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Second Physics Conference May, 2007 A confined N – dimensional harmonic oscillator. Sami M. Al – Jaber Department of physics An- Najah National University, Nablus , Palestine. Abstract. - PowerPoint PPT Presentation
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Second Physics ConferenceMay, 2007
A confined N – dimensional harmonic oscillator
Sami M. Al – JaberDepartment of physics
An- Najah National University,
Nablus , Palestine
Abstract
We compute the energy eigenvalues for the N- dimensional harmonic oscillator confined in an impenetrable spherical cavity. The results show their dependence on the size of the cavity and the space dimension N. The obtained results are compared with those for the free N- dimensional harmonic oscillator, and as a result, the notion of fractional dimensions is pointed out. Finally, we examine the correlation between eigenenergies for confined oscillators in different dimensions.
2- The N- dimensional harmonic oscillator
The radial part solution, , satisfies
)4(,21
211
222
22
22
rERrRrmmrN
drd
rN
drd
m
Connection between a confined harmonic oscillator and a free one
The value of λ is
We consider the case a = -1and
)19(24 Na
.0
Table 1: a = -1 andNZ0SR0s/r0
2111.58110.6324
31.51.22471.73210.7071421.41421.87080.759952.51.581120.7906631.73212.12130.816573.51.87082.23610.83668422.34520.852894.52.12132.44950.8660
1052.23612.54950.8771100507.07117.17640.9853
100050022.360722.23940.9985
.0
Table 2: The case a = -1 and NZ0Sr0S/r0
221.41421.87080.7559
32.51.581120.7906431.73212.12130.816553.51.87082.23610.83666422.34520.852874.52.12132.44950.8660852.23612.54950.877195.52.34522.64580.88641062.44952.73860.8944
100517.14147.24570.9856100050122.383022.41650.9985
1
For a = -2, 1F1(a,b,z) has two roots. and For the ground state we have
and b =N /2, and thus The two roots z1 and z2 yield two radii for thecavity , S1 and S2, which correspond to the states = ( 4 ,0) and ( 4 , 2) for thefree harmonic oscillator
111 bb 112 bb
,0 8N
.2
4
NE
,n
Table 3: a = -2 and
NZ1S1=Z2S2
20.58580.76533.41421.8477
30.91880.95854.08112.020041.26791.12604.73212.175351.62921.27645.37082.3175621.414262.449572.37871.54236.62132.573282.76401.66257.23612.690093.15481.77627.84522.8010
103.55051.88438.44952.906810043.85866.622658.14147.62511000478.617021.8773523.383022.8776
r0
2.1213
2.2361
2.34522.44952.54952.64572.73862.82842.91557.3144
22.4388
0
General Case
= even integer implies even or odd.
The n = even case yields mod 4, a = negative integer, and this
energy corresponds to the states for the free harmonic oscillator with whose number is n /2.
,cN .22
NcE
c nc
24N
.2
NnE
,n ,2,.....2,0 n
•n= odd case yields mod 4, a = negative half – integer, and the energy corresponds to the states
for the free harmonic oscillator with whose number is (n+1)/2. n = odd integer : using n =half odd integer. This energy could be written in two
ways:The first: which is the same as that of the states of the free harmonic
oscillator in (N-1) dimensions. The values of are given by
2N
2/NnE ,n
............,3,1 n
,2c
n
2NnE
2
12
12 NnE
,2
12n
The second: which is the same as that of the states
of the free harmonic oscillator in (N+1) dimensions. The values of are given by
4mod30....,..........,
252,
232
4mod1.1..,..........,2
32,2
12
cfornn
cfornn
2
12
12 NnE
,2
12n
4mod3.1,............,
252,
212
4mod10.,..........,2
52,
212
cfornn
cfornn
defined
C(n+ , )in (N-1)dim(n- , )in (N+1)dim
1(1,1)(0,0)3(2,0)(1,1)5(3,1), (3,3)(2,0), (2,2)7(4,0), (4,2)(3,1), (3,3)9(5,1), (5,3), (5,5)(4,0), (4,2), (4,4)11(6,0), (6,2), (6,4)(5,1), (5,3), (5,5)13(7,1), (7,3), (7,5), (7,7)(6,0), (6,2), (6,4), (6,6)15(8,0), (8,2), (8,4), (8,6)(7,1), (7,3), (7,5), (7,7)
,2
12
nn
For computational purposes, we choose c=1 and
and it is just the energy of the state ( 0 ,0) or ( 1 , 1) for the free harmonic oscillator in ( N+1) or ( N-1) dimensions
respectively,
1N .25.04/ Na
,22
1
NE
Table 5. c = 1NZ0Sr0
22.30131.51700.707133.13611.77091.2247
43.91291.97811.414254.65362.15721.581165.36952.31721.732176.06672.46311.870886.74942.5980297.42032.72402.1213108.08142.84282.23612014.37063.79093.16235031.90145.64815
10059.73947.72917.0711
c=half odd integer: For example if c= , then the energy of the confined oscillator becomes
This energy could be written as which is just the ground – state energy of the state
(n' , ) (0 ,0 ) for the free harmonic oscillator in the fractional dimension
if we let with n being odd, then the energy of the confined harmonic oscillator becomes
This energy could be written as which is exactly the energy of the state (n' , )= for the free harmonic oscillator in the fractional
dimension
21
.24
1
NE
22
10
NE
2
1N
2nc
.24
NnE
22
1
41 Nn
E
,4
1n
21N
4- Energy levels and eigenfunctions for a confined harmonic oscillator
Letting the value of, for a given at which 1F1 has its zero at
We numerically compute E10, E21, and E32 for two cavities.
na thn thn22SZ
24 Nann
.2
22
NaE nn
)22(./,2
,422
1 2211
2/,
22
SrZNNFerAr nSrz
nimn
Table 6: computed energy levels
NE10E21E32E10E21E32
21.12226.825817.57741.00084.16228.546231.76488.279519.69741.50284.75849.418042.47189.582821.90632.00705.383610.330053.246911.093424.20282.51566.040411.278264.092612.689426.58623.03126.728412.261075.010114.370029.05503.55147.448413.278686.000016.133631.60924.08868.200014.330497.062917.979234.24764.63688.983215.4156
108.198719.906336.96965.20089.797616.5342
ASforEn 21
ASforEn 102
Table 7:Results for E10 & E20 for S=4A0 .
N E10 E20
21.0000033.000640.000150.02133
31.5000153.501700.001000.04857
42.000054.003790.002500.0948552.500144.508320.005760.1848863.000375.016050.012200.3210073.500845.528700.024000.5218084.001786.048000.044500.8000094.503496.576000.077601.16923
105.006447.114000.128801.628572010.2800013.420002.800011.00005033.6680042.2900034.67256.6300
%100%1002
220
2
10 xEEE
EEE
N
N
N
N
•It is observed that E10 corresponds to the state (n , ) = (0 , 0) for the free N –dimensional harmonic oscillator.E20 corresponds to the state ( 2, 0) with energy
One also observes that the effect of the boundary is relatively larger on the energy level E20 compared to its effect on E10.
Mapping of energy eigenvalues For example, the energies for in dimension
N =5 are identical to the three- dimensional solutions for the case
0
,1
•For N = odd and angular momentum the energies correspond to solutions for the roots of 1F1
These correspond exactly to those either for dimension (N-2) and angular momentum
or they correspond to the three dimensional case with
For N = even and angular momentum the energies correspond to those in dimension (N-2) with angular momentum or they correspond to those for two- dimensional case (N=2) with
,
22,
2,
4221
rNN
,1
2/3 N,
,1
.2/2' N
Conclusion
The energy eigenfunctions were computed numerically using mathematica.
We pointed out the connection between solutions for the confined oscillator and the free one.
The effect of the boundaries and the dimension N were discussed.
The notion of fractional dimensions was explored. Mapping between energy eigenvalues in different
dimensions was examined.