Consider a particle of mass m inside a hollow spherical cavity of radio a. The potential...

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Consider a particle of mass m inside a hollow spherical cavity of radio a. The potential energy is zero inside the cavity but infinity otherwise. Answer the following questions, knowing that the radial wave function R(r) of bound states in spherically symmetric potentials obeys the following differential equation: ħ² d² u 2m dr² + [V+ ħ² l (l + 1). 2m 7-2 u= Eu where is the angular momentum quantum number, r is the radial coordinate and R(r)=u(r)/r. (a) Find the ground state energy of a particle confined in this cavity. [10 marks] (b) Obtain the eigenvalue and the corresponding eigenfunction of all bound states that carry no angular dependence, i.e., bound states whose wave functions depend on r only. [10 marks] (c) Without doing any calculations, can you tell how the eigenvalues of the angular- dependent states compare to those of the angular-independent states? Justify your answer. [5 marks] Consider a particle of mass m inside a hollow spherical cavity of radio a. The potential energy is zero inside the cavity but infinity otherwise. Answer the following questions, knowing that the radial wave function R(r) of bound states in spherically symmetric potentials obeys the following differential equation: ħ² d² u 2m dr² + [V+ ħ² l (l + 1). 2m 7-2 u= Eu where is the angular momentum quantum number, r is the radial coordinate and R(r)=u(r)/r. (a) Find the ground state energy of a particle confined in this cavity. [10 marks] (b) Obtain the eigenvalue and the corresponding eigenfunction of all bound states that carry no angular dependence, i.e., bound states whose wave functions depend on r only. [10 marks] (c) Without doing any calculations, can you tell how the eigenvalues of the angular- dependent states compare to those of the angular-independent states? Justify your answer. [5 marks]