H* PHY 326 - 3rd homework - Optional complement to the second exercise - H. Panagopoulos, December 2014 *L H* We shall use the quantity u =Π 2 2 I2ma 2 M as our unit of energy. In this unit: E = z * u, where z is a dimensionless unknown. Thus, e.g., when z = 1 the energy assumes its unperturbed value. Denoting also: ¶ = V 0 u, the quantization condition takes the form: z-¶ tanIHΠ4L z-¶ M = z cotIHΠ4L z M For a given value of ¶ this condition can be solved numerically for z. Let us plot the left hand side HblueL and the right hand side HredL of the condition. The abscissa of the leftmost intersection point is the smallest value of z Hthat is, of the energyL which satisfies the quantization condition. *L H* Let us give a particular value to the perturbation V 0 : ¶ = 0.3 *L In[1]:= Plot@8Sqrt@z - 0.3D Tan@Pi 4 Sqrt@z - 0.3DD, Sqrt@zD Cot@Pi 4 Sqrt@zDD<, 8z, 0, 30<D Out[1]= 5 10 15 20 25 30 -20 -10 10 20 H* Let us compute the energy perturbatively IE H0L +E H1L M for various values of V 0 : ¶ = 0, 0.1, 0.2, ..., 0.9, 1.0 *L In[2]:= Table@8e, 1 + e H1 2 + 1 PiL<, 8e, 0, 1, 0.1<D Out[2]= 880., 1.<, 80.1, 1.08183<, 80.2, 1.16366<, 80.3, 1.24549<, 80.4, 1.32732<, 80.5, 1.40915<, 80.6, 1.49099<, 80.7, 1.57282<, 80.8, 1.65465<, 80.9, 1.73648<, 81., 1.81831<< H* Now let us compute the energy variationally for the same values of V 0 : ¶ = 0, 0.1, 0.2, ..., 0.9, 1.0 *L
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H PHY 326 3rd homework Optional complement to the second ... · H* PHY 326 - 3rd homework - Optional complement to the second exercise - H. Panagopoulos, December 2014 *L H* We shall
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H* PHY 326 - 3rd homework - Optional complement to the second exercise -
H. Panagopoulos, December 2014 *L
H* We shall use the quantity u = Π2Ñ2�I2ma2M as our unit of energy.
In this unit: E = z * u, where z is a dimensionless unknown.
Thus, e.g., when z = 1 the energy assumes its unperturbed value.
Denoting also: ¶ = V0�u, the quantization condition takes the form:
z-¶ tanIHΠ�4L z-¶ M = z cotIHΠ�4L z MFor a given value of ¶ this condition can be solved numerically for z.
Let us plot the left hand side HblueLand the right hand side HredL of the condition.
The abscissa of the leftmost intersection point is
the smallest value of z Hthat is, of the energyLwhich satisfies the quantization condition. *L
H* Let us give a particular value to the perturbation V0: ¶ = 0.3 *L