rule hen #is ininite.8

#8 In order to nor'alie the ave unctions5 the/ 'ust approach ero as$

approaches ininit/.

Solutions that do not satis/ these properties do not generall/

correspond to ph/sicall/ realiale circu'stances.


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Ti'e!Independent Schrdinger Wave

Equation The potential in 'an/ cases ill not depend explicitl/ on ti'e. The dependence on ti'e and position can then e separated in the

Schrdinger ave equation. :et 5

hich /ields0

4o divide / the ave unction0

The let side o Equation 76.1;8 depends onl/ on ti'e5 and the right sidedepends onl/ on spatial coordinates. (ence each side 'ust e equal toa constant. The ti'e dependent side is


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We integrate oth sides and ind0

here &is an integration constant that e 'a/ choose to e ;. Thereore

This deter'ines f to e ree particle

This is


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Stationar/ State

The ave unction can e ritten as0

The proailit/ densit/ eco'es0

The proailit/ distriutions are constant in ti'e. This is a standing

ave pheno'ena that is called the stationar/ state.

2( )x=


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,o'parison o ,lassical and =uantu'

>echanics

4eton?s second la and Schrdinger?s ave equation are

oth dierential equations.

4eton?s second la can e derived ro' the Schrdinger

ave equation5 so the latter is the 'ore unda'ental.

,lassical 'echanics onl/ appears to e 'ore precise ecause

it deals ith 'acroscopic pheno'ena. The underl/ing

uncertainties in 'acroscopic 'easure'ents are @ust too s'all

to e signiicant.


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6.20 Expectation Values

The expectation valueis the expected result o the average o'an/ 'easure'ents o a given quantit/. The expectation value

o$is denoted / A$B

-n/ 'easurale quantit/ or hich e can calculate the

expectation value is called a physical observable. The

expectation values o ph/sical oservales 7or exa'ple5position5 linear 'o'entu'5 angular 'o'entu'5 and energ/8

'ust e real5 ecause the experi'ental results o

'easure'ents are real.

The average value o$is


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,ontinuous Expectation Values

We can change ro' discrete tocontinuous variales / using the

proailit/ %7$5t8 o oserving the

particle at a particular$.

Csing the ave unction5 theexpectation value is0

The expectation value o an/

unction g7$8 or a nor'alied aveunction0


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+ra!Det 4otation

*( ) ( ) ( ) ( ) | |g x x g x x dx g=

This expression is so i'portant that ph/sicists have a special

notation or it.

The entire expression is thought to e a rac


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>o'entu' )perator To ind the expectation value op5 e irst need to representpin ter's

o$and t. ,onsider the derivative o the ave unction o a ree particle

ith respect to$0

With kGpH ' e have

This /ields

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

The expectation value o the 'o'entu' is


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The position$is its on operator as seen aove. The ti'e derivative o the ree!particle ave unction is

Sustituting (G "H ' /ields

The energ/ operator is

The expectation value o the energ/ is

"osition and Energ/ )perators


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6.30 Ininite Square!Well "otential

The si'plest such s/ste' is that o a particle trapped in a ox ith

ininitel/ hard alls that the particle cannot penetrate. This potentialis called an ininite square ell and is given /

,learl/ the ave unction 'ust e ero here the potential isininite.

Where the potential is ero inside the ox5 the Schrdinger ave

equation eco'es here .

The general solution is .


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Sustituting operators0

E0

K+V0

&eriving the Schrodinger Equation

using operators2

2

pE K V V

m= + = +The energ/ is0

E i t

= h22 1

2 2

pV i V

m m x

+ = + h

2

2

pE V

m = +

2 2

22V

m x = +

h

2 2

22i V

t m x

= +

h

hSustituting0


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=uantiation +oundar/ conditions o the potential dictate that the ave unction 'ust

e ero at$G ; and$G ). This /ields valid solutions or integer values o

nsuch that k)G n*.

The ave unction is no

We nor'alie the ave unction

The nor'alied ave unction eco'es

These unctions are identical to those otained or a virating string ith

ixed ends.


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6.30 Ininite Square!Well "otential

The si'plest such s/ste' 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 /0

,learl/ the ave unction 'ust e ero here the potential is

ininite.

Where the potential is ero 7inside the ox85 the ti'e!independentSchrdinger ave equation eco'es0

The general solution is0

x0 L

here

The energ/ is entirel/


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+oundar/ conditions o the potential dictate

that the ave unction 'ust e ero atx= 0andx=L. This /ields valid solutions or

integer values o nsuch that kL= n.

The ave unction is0

We nor'alie the ave unction0

The nor'alied ave

unction eco'es0

The sa'e unctions as those or a virating string ith ixed ends.

=uantiation

x0 L

2 /A L=

cos(2nx/L)


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=uantied Energ/

The quantied ave nu'er no eco'es

Solving or the energ/ /ields

4ote that the energ/ depends on the integer values o n. (ence the

energ/ is quantied and nonero.

The special case o nG 1 is called the ground state energ/.


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6.#0 $inite Square!Well "otential

The inite square!ell potential is

The Schrdinger equation outside the inite ell in regions I and III is

or using

/ields . ,onsidering that the ave unction 'ust e ero at

ininit/5 the solutions or this equation are


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Inside the square ell5 here the potential #is ero5 the ave equation

eco'es here

Instead o a sinusoidal solution e have

The oundar/ conditions require that

and the ave unction 'ust e s'ooth here the regions 'eet.

4ote that the

ave unction is

nonero outsideo the ox.

$inite Square!Well Solution


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"enetration &epth

The penetration depth is the distance outside the potential ell here

the proailit/ signiicantl/ decreases. It is given /

It should not e surprising to ind that the penetration distance that

violates classical ph/sics is proportional to "lanc


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The ave unction 'ust e a unction o all three spatial coordinates.

We egin ith the conservation o energ/ >ultipl/ this / the ave unction to get

4o consider 'o'entu' as an operator acting on the ave

unction. In this case5 the operator 'ust act tice on each

di'ension. iven0

The three di'ensional Schrdinger ave equation is

6.%0 Three!&i'ensional Ininite!"otential Well


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&egenerac/

-nal/sis o the Schrdinger ave equation in three di'ensions

introduces three quantu' nu'ers that quantie the energ/.

- quantu' state is degenerate hen there is 'ore than one ave

unction or a given energ/.

&egenerac/ results ro' particular properties o the potential energ/

unction that descries the s/ste'. - perturation o the potential

energ/ can re'ove the degenerac/.


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6.60 Si'ple (ar'onic )scillator Si'ple har'onic oscillators descrie 'an/ ph/sical situations0 springs5

diato'ic 'olecules and ato'ic lattices.

,onsider the Ta/lor expansion o a potential unction0

edeining the 'ini'u' potential and the ero potential5 e have

Sustituting this into the ave equation0

:et and hich /ields .


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"araolic "otential Well

I the loest energ/ level is ero5 this violates the uncertaint/ principle.

The ave unction solutions are here +n7$8 are (er'ite

pol/no'ials o order n.

In contrast to the particle in a ox5 here the oscillator/ ave unction is a

sinusoidal curve5 in this case the oscillator/ ehavior is due to the pol/no'ial5

hich do'inates at s'all$. The exponential tail is provided / the aussian

unction5 hich do'inates at large$.


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-nal/sis o the "araolic "otential Well

The energ/ levels are given /

The ero point energ/ is called the (eisenerg

li'it0

,lassicall/5 the proailit/ o inding the 'ass is

greatest at the ends o 'otion and s'allest at the

center 7that is5 proportional to the a'ount o ti'ethe 'ass spends at each position8.

,ontrar/ to the classical one5 the largest proailit/

or this loest energ/ state is or the particle to e

at the center.


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The "araolic

"otential WellThe ave unction solutions

are

hereHn(x)are ermite

polynomialso order n.


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The "araolic

"otential Well,lassicall/5 the

proailit/ o inding the

'ass is greatest at the

ends o 'otion ands'allest at the center.

,ontrar/ to the classical

one5 the largest

proailit/ or this loestenerg/ state is or the

particle to e at the

center.


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-nal/sis o the "araolic "otential Well

-s the quantu' nu'er increases5 hoever5 the solution

approaches the classical result.

!"v


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The "araolic "otential Well

The energ/ levels are given /0

The ero point

energ/ is

called the

(eisenerg

li'it0


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6.*0 +arriers and Tunneling ,onsider a particle o energ/ "approaching a potential arrier o height #; and

the potential ever/here else is ero.

We ill irst consider the case hen the energ/ is greater than the potential

arrier.

In regions I and III the ave nu'ers are0

In the arrier region e have


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election and Trans'ission

The ave unction ill consist o an incident ave5 a relected ave5 and a

trans'itted ave.

The potentials and the Schrdinger ave equation or the three regions areas ollos0

The corresponding solutions are0

-s the ave 'oves ro' let to right5 e can si'pli/ the ave unctions to0


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"roailit/ o election and Trans'ission

The proailit/ o the particles eing relected or trans'itted -is0

+ecause the particles 'ust e either relected or trans'itted e have0

9 -G 1.

+/ appl/ing the oundar/ conditions$ JK5$G ;5 and$G )5 e arrive

at the trans'ission proailit/0

4otice that there is a situation in hich the trans'ission proailit/ is 1.


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Tunneling 4o e consider the situation here classicall/ the particle does not have

enough energ/ to sur'ount the potential arrier5 "A #;.

The quantu' 'echanical result5 hoever5 is one o the 'ost re'ar


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Cncertaint/ Explanation

,onsider hen )BB 1 then the trans'ission

proailit/ eco'es0


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-nalog/ ith Wave )ptics

I light passing through a glass pris' relects ro' an

internal surace ith an angle greater than the criticalangle5 total internal relection occurs. (oever5 the

electro'agnetic ield is not exactl/ ero @ust outside

the pris'. I e ring another pris' ver/ close to the

irst one5 experi'ents sho that the electro'agnetic

ave 7light8 appears in the second pris' The situation

is analogous to the tunneling descried here. This

eect as oserved / 4eton and can e

de'onstrated ith to pris's and a laser. The

intensit/ o the second light ea' decreases

exponentiall/ as the distance eteen the to pris's

increases.


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"otential Well

,onsider a particle passing through a potential ell region rather than through a

potential arrier. ,lassicall/5 the particle ould speed up passing the ell region5 ecause /G mv2H 2 G

"9 #;.

-ccording to quantu' 'echanics5 relection and trans'ission 'a/ occur5 ut the

avelength inside the potential ell is s'aller than outside. When the idth o the

potential ell is precisel/ equal to hal!integral or integral units o the avelength5 therelected aves 'a/ e out o phase or in phase ith the original ave5 and

cancellations or resonances 'a/ occur. The relectionHcancellation eects can lead to

al'ost pure trans'ission or pure relection or certain avelengths. $or exa'ple5 at the

second oundar/ 7$ G )8 or a ave passing to the right5 the ave 'a/ relect and e

out o phase ith the incident ave. The eect ould e a cancellation inside the ell.


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-lpha!"article &eca/ The pheno'enon o tunneling explains the alpha!particle deca/ o heav/5

radioactive nuclei.

Inside the nucleus5 an alpha particle eels the strong5 short!range attractive

nuclear orce as ell as the repulsive ,oulo' orce.

The nuclear orce do'inates inside the nuclear radius here the potential is

approxi'atel/ a square ell.

The ,oulo' orce do'inates

outside the nuclear radius. The potential arrier at the nuclear

radius is several ti'es greater than

the energ/ o an alpha particle.

-ccording to quantu' 'echanics5

hoever5 the alpha particle can

tunnelF through the arrier. (ence

this is oserved as radioactive deca/.

modern physics 7

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