# Momentum representation, momentum space wave function

We study the relationship between the position space wave function and the momentum space wave function in quantum mechanics. We show that they are related by the Fourier transform, more specifically we show that a momentum wave function given by the Fourier transform of the space wave function satisfies the requirements to be a momentum wave function, that is, normalization (the integral of its magnitude square equals one) and gives the correct formula for the expected value of the momentum.

See the attached file.

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#### Solution Preview

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The momentum in quantum mechanics is defined as the operator

p=-i*hbar*(d/dx)=(hbar/i)*(d/dx)

(this is in one variable, and p=-i*hbar*Gradient for more variables). The expected value of p, denoted by <p> is then

<p>=int_{-infinity}^{infinity} psi^*(x) (hbar/i) (d psi/dx)(x)dx,

and this is in space representation.

In the momentum representation, we look for g(p) that satisfies (15.60) and (15.61). As it is claimed there in the attachment, such g can be taken to be the Fourier transform of psi (with the hbar normalization in the definition of Fourier ...

#### Solution Summary

The solution discusses the momentum representation and the relationship between space wave function and momentum wave function.

Vibrational frequency, Earth mass, harmonic function, hydrogen atom degeneracy

See attached file for full problem description and clarity in symbols.

1. a) In the infrared spectrum of H79Br, there is an intense line at 2630 cm¡1. Calculated the force constant

of H79Br and the period of vibration of H79Br.

b) The force constant of 79Br79Br is 240 N ¢m¡1. Calculated the fundamental vibrational frequency

and zero-point energy of 79Br79Br

c) Calculate the average z component of angular momentum, hLzi, for an electron in a ring of constant

potential when it has the wavefunction:

i) y(f ) = r1

p sin3f

and

ii) y(f ) = r 1

2p e¡3if :

2. The Earth (mass »=6£1024 kg) rotates about the sun (at an average distance of r »=1:5£1011 m) once

per year (one year = 10p million seconds).

a) What is the Earth's angular momentum with respect to the sun in units of ¯h?

b) What is the smallest principle quantum number n that allows the electron in a hydrogen atom to

have this value of angular momentum?

c) How large is such an atom? Use hˆri for the H atom in the 1s state to express this size.

3. The ve l = 2 spherical harmonic functions are:

Y2;0(q ;f ) =

1

4r5

p ¡2cos2q ¡sin2q ¢

Y2;§1(q ;f ) = ¨

1

2r15

2p cosq sinq e§if

Y2;§2(q ;f ) =

1

4r15

2p sin2q e§2if

a) Prove that these functions are eigenfunctions of ˆL2 and ˆLz with appropriate eigenvalues.

b) Write Y2;0(q ;f ) in terms of xr

, yr

, and z

r .

c) Use superposition to relate Y2;1 and Y2;¡1 to real functions of xr

, yr

, and z

r . Hint: Remember that

e§if = cosf §i sinf

Chem 120A, Spring 2006 1

d) Show that the smplitude,

y(q ;f ) =

1

p2

[Y1;0(q ;f )+Y2;0(q ;f )]

is an eigenfunction of ˆLz but not ˆL2. Y1;0(q ;f ) =q 3

4p cosq .

4. In class you were shown that not only are the 2l +1 m states of each l degenerate for the hydrogen

atom, but there is an additional degeneracy for each n from the fact that 0 · l · n¡1.

a) Given that under parity

ˆPYlm(

q

;

f

)

=

(

¡

1

)

¡

lYlm(

q

;

f

)

;

what is the rst n for which a superposition of two different l states of even parity can be constructed?

What are the values of l?

b) The same question as in a), but for the case of odd parity.

5. This is a subtle question. When (in physical space, a.k.a. position representation) an eigenfunction of

some Hamiltonian, ˆH , is ys(~r) with a corresponding eigenvalue Es (i.e. ˆHys(~r) = Esys(~r)), then we

know that

Es = Z y¤s (~r) ˆHys(~r)d3r;

where here we assume that ys(~r) is normalized. Now, the amazing thing about the hydrogen atom

(or any tow-body coulomb interaction) is that the 2s and 2p states are degenerate. Suppose instead of

having a coulomb interaction that the relative interaction in atomic units is actually

V0(r) = ¡e¡gr

r ;

where g << 1.

a) Show that the new relative Hamiltonian, ˆH 0, may be written as,

ˆH

0 = ˆH +µ¡e¡gr

r +

1

r ¶;

where ˆH is the hydrogen atom Hamiltonian.

b) Assume that g is so small that the eigenfunctions of the hydrogen atom Hamiltonian ˆH are also

eigenfunctions of ˆH0. Show that no matter how small g is, the energies of the 2s and 2pz are no longer

degenerate. Are the three 2p states still degenerate?

Extra credit: Suppose instead of a regular Coulomb interaction that the potential was given by ¡1

r1+e ,

with e << 1. Show that the 2s and 2pz states are not degenerate using the ideas advanced in the

previous part of this problem.

6. Starting with the relationship [ ˆ x; ˆ p] = i¯h and the denition~L =~r£~p, show that:

£ˆLx;ˆLy¤ = i¯hˆLz

and

£ˆLz;ˆL2¤ = 0

The following commutator relations may be useful: [A;BC]=B[A;C]+[A;B]C and [AB;C]=A[B;C]+

[A;C]B.

7. Consider a U atom stripped of all but one electron: U91+.

a) What is the ion's ionization energy in both eV and in atomic units?

b) What is hˆri for the 1s state of this ion in both Angstroms and atomic units?

c) For some principle quantum number n¤, U91+ will have hˆrin¤s ¸ hˆri1s for H. What is the smallest

value that n¤ can be?

d) What does the plot of hˆri1s look like for all the one electron species from H to U91+?

8. Consider two possible spin states:

¯¯

Y1® = ¯¯

0®

¯¯

Y2® = r1

4¯¯

0® +ir3

4¯¯

1®

a) What is the expectation value of Sz for each of these states? (Expectation value = hSzi=Y¯¯ Sz¯¯

Y®)

b) What is hSxi for each of these states?

c) What is hSyi for each of these states?