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