chemstiy homework need help

CHEM 133 – Homework #6 (due 05/16/2014)

1. For the hydrogen atom

(a) What is the average value of the electron-nucleus separation in the 1s and 2s states of a hydrogen atom?

(b) What is the most probable electron-nucleus distance for 1s and 2s states?

2. By evaluating the appropriate integrals in the 2s and 2p states of the hydrogen atom

(a) Find 〈r〉 and 〈r2〉. (b) Find 〈x〉 and 〈x2〉. (c) Compute 〈1/r〉. (Use both direct computation and the virial the-

orem.) Show that 1/〈r〉 6= 〈1/r〉

Express your answer in terms of Bohr radius. Hint:

∫∞ 0 dx xn e−αx = n!/αn+1

3. At the time t = 0 the wave function for hydrogen atom is

ψ(~r, 0) = 1√ 10

( 2ψ100(~r) + ψ210(~r) +

√ 2ψ211(~r) +

√ 3ψ21−1(~r)

) where the subscripts are values of the quantum numbers n, `, m.

(a) What is the expectation value for the energy of this system?

(b) What is the probability of finding the system with ` = 1, m = +1 as a function of time?

(c) What is the probability of finding the electron within 10−10 cm of the proton at time t = 0? (a good approximate result is acceptable here.)

(d) How does this wave function evolve in time; i.e. what is ψ(~r, t)?

(e) Suppose a measurement is made on the state described by the wave function ψ(~r, 0) above which shows that ` = 1. Describe the wave functions immediately after such a measurement in terms of the ψn`m.

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4. Find the angle between the xy plane and the angular momentum vector ~L of an electron in the (4, f, 2) orbital.

5. What is the degeneracy of each of the hydrogen atomic levels? Hint:

n2∑ i=n1

i = n2 + n1

2 (n2 − n1 + 1)

6. What is the most probable point (not radius) at which 2p electron will be found in the hydrogen atom?

7. True or false? (explain)

(a) For a one-particle problem with V = br3, where b is a posi- tive constant, the stationary-state wave functions have the form ψ(~r) = f(r)Y m` (θ, φ).

(b) Every one-particle Hamiltonian operator commutes with ~̂L 2 and with L̂z.

(c) For a system of N non-interacting particles, each stationary-state wave function has the form ψ(~r1, ~r2, . . . , ~rN) = ψ1(~r1) + ψ2(~r2) + . . .+ ψN(~rN)

(d) The energy of a system of noninteracting particles is the sum of the energies of the individual particles.

(e) The reduced mass of a two-particle system is always less than m1 and less than m2.

(f) The value zero is never allowed as for an eigenvalue.

(g) In the equation 〈Â〉 = ∫ ψ∗(~r) Â ψ(~r) d~r, where d~r = r2 sin θ dr dθ dφ,

 operates on ψ(~r) only and does not operate on r2 sin θ.

(h) The electron probability at the nucleus is zero for all H-atom states.

(i) All the spherical harmonics Y m` are constant on the surface of a sphere centered at the origin.

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8. ψnlm(~r) are eigenfunctions of the time-independent Schrödinger equa- tion for a hydrohen atom

(a) What is the energy corresponding to ψ(~r) = c1ψ210(~r) + c2ψ211(~r), where |c1|2 + |c2|2 = 1?

(b) What is the energy corresponding to ψ(~r) = c1ψ210(~r) + c2ψ310(~r), where |c1|2 + |c2|2 = 1?

(c) What do these results tell you about the uniqueness of the the three p orbitals: px, py, and pz?

9. In 1976 it was mistakenly believed that the first of the superheavy elements had been discovered in a sample of mica. Its atomic number was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element?

10. Consider the earth-sun system as a gravitational analog to the hydrogen atom.

(a) What is the potential energy function? (Let m be the mass of the earth, and M the mass of the sun.)

(b) What is the “Bohr radius,” ag, for this system? Work out the actual number.

(c) By equating the gravitational En to the classical energy of a planet in a circular orbit of radius r0, show that n =

√ r0/ag. From this,

estimate the quantum number n of the earth.

(d) Suppose the earth made a transition to the next lower level (n−1). Ho much energy (in Joules) would be released? What would the wavelength of the emitted photon (or, more likely, graviton) be? (Express your answer in light years.)

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