Homework 8 (due Monday, Nov. 14)

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Homework 8

(due Monday, Nov. 14)

This is the homework assignment on the radial Schrodinger equation. We are focusing on the bound states of the Hydrogen atom and also the 3D particle in a box.

1.

Hydrogen atom:

Solve for the radial wave functions of the hydrogen atom for possible l values for n = 2: l = 0 and l = 1.

n = 2. There are two

(a) The first step is to solve for the coefficients of the power series using the recursion relation c j +1

=

( j

2( j + l

+ 1)( j

+ 1

+ 2

− l n )

+ 2) c j

(1)

For n = 2 and l = 1 show that c

1

= 0. For n = 2 and l = 0 show that c

2

= 0.

(b) Using the book’s notation, the function u ( ρ ) is then u ( ρ ) =

ρ l +1 e

ρ X c j

ρ j

.

j

=0

(2)

Express n = 2.

u in terms of r instead of ρ , making sure to use the correct energy for

(c) Apply the normalization condition

Z

0

( u ( r ))

2 dr = 1 (3) to determine the coefficient c

0

.

(d) Finally compute the book.

R ( r ) = u ( r ) /r and compare your results to those in Table 4.7 in

2.

3D Particle in a box:

(Problem 4.9 in book)

In three dimensions a particle in a finite box has the potential V ( r ) = − V

0 for r < a and V ( r ) = 0 for r > a

Schrodinger equation for with

− V o

V

0

> 0. We are going to look for solutions to the radial

< E < 0 and for l = 0. By using l = 0 we will readily be able to solve the radial equation for u .

Hint: For this problem if you define k = s

2 m (( E

2

+ V

0

)

= s

2 m (( V

0

2

− | E | )

κ = s

2 m | E |

, h

2 then the relation k

2

+ κ

2

=

2 mV

0 h

2 will allow you to eliminate κ in part (d).

(4)

(5)

(6)

(a) Write down the equation for u ( r ) for r < a with l = 0 and solve it. (See Eq.

4.37 in the book.) It is a second order differential equation so there will be two independent solutions. Given that physical one?

R ( r ) = u ( r ) /r which of the solutions is the

(b) Write down the equation for u ( r ) for r > a with l = 0 and solve it. Again there will be two independent solutions, but only one of them will be physical

(normalizable). Indicate which solution is physical.

(c) As in one dimensional problems, match the solutions for that both u and its first derivative are continuous at to determine the bound state energies.

r u (

= r ) at r = a by assuming a . Derive an equation

(d) Show that there is no bound state if V

0 a

2

< π

2 h

2

/ 8 m , where a is the Bohr radius.

(e) For a general V

0 how many bound states are there?

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