Download Hydrogen Mastery Answers

Hydrogen Atom “Mastery of Fundamentals” Questions
Brief Answers
CH351 – Prof. Wu
Here are some questions to test your mastery of the fundamentals of the quantum
mechanical description of the hydrogen atom. Once you’ve mastered the material, you
should be able to answer these questions without reference to your notes or textbook.
1. What are the kinetic, potential and Hamiltonian operators for the hydrogen atom?
!2 2
e2
ˆ
ˆ
ˆ
H =K +V =! " !
2µ
4 #$0 r
2. What is the reduced mass of the H-atom, and is it close to the electron mass?
µ=
m p me
mp
=
me = 0.9995 me
m p + me m p + me
3. What is the Schroedinger equation for the H-atom?
Hˆ ! = E!
!2 2
e2
" #!"
! = E!
2µ
4 $%0 r
4. What are the energy eigenstates and eigenvalues?
! nlm = Rnl (r)Yl m (" , # )
R%
me e 4
En = $ 2 = $
n
8& 0 2 h 2 n 2
5. What are the quantum numbers for the H-atom orbitals (stationary states)? What
values can these quantum numbers assume? Where do these quantum numbers
appear in the separable form of the stationary states?
The principal quantum number is n = 1,2,3…
The angular momentum quantum number is l = 0,1,…n-1
The azimuthal number is m = -l, …0,…l
They appear in the stationary states as shown above.
6. Which physical properties are associated with each quantum number?
n controls the energy, En = !
R"
me e 4
=
!
n2
8# 0 2 h 2 n 2
l controls the magnitude of the angular momentum L = L2 and L2 = l(l + 1)! 2
m controls the z-component of the angular momentum, Lz = m!
Hydrogen Atom “Mastery of Fundamentals” Questions
Brief Answers
CH351 – Prof. Wu
7. What is the degeneracy of a given H-atom energy level? For orbitals sharing the
same energy, how many share the same angular momentum?
The degeneracy of the nth energy level is n2. For orbitals with the same energy, 2l+1
have the same angular momentum l.
8. What is the Bohr radius, a0, and what is its approximate value in angstroms?
a0 =
4 !"0 ! 2
# 0.53Å
mee 2
9. What does the “x” mean in the Legendre functions Plm (x) , or in the Laguerre
polynomials L2ln +l+1 (x) ?
“x” is just a dummy variable used to define the functions, and don’t correspond to
physical variables.
10. What shape does a radial node have? What shape does an angular node have?
How are these shapes represented as equations in spherical coordinates?
Radial nodes are spheres, and are represented as r = constant. Angular nodes are
planes or cones, and are represented by ! = constant or ! = constant.
11. What linear combination of orbitals give rise to the 2px and 2py orbitals?
1
(! 211 + ! 21"1)
2
1
=
(! 211 " ! 21"1)
2i
!2 px =
!2 py
12. Which pairs of operators commute in the set Lˆ2 ,Lx ,Ly ,Lz ? How is this related to
which quantities can be simultaneously determined to arbitrary precision?
Lˆ2 commutes with each of Lx ,Ly ,Lz individually, but no other pair commutes. The
quantities whose operators commute can be simultaneously determined to arbitrary
precision.
In addition, you should feel comfortable doing problems like those that have been
assigned in homework. Here are some additional problems you should feel comfortable
doing once you’ve mastered the material.
Hydrogen Atom “Mastery of Fundamentals” Questions
Brief Answers
CH351 – Prof. Wu
1. Given an orbital (such as 2s, 3pz), write down the wavefunction (given the general
formula). From this wavefunction, solve for the equations for the nodes. Sketch
the orbital and its nodes.
First, we determine the quantum numbers n,l,m. We then plug these values into the
equation above for wavefunction ! nlm = Rnl (r)Ylm (",# ) . The node is given by when
the wavefunction is zero, which can occur either when the radial term is zero, or
when the angular term is zero. These equations can be solved to produce equations
for the nodes such as r = constant, ! = constant or ! = constant.
2. For a given wavefunction (such as 1s, 2px, etc.), what is the average distance <r>
of the electron from the nucleus? What is the most probable distance? What is
the probability that the electron will be found within a distance d? What is the
energy, the angular momentum, and the z-component of the angular momentum?
< r >=
!
"r
2
2
Rnl (r) rdr . The most probable distance is found at a maximum (setting
0
2
derivative with respect to r, equal to zero) in r 2 Rnl (r) . The probability of finding an
d
electron within some distance “d” is
!r
2
2
Rnl (r) dr . The energy, angular momentum,
0
and z-component of the angular momentum are given above.
3. Demonstrate that any two different H-atom orbitals are orthogonal.
For two different wavefunctions ! nlm (r) and ! n' l' m' (r) , we can show they are
*
(r)! n' l' m' (r)dr = 0 , where the integration is over 3orthogonal by showing " ! nlm
dimensional space (usually performed in spherical coordinates).