Question

A particle of mass m is exposed to the spherically symmetric potential: V(r) = –6U8(r – b), parameters U > 0, b>0, , (3) which, perhaps, can be called an attractive S-function potential shell of radius b. In our studies of one-dimensional quantum mechanics, we found that the ground state for a particle exposed to a S-function attractive potential in 1D is always a bound state. You are now asked to explore this question for our particle in 3D exposed to the attractive S-function shell (3). 1 (a) Would the ground state be a bound state for arbitrarily small (but, of course, still positive) energy parameter U? (b) If not, find the minimal required value Um, such that that the ground state is bound at U > Um. (If the bound state is formed for arbitrarily small U, that would mean Um = 0.) (c) Furthermore, find (approximately) energy E of this bound ground state in the limit of very small values of (U – Um) above the minimal Um.

Answer 1

For arbitrarily small positive U, the ground state might not remain bound, and the minimal required value
`\(U_m\)` for a bound state would be E/6, where E is the energy of the ground state for
`\(U > U_m\)`. Obtaining the energy E for small
`\(U - U_m\)` would involve advanced analysis or numerical methods.

The ground state for a particle in a spherically symmetric potential can be determined by examining the behavior of the potential at small distances compared to the characteristic length scales involved.

(a) Bound State for Small Energy Parameter U:

For arbitrarily small positive values of U, the ground state might not be a bound state. As U decreases, the potential well becomes shallower, and the binding energy might become insufficient to keep the particle bound.

(b) Minimal Required Value
`\(U_m\)` for a Bound State:

To determine the minimal
`\(U_m\)` for which the ground state is bound, we analyze when the binding energy matches the depth of the potential well. When the binding energy is equal to the depth of the potential well
`(\(E = |V_0|\), where \(V_0 = -6U\))`, the particle remains bound.

Therefore, for the ground state to be bound at
`\(U > U_m\):`

`\[ E = |V_0| = 6U \]`

`\[ E = 6U_m \]`

The minimal required value
`\(U_m\) would be \(E/6\).`

(c) Energy E of the Bound Ground State for Small
`\(U - U_m\):`

To approximate the energy E of the bound ground state for very small values of
`\(U - U_m\)`, you can consider using perturbation theory or expansion methods to derive an approximate expression for the energy in terms of
`\(U - U_m\)`. However, this might require solving the Schrödinger equation in the presence of the S-function potential and performing calculations involving the small parameter
`\(U - U_m\).`

The energy E in the limit of very small values of
`\(U - U_m\)` would likely be a function depending on the difference
`\(U - U_m\)` and the parameters of the potential, possibly requiring numerical or advanced analytical techniques to obtain precise values.