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• 8/17/2019 Townsend, Problem 10

1/22

Physics

412

Homework Set No.

t.

Townsend,

Problem

10.3,

page

303;

Townsend,

Problem

10.6,

page

303.

3

. Evaluate explicitly the energy eigenfunctions for the

|

ls)

,

I

Zsl

,lZ

pl,

|

3s)

bound

states

of

the

hydrogen atom.

Verify

that

these

eigenfunctions are

orthogonal to each other

(it's

enough

to

check

this result for

just

two of

these eigenfunctions). Don't

forget to

normalize

the eigenstates Note: You must

use

the

recursion relation

derived

in

class.

4. Evaluate the mean value of the

(measurements

of

the)

position

of the

electron for

the

ground

state

(n:l)

and first

excited

state(s)

(n:2)

of

the hydrogen

atom.

(We

can

,

interpret

the result

as

providing

a

measure

of

the

"mean distance"

between

the

electron

and the

proton,

for both

states.)

Note the dependence

of

your

result on the

state.

5. a)

Wite

down

the

energy eigenfuncti ons

Qn1.(;)

=

(;

I

nlml for the

bound

states

of

a

spinless

particle

in

the

presence

of

a

three-dimensional

"spherical

square-well

,"

i.e.,

a

central

potential

ofthe

form

,

-Vo,

r

Ea

v

(;):

0, rla,

where Vo > 0

(i.e.,

the

potential

is attractive

-it

tries to bind the

particle).

No

need

to

normalize

the

eigenfunctions. Obtain

the

transcendental equation from which

the

corresponding

energy

eigenvalues

would

be obtained

(numerically).

Note:

Remember

that

the boundary

condition

which

must be satisfied by

function

at

the

origin

is

equivalent

to thinking

that there

is

an

infinite

barrier

for

r

=0.

b)

Obtain

an

equation

relating the

parameters

of

the

well

(its

depth and

its

width)

such that the

ground

state

is barely

bound, i.e., its

binding energy

approaches zero.

(Hint:

Obtain first the

general

transcendental

equation from which

this eigenvalue

is

to

be determined,

and

then

take the appropriate

limit.) Recall

that a similar

problem

was solved in the case

of

a one-dimensional square

well

potential

9'

2.

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