Two-dimensional quantum well

In three dimensions, the Shrödinger Equation is:

),,(),,(),,(),,(),,(),,(

2 2

2

2

2

2

22

zyxEzyxzyxUz

zyx

y

zyx

x

zyx

m

The 3D potential well is an important system of many applications –for instance, in the modern approach to the theory of gases, sucha well is the “starting point” for all considerations.

But we will talk now only about the 2D potential well.

x

y

),(),(),(),(),(

2 2

2

2

22

yxEyxyxUy

yx

x

yx

m

Lo

L0

0

)cos()sin()(

)cos()sin()(

: trysLet'

)()(),(

:solution separable aseek We

2

:so ,0),( :Inside

0),( or 0 and

or 0for :meansit Outside,

2

2

2

22

0

0

ykDykCyg

xkBxkAxf

ygxfyx

Eyxm

yxU

yxLyy

Lxx

yy

xx

.2

ifsolution good"" a issolution our So,

)()( sideright The

);()(2

sideleft theSo,

)()()(

)(

:Similarly

).()(

)cos()sin()cos()sin(

)()( ;

)()(

222

222

22

2

2

2

2

22

2

2

2

2

yx

yx

y

x

xxxxyy

kkm

E

ygxfE

ygxfkkm

ygxfky

ygxf

y

ygxfk

xkBkxkAkykDykC

x

xfyg

xx

xfyg

x

)cos()sin()( );cos()sin()(

where),()(),(

ykDykCygxkBxkAxf

ygxfyx

yyxx

Along the walls marked red the wave function must be zero:

)sin()sin(),(

: tosimplifies So,

.0 and 0 : thatrequiresIt

:0or 0for 0)()(

ykxkCAyx

DB

yxygxf

yx

Along the lines marked blue the wave functionalso must be zero, which requires:

yyxx

yyy

xxx

nL

knL

k

nnLk

nnLk

00

0

0

and :So

,3,2,1

and ,,3,2,1

sinsin2

),( and ,2

:Then

well).1D for then integratio the(recall 422

sinsin),(1

:conditionion Normalizat

,3,2,1 ;,3,2,1 with sinsin),(

:bemust function wave theSo,

0000

20220022

0 0

2

0 0

2

0 0

222

00

000 0

yL

nx

L

n

Lyx

LAC

LCA

LLCA

dyyL

ndxx

L

nCAdxdyyx

nnyL

nx

L

nACyx

yx

Ly

L

x

L L

yxyx

Having found the wavefunction, we can write the probability density:

sinsin4

),(),(0

2

0

220

2

y

L

nx

L

n

LyxyxP yx

So that the probability of finding the particle within a dxdy surfaceelement in the well interior is:

dxdyyL

nx

L

n

LdxdyyxP yx sinsin

4),(

0

2

0

220

Now, energy! We have obtained before:

And:

222

2 yx kkm

E

; 00

yyxx nL

knL

k

)( :can write we,2

:gIntroducin

,3,2,1 ;,3,2,1 : where)(2

2202

0

22

0

2220

22

yx

yxyx

nnEEmL

E

nnnnmL

E

So, we obtain the following expressions for the energies of the allowed quantum states in the 2D well:

Let’s now examine the possible energy values:

Applying the commonly usedplotting scheme,we can display the possible Evalues using ho-rizontal “bars”:

For different pairs of the nx, ny quantum numbers,one can plot the probability density function profi-les in the LxL area:

sinsin4

),(0

2

0

220

y

L

nx

L

n

LyxP yx

Now, we want to introduce an extremely important notion inQuantum Mechanics – namely, the notion of DEGENERACYIts meaning is not exactly the same as of “degeneracy” in common life situations:

What does “degeneracy” mean in Quantum Mechanic? Well, if there are two or more different quantum states that happento have exactly the same energy, we call such states “degenerate”. In the 2D quantum well most states are!

.50 isenergy Its ?5 with about thisWhat

ate.nondegener are 4 and 3 ,2 ,1 with states The

them!of allnot But .for which thoseYes,

well?2D in the states degenerate-non thereAre

".degeneracy twofold" a hasit or that ,"degenerate

doubly " is levelenergy 13 hesay that t We

.13 ofenergy same eexactly th hasit but

state,different a is 2 ,3 with state The

13)32( isenergy Its

.3 ,2 with state the takeinstance,For

0

0

0

022

0

Enn

nn

nn

E

E

nn

EEE

nn

yx

yx

yx

yx

yx

:belowshown are 5 theof and state,

1 ,7 theof functionsdensity y probabilit The

.7

,1 that withand ,1 ,7 with one The

energy! same with thestates more twoare There

yx

yx

y

xyx

nn

nn

n

nnn

We say: “The 50E energy level has a threefold degeneracy”