3. Ground-state calculations for various potentials (a) Consider the spherically-symmetric potential -T, rTO

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3 Ground State Calculations For Various Potentials A Consider The Spherically Symmetric Potential T R Ro Ur T To 1 (51.14 KiB) Viewed 22 times

3. Ground-state calculations for various potentials (a) Consider the spherically-symmetric potential -T, r<ro, ur) T>TO in three dimensions, with I is a positive constant. Assume the existence of a separable wavefunction of the form (1,0,0) = Yem(0,4) (1) Hence, show that the ground-state energy is h? Lo = -1 + 2mrz where zo is the smallest root of the equation – cot z = 72 - z2, and where y= 2m1/12r. [10 marks) (b) Consider the following singular potential (u) = -18() in one dimension, where I is a positive constant and (2) is the Dirac delta function. Show that the potential (3b) admits precisely one bound state and compute the associated energy and wavefunction. Hint: The eigenvalue satisfies –ml/?<E<0. [10 marks]