Comparison of the energy levels of an infinite and finite potential well

Infinite well• number of bound states is infinite

Finite well• number of bound states is finite• energy of bound states must be <Vo

• for given n the energy of the state is somewhat lower than for infinite well

• wave function is more spread out

Comparison of the energy levels of an infinite and finite potential wellatomic physics case (1-dimensional)

Bound state energy:

][00512.0

2 102

0

fmVEVEm

k e

2

2

Eme

200512.0

2

200512.0

2 0

aE

a

aVE

ak

2

2

0522

41062.2 R

aV

)(cos.tan likesolutionsym

)(sin.cot likesolutionsymanti

Test case:

V0=300 eV 0.0003 MeVa=0.2 nm 200000 fm

-E (eV)

E+V0 (eV)

ξ η=

(R2-ξ2)1/2

η=

ξ tanξ

η=

-ξ cotξ

9.4 292.4 7.6 1.41 8.78 8.78 -0.23

37.6 269.7 30.3 2.82 8.41 -0.94 8.44

84.6 232.2 67.8 4.22 7.81 7.84 -2.27

150.4 180.4 119.6 5.60 6.88 -4.54 6.91

235.0 115.7 184.3 6.96 5.51 5.54 -8.73

41.4 258.6 8.24 3.30 -20.36 3.33

2

222

2 am

n

e

2

222

2 am

nE

en

infinite potential well:

me=511 keV/c2

Comparison of the energy levels of an infinite and finite potential wellnuclear physics case (1-dimensional)

Bound state energy:

][2187.0

2 102

0

fmVEVEm

k n

2

2

Emn

22187.0

2

22187.0

2 0

aE

a

aVE

ak

2

2

0222

41078.4 R

aV

)(cos.tan likesolutionsym

)(sin.cot likesolutionsymanti Test case:

V0=54.7 MeV a=3.96 fm

-E (MeV)

E+V0 (MeV)

ξ η=

(R2-ξ2)1/2

η=

ξ tanξ

η=

-ξ cotξ

13.15 47.15 7.55 1.190 2.974 2.973 -0.4763

52.60 25.80 28.90 2.328 2.200 -2.462 2.202

118.35 0.18 54.52 3.198 0.1818 0.1805 -56.67

2

222

2 am

n

n

2

222

2 am

nE

nn

infinite potential well:

mn=931.5 MeV/c2

Energy levels of an infinite square well potentialnuclear physics case (3-dimensional)

Schrödinger equation:

)()()(2

22

rErrV

),()()( mn Yrur

01

)(22

222

2

rur

rVEdr

du

rdr

ud

2

2

2

22

90.202 R

X

R

XE nnn

Orbital

nℓ

Xnℓ Enℓ *R2

(MeV fm2)

Enℓ (MeV)36Ca R=3.96fm

Nnℓ=

2(2ℓ+1)

parity

1s 3.142 206.33 13.16 2 +

1p 4.493 421.90 26.90 6 -

1d 5.763 694.12 44.26 10 +

2s 6.283 825.04 52.61 2 +

1f 6.988 1020.57 65.08 14 -

J.M.Eisenberg, W.Greiner: Nuclear Theory 1, p.188

V(r)

R r

2,,, 478.931

c

MeVm

M

mMmwith np

A

npAnp

Comparison of the energy levels of an infinite and finite potential wellnuclear physics case: 36Ca, 36S (3-dimensional)

ℓ=0 energies:

Orbital

nℓ

Enℓ (MeV)36Ca R=3.96fm

Enℓ (MeV)36Ca V0=54.7MeV

Enℓ (MeV)36S V0=47.3MeV

1s 13.16 9.75 9.55

1p 26.90 19.77 19.31

1d 44.26 32.20 31.32

2s 52.61 37.55 36.25

1f 65.08

RER

RVERk

2187.0

2187.0 0

ns

nsns EV

EEVR

002187.0cotcot

ℓ=1 energies:

RkRkRk

11cot

2

2

R

k

RkRkRk

1

11

3

cot1

12

2

22

ℓ=2 energies:

MeVA

ZNV

1.33510

][2.1 3/1 fmAR

Depth of the potential square well deuteron case (3-dimensional)

ℓ=0 energies:

RER

RVERk

2187.0

2187.0 0

cot

2

Rk

202

222

02 22RVRk

Vk

][103 220 fmMeVRV

][2.1][45 3/10 fmARwithMeVV

202

2 2

4RV

20

22

8RV

deuteronfor

mm

mmwith

pn

pn

nsns

ns EVforEV

ERk

0

0

0cot

Energy levels of finite square well potentialsfor ℓ=0 bound states of 4He, 16O, 40Ca and 208Pb (3-dimensional)

Wave function in a finite square well potentialwave function of deuteron

snrIIsn

snIsn

EwitheBru

EVkwithrkAru

,

,

,0

,

2

2sin)(

RkRkRkk

RA

2sincossin

2

MeVV 510

fmR 65.1

RkeAB R sin

12

0

,

drrun normalisation:

Ι ΙΙ

5.0

MeVE s 224.2,1

Mean square radius – a measure of the nuclear size

drerBdrkrrAdrdr

drdrrr

R

rR

22222

0

2

2*

22*

2 sin

RkRkRkk

RA

2sincossin

2

RkeAB R sin

MeVV 510

fmR 65.1

5.0

MeVE s 224.2,1

fmr 7.32

outer region

inner region