130_notes.dvi

For the case of a constant potentialV 0 , we definek=

√

2 μ(E−V 0 )
̄h^2 andρ=kr, and the radial equation
becomes.

d^2 u(r)

dr^2

+

2 μ

̄h^2

[

E−V 0 −

ℓ(ℓ+ 1) ̄h^2

2 μr^2

]

u(r) = 0

d^2 u(r)

dr^2

+k^2 u(r)−

ℓ(ℓ+ 1)

r^2

u(r) = 0

d^2 u(ρ)

dρ^2

−

ℓ(ℓ+ 1)

ρ^2

u(ρ) +u(ρ) = 0

Forℓ= 0, its easy to see that sinρand cosρare solutions. Dividing byrto getR(ρ), we see that
these arej 0 andn 0. The solutions can be checked for otherℓ, with some work.

15.9 Sample Test Problems

1. A particle has orbital angular momentum quantum numberl= 1 and is bound in the potential
   wellV(r) =−V 0 forr < aandV(r) = 0 elsewhere. Write down the form of the solution (in
   terms of known functions) in the two regions. Your solution should satisfy constraints at the
   origin and at infinity. Be sure to include angular dependence. Now usethe boundary condition
   atr=ato get one equation, the solution of which will quantize the energies.Do not bother
   to solve the equation.


2. A particle of massmwith 0 total angular momentum is in a 3 dimensional potential well
   V(r) =−V 0 forr < a(otherwiseV(r) = 0).

a) Write down the form of the (l= 0) solution, to the time independent Schr ̈odinger equa-

tion, inside the well, which is well behaved at atr= 0. Specify the relationship between

the particles energy and any parameters in your solution.

b) Write down the form of the solution to the time independent Schr ̈odinger equation, outside

the well, which has the right behavior asr→ ∞. Again specify how the parameters

depend on energy.

c) Write down the boundary conditions that must be satisfied to match the two regions.

Useu(r) =rR(r) to simplify the calculation.

d) Find the transcendental equation which will determine the energy eigenvalues.

3. A particle has orbital angular momentum quantum numberl= 1 and is bound in the potential
   wellV(r) =−V 0 forr < aandV(r) = 0 elsewhere. Write down the form of the solution (in
   terms of known functions) in the two regions. Your solution should satisfy constraints at the
   origin and at infinity. Be sure to include angular dependence. Now usethe boundary condition
   atr=ato get one equation, the solution of which will quantize the energies.Do not bother
   to solve the equation.


4. A particle is confined to the inside of a sphere of radiusa. Find the energies of the two lowest
   energy states forℓ= 0. Write down (but do not solve) the equation for the energies forℓ= 1.

5.