Homework #9

1. For an electron in the ground state of a hydrogen atom potential, calculate the expectation value of:
a) 1/r
b) r
c) x^2
d) the vector r = (x,y,z)
e) p^2
[what is the relationship between the expectation value of r and of x^2 ?

2. a) (Using your results from problem 1) what are the expectation values of the K.E. and potential energy, respectively. Calculate your answer in e.V. and make sure the signs are correct.
How does the K.E. compare with the P.E.? (xc) How does this relate to the virial theorem?
b) Show that the P.E. can be expressed very simply using e^2 and the characteristic length scale, a_0 = hbar^2/m e^2.
c) Show that your result for the K.E. can be expressed very simply using hbar, m and the characteristic length scale, a_0 = hbar^2/m e^2.
d) why does this make sense?

3. For many potentials, there is no intrinsic length scale associated with V(r) and yet there can be quantum length scale which emerges when a particle of mass m is localized by the potential. What are the characteristic length scales for each of the following?:
a) V(x)= -alpha delta(x)
b) V(x) = (1/2) k x^2
c) V(x) = -e^2/r (3D)

4. Using the Psi_n,l,m as a basis, what is the represention of the state Psi = A (x/2a_0) exp[-r/2a_0], where A is a normalization constant independent of r, theta and phi or x, y and z). In other words, write the state Psi = A (x/2a_0) exp[-r/2a_0] as a linear combination of Psi_n,l,m eigenstates. How many states do you need? Is this a mixed state?

5. For the 3D H.O., the 1st excited state manifold contains 3 states: which could be called 100,
010, 001; where 100 includes a term proportional to x, 010 includes a term proportional to y, ...

Show, by construction, that the 1st-excited state manifold of H can be arranged to also include real eigenstate which are proportional to x, y and z (and orthogonal to each other). What else does the 1st excited state manifold of H include? Discuss the differences and similarities of the 1st excited state manifolds of H and a symmetric 3DH.O..

6. a) Use appropriate indentities and relationships to express Psi3,2,2 + Psi3,2,-2 as a relatively simple function of the cartesian variables x, y and z.

b) Show that a 45 degree rotation of this state (around the z axis) will give another state that: is a linear combination of 2 Psi n,l,m states, and is a relatively simple function of x,y and z. What is the linear combination? What would be good names for these states?

7. Use that state from problem 4, let's call it Psi_2x, to make the state (Psi 100 + Psi_2x)/sqrt(2)
a) calculate the expectation value of x for an electron in this state? Is this a mixed state? Is x a function of time?

8. With or without doing any calculations (calculation is extra-credit of you have time), describe the nature of the trajectory of the expectation value of the vector r:
a) in the state: (Psi 100 + Psi 211)/ sqrt(2)
b) in the state: (Psi 100 + Psi 210)/ sqrt(2)

c) Are these qualitatively similar or different? Discuss?

9. a) What is the degeneracy of the 2nd excited state (manifold) of a symmetric 2-D H.O. ?
(extra credit) b) What angular momentum eigenstates are contained within this N-dimensional subspace?

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(extra credit) a) Figure out how to make the state, B (x^2 - y^2) exp[-r/3a_0] from a combination of Psi_n,l,m and/or how to make C x y exp[-r/3a_0].
b) What is the difference (relationship) between these two states? Are they eigenstates or not? Are they degenerate?