Townsend, Problem 10

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    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

    the radial wave

    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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