–schrödinger equation i€¦ · schrödinger’s equation of motion for particles moving in...

Experimental Physics 4 - Schrödinger equation 1

Experimental Physics EP4 Atoms and Molecules

– Schrödinger equation I –Derivation, stationary states,

potential barrier, tunnel effect

https://bloch.physgeo.uni-leipzig.de/amr/


Empirical derivation( )tie w-Y=Y kr

0Y=

¶Y¶

xikxY-=

¶Y¶ 22

2

xkxY-=Y-=YÑ 2

222

!pk

Equation of motion with constant momentum

Y-=Y-=Y-=¶Y¶

!! mpiEii

t 2

2

w

Equation of motion with constant (kinetic) energy YÑ-=¶Y¶ 2

2

2mti !!

Equation of motion for freely moving particles

JssJ=

×=úû

ùêëé¶¶t

!

Jskgm

kgmssmkg

2

2

24

2422

2

===úû

ùêë

éÑ

m!

( )Y+YÑ-=¶Y¶ rU

mti 2

2

2!

!

Schrödinger’s equation of motion for particles moving in potential fields

Wave-matter duality: Non-localized particle Function of localized particle in the potential field U


Stationary states

( )Y+YÑ-=¶Y¶ rU

mti 2

2

2!

!If Y in the whole space is known at certain instant of time,then Y can be determined at any other instances.

If Y1 and Y2 are two solutions of the Schrödinger equation,then a1Y1 + a2Y2 is also solution (superposition).

Stationary states: tie wyy -= )(r function of time only

function of space only)(*)(* rr yyyyr ==

( )yy úû

ùêë

é+Ñ-= rU

mE 2

2

2!

Schrödinger’s equation for stationary states

( )[ ]yy rUEE kintotal +=Energy conservation in quantum mechanics

Superposition principle applies too! But superposition of two stationary state solutions will not be stationary state!

tie 1)(11wy -=Y rtie 2)(22

wy -=Y r( )( )titititi eeee 2

2

1

1

21 )()()()( **21

221

wwww yyyy rrrr ++=Y+Y --

titi ee )(*12

)(*21

22

21

1221 )()()()()()( wwww yyyyyy ---- +++= rrrrrr1

11 )( dy ieA -=r2

22 )( dy ieA -=r( ))()(cos2 121221

22

21 ddww -+-++= tAAAA


Free particles

( ) 0=rU( ) yy EUm

=úû

ùêë

é+Ñ- r2

2

2!

yy Em

=Ñ- 22

2!

yymk

m 22

222

2 !!=Ñ- yy 22 k-=Ñ

( ) ikxikx BeAex -+=y

yy 22

2

kdxd

-=

Stationary states: tie wyy -= )(r

( ) )()(, tkxitkxi BeAetx wwy +-- +=x

( ) )(2/1, tkxieLtx wy --=

( )òD+

D-

-=2/

2/

0

0

)(),(kk

kk

tkxi dkekAtx wy

)( ykinE=

From Demtröder


Potential barrier

x

U(x)

I II

U0

ikxikxI BeAex -+=)(y

Incident wave Reflected wave

( ) IIII UE

dxd

myy

02

22

2-=-

!

( )II

II EUmdxd yy

20

2

2 2!-

= IIya 2º

xxII DeCex aay -+=)(

Solutions of the Schrödinger equation must be (i) finite, (ii ) continuous and (iii) unambiguously defined for any spatial coordinates.

{)(xy

)0()0( === xx III yyDCBA +=+

00

)()(==

=x

II

x

I

dxxd

dxxd yy

( ) ( )DCBAik -=- a


Potential barrier: E < U0

x

U(x)I II

U0

( ) 02202 >-

=!

EUma


xxII DeCex aay -+=)({)(xy


A DCBA +=+

( ) ( )DCBAik -=- a

ikikAB

-+

-=aa

ikikAD-

-=a2

÷øö

çèæ

-+

-= -ikxikxI e

ikikeAx

aay )(

B

2

2

AB

=2

ikik

-+

=aa 1=

Reflection coefficient

E

I

R

FFR = 2

2

IA

IB

yuyu

=


Potential barrier: E < U0

( ) 02202 >-

=!

EUma


xxII DeCex aay -+=)({)(xy


DCBA +=+

( ) ( )DCBAik -=- a

ikikAB

-+

-=aa

ikikAD-

-=a2

xII e

ikikAx a

ay -

--=

2)( xII e

kkAx a

ay 2

22

222 4)( -

+= xeA

mUk a22

0

222 -=!

Probability to find a particle in the region II is non-zero!

x

U(x)

I II

U0AB D

E


Potential barrier: E > U0

( ) 02202 <-

=!

EUma


xikxikII DeCex '')( -+=y{)(xy


DCBA +=+

( ) ( )DCikBAik -=- '

( )2

02'!UEmk -

º

x

U(x)III

U0

A

E

÷øö

çèæ

+-

-= -ikxikxI e

kkkkeAx'')(y 2

''kkkk

+-

=

Reflection coefficient

2

2

AB

R =

''kkkkAB

+-

='

2kkkAC+

=

2

2'AC

Tuu

= 2''4kkkk+

=

Transmission coefficient

xikII e

kkkAx '

'2)(+

=y


Tunnel effect

x

U(x)

I II

U0A

E

0 b

IIIikxikx

I BeAex -+=)(y{)(xyxikxik

III eBeAx '' '')( -+=y

xxII DeCex aay -+=)(

DCBA +=+

( ) ( )DCBAik -=- a

ikbbb eADeCe '=+ -aa

ikbbb eikADeCe '=- -aa aa( ) úû

ùêë

é÷ø

öçè

æ÷øö

çèæ +--=

aaaa kk

ibeea bikb

21sinh

2

2''AA

Tuu

= 21a

=

úúû

ù

êêë

é÷øö

çèæ ++=

222

161

41

aaa kk

ea b

( ) beb aa 21sinh »

( )202 2!

EUm -=a

22 2!mEk =

<<222

2

161

÷÷ø

öççè

æ +»

kke b

aaa

EEUUe b

)(161

0

202

-= a

beU

EEU a220

0 )(16 --» - The tunnel effect

! = ##′


Antireflecting barrier

x

U(x)I

II

U0

A

E

0 b

III

B

ikxikxI BeAex -+=)(y{)(xy

ikxikxIII eBeAx -+= '')(y

xikxikII DeCex '')( -+=y

DCBA +=+

( ) ( )DCkBAk -=- '

ikbbikbik eADeCe ''' =+ -

ikbbikbik ekADekCek ''' '' =- -

( )[ ]bkyya '2cos)4(481 222 -++=

kk

kky ''+º

nbky

T

nbkT

´==

´==

p

p

'242'21

2

b/ldB

Tran

smis

sion


Barrier of an arbitrary shape

U(x)I II

A

E

III

dx

dxdx e

UEEUTT a2

20

00

)(16 -÷÷ø

öççè

æ -»=

a b

( )202 2!

EUm -=a

þýü

îíì

--» òb

a

dxExUmTT )(22exp0 !

U0

U0 -eEx

xE

'x0

U0-eEx’=Ex

( ) ( )eEEUdxEeExU x

x

x

2/30

'

00 3

2 -=--ò

( )þýü

îíì --»

eEEUmTT x

2/30

0 324exp!

EEKeTII /0

0-==


Field emission microscope

rRA = Up to 106

~ 100 nm


To remember!

Ø Generalized Schrödinger equation is (postulated) equation of motion for particles in an arbitrary potential field.

Ø Schrödinger equation for stationary states is obtained if spatial and temporal coordinates are separated (energy conservation).

Ø Superposition of stationary-state solutionsdescribes a non-stationary process.

Ø Wave function is not a measurable quantity.

Ø Due to the wave-matter duality principle, thereis a non-zero probability to find a particle in regions forbidden in classical mechanics.

Ø One of such examples is the tunnel effect.

–schrödinger equation i€¦ · schrödinger’s equation of motion for particles moving in...

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