A Particle in a Time-Independent Scalar Potential

Problem:
Assume that the potential energy of the deuteron is given by

.

(a)  Show that the ground state of the deuteron possesses zero orbital angular momentum (l=0).   Since this is true for any central potential, you may not need the detailed nature of the square well potential.

(b)  Assume that l=0 and estimate the value of V0 under the additional condition that the value of the binding energy is much smaller than V0.

Solution:

Concepts, principles, relations that apply to the problem:
Three-dimensional square potentials.

Why do they apply?
We have to investigate the properties of the ground state in a three dimensional square potential.

How do they apply?
(a)  The Hamiltonian of the relative motion of the two particles is   .  .

Let yl be the lowest energy eigenfunction of H with orbital angular momentum quantum  number l.

.

Let yl+1 be the lowest energy eigenfunction of H with orbital angular momentum quantum number l+1.

.

.

since yl is assumed to be the lowest energy eigenfunction of the Hamiltonian .

Therefore   and the ground state possesses zero orbital angular momentum.

(b)  The relative motion of the two particles is described in the same way as the motion of a fictitious particle of reduced mass m in a central potential. Therefore we have  .    .   The problem is reduced to a one dimensional "square well problem" with E<0.

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Details of the calculations:

Let   .

Let r<r0 define region 1 and r>r0 define region 2. The coordinate r is never negative. Therefore  .  .

The boundary conditions are   .

. 

, in regions where   .

f we plot we find solutions at the intersections of the two curves in regions where , i.e. .

For only one solution to exist we need  . If |E|<<|V0|   then  .

Then   .

.

Problem:

Find the condition that must be satisfied by the spherically symmetric square well potential if it is just barely deep enough to contain one bound state.

Solution:

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Problem:

In an atom, a valence electron experiences a long range Coulomb force and the potential well representing the interaction supports an infinite spectrum of bound states. In contrast, the interaction between the outermost electron in a negative ion and the neutral atomic core, which is weak and short ranged, results in only a finite number of bound states.

For the case of s states (l=0), the negative ion may be approximated by a model in which the interaction between the outermost electron of mass m and the core is represented by an attractive one-dimensional central potential of the form

.

(a)  Solve the time independent Schroedinger equation and determine an expression, in the form of a transcendental equation, relating the eigenvalues of this system to the quantities V0, a, and m. Solve this equation graphically.

(b)  Show, graphically or otherwise, that there will exist no bound states unless  

(c)  Determine how many bound states exist if R=p.

(d) T he condition for the existence of bound states depends on the product of the depth and the square of the width of the potential well. Explain this in terms of the uncertainty principle.

Solution:

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Problem:

Let us represent the interaction between two He atoms by the sum of a short-range repulsive part and a long-range attractive part.

V(r)=+ ¥   for r<a,
V(r)=-|V0|   for a<r<b,
V(r)=0   for r>b.

(a)  Find the condition satisfied by a, b, and V0 for a bound state to exist in the relative motion of two He atoms.
(b)  Experimental evidence indicates that He does not solidify at atmospheric pressure, even at ~0K. What do you think this implies about the effective helium-helium interaction?

Student solution: