Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have...

Question:

Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have not discussed multi-dimensional systems, but please keep in mind that in order to answer this question all you need is the knowledge about a particle moving in one dimension in a finite square well. Consider a particle of mass m moving in a three-dimensional spherically symmetric square-well potential of radius a and depth V. (see also figure on pag. 3): V(r) = { S-Vo o if 0 <r <a, if a <r <oo (0.3) (r being the distance from the origin of the coordinate system). The Schroedinger equation happens to have the form: 12 1 dl,d 2mo? dor ( v(n)) + V()(r) = Ee(r), (0.4) where E is assumed to be negative (we want to focus on bound states only).
(a) Find y(r). Hint: write (r) = x(r)/r and replace it in Eq. (0.4) to obtain a differential equation for x (the equation that you obtain for x should be similar to the typical differential equations we solved in lecture 5). You do not need to normalise the wavefunction. [3 marks] (b) Obtain the transcendental equation for the energy. Hint: the equation for the energy should look a lot like one we derived in Lecture 5 for the finite square well. Please show all the work. [5 marks] (c) What is the minimum depth of the potential needed to achieve two bound states? Hint: again, please refer to Lecture 5 for guidance on answering this question. [4 marks]

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