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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.

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Presentation on theme: "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."— Presentation transcript:

1 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 <V o for given n the energy of the state is somewhat lower than for infinite well wave function is more spread out

2 Comparison of the energy levels of an infinite and finite potential well atomic physics case (1-dimensional) Bound state energy: Test case: V 0 =300 eV 0.0003 MeV a=0.2 nm 200000 fm -E (eV) E+V 0 (eV) ξη= (R 2 -ξ 2 ) 1/2 η= ξ tanξ η= -ξ cotξ 9.4292.47.61.418.78 -0.23 37.6269.730.32.828.41-0.948.44 84.6232.267.84.227.817.84-2.27 150.4180.4119.65.606.88-4.546.91 235.0115.7184.36.965.515.54-8.73 41.4258.68.243.30-20.363.33 infinite potential well: m e =511 keV/c 2

3 Comparison of the energy levels of an infinite and finite potential well nuclear physics case (1-dimensional) Bound state energy: Test case: V 0 =54.7 MeV a=3.96 fm -E (MeV) E+V 0 (MeV) ξη= (R 2 -ξ 2 ) 1/2 η= ξ tanξ η= -ξ cotξ 13.1547.157.551.1902.9742.973-0.4763 52.6025.8028.902.3282.200-2.4622.202 118.350.1854.523.1980.18180.1805-56.67 infinite potential well: m n =931.5 MeV/c 2

4 Energy levels of an infinite square well potential nuclear physics case (3-dimensional) Schrödinger equation: Orbital nℓ XnℓXnℓ E nℓ *R 2 (MeV fm 2 ) E nℓ (MeV) 36 Ca R=3.96fm N nℓ = 2(2 ℓ +1) parity 1s3.142206.3313.162+ 1p4.493421.9026.906- 1d5.763694.1244.2610+ 2s6.283825.0452.612+ 1f6.9881020.5765.0814- J.M.Eisenberg, W.Greiner: Nuclear Theory 1, p.188 V(r) R r

5 Comparison of the energy levels of an infinite and finite potential well nuclear physics case: 36 Ca, 36 S (3-dimensional) ℓ=0 energies: Orbital nℓ E nℓ (MeV) 36 Ca R=3.96fm E nℓ (MeV) 36 Ca V 0 =54.7MeV E nℓ (MeV) 36 S V 0 =47.3MeV 1s13.169.759.55 1p26.9019.7719.31 1d44.2632.2031.32 2s52.6137.5536.25 1f65.08 ℓ=1 energies: ℓ=2 energies:

6 Depth of the potential square well deuteron case (3-dimensional) ℓ=0 energies:

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

8 Comparison of the energy levels of an infinite and finite potential well nuclear physics case: 36 Ca, 36 S, 34 Si, 32 Mg ℓ=0 energies: Orbital nℓ E nℓ (MeV) 36 Ca R=3.96fm E nℓ (MeV) 36 Ca V 0 =54.7MeV E nℓ (MeV) 36 S V 0 =47.3MeV E nℓ (MeV) 34 Si V 0 =45.2MeV E nℓ (MeV) 32 Mg V 0 =42.7MeV 1s13.169.759.55 1p26.9019.7719.31 1d44.2632.2031.32 2s52.6137.5536.25 1f65.08 ℓ=1 energies: ℓ=2 energies:

9 Wave function in a finite square well potential wave function of deuteron normalisation: ΙΙΙ

10 Mean square radius – a measure of the nuclear size outer region inner region

11 Deformed Nuclear Shell Model

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