Homework Assignment 9, due 29 March 2006 ... - HMC Physics

Homework Assignment 9, due 29 March 2006Problem 1 – Townsend 9.12 The wave function for a particle is of the form ψ(r) =(x + y + z)f(r). What are the values that a measurement of L 2 can yield? What valuescan be obtained by measuring L z ? What are the probabilities of obtaining these results?Suggestion: Express the wave function in spherical coordinates and then in terms of theY l,m ’s.Problem 2 – Townsend 9.19Ĥ = ˆL 2 /2I is given byThe wave function of a rigid rotator with a Hamiltonian〈θ, φ|ψ(0)〉 =√3sin θ sin φ4π(a) What is 〈θ, φ|ψ(t)〉? Suggestion: Express the wave function in terms of the Y l,m ’s.(b) What values of L z will be obtained if a measurement is carried out and with whatprobability will these values occur?(c) What is 〈L x 〉 for this state?(d) If a measurement of L x is carried out, what result(s) will be obtained? With whatprobability?Problem 3 – Townsend 9.21 Treat the ammonia molecule, NH 3 , shown in Fig. 9.12 asa symmetric rigid rotator. Call the moment of inertia about the z axis I 3 and the momentsabout the pair of axes perpendicular to the z axis I 1 .(a) Express the Hamiltonian of this rigid rotor in terms of ˆL, I 1 , and I 3 .(b) Show that [Ĥ, ˆL z ] = 0.(c) What are the eigenstates and eigenvalues of the Hamiltonian?(d) Suppose that at time t = 0 the molecule is in the state|ψ〉 = 1 √2|0, 0〉 + 1 √2|1, 1〉What is |ψ(t)〉?Problem 4 – Townsend 10.1 The position-space representation of the radial componentof the momentum operator is given byˆp r → ( ∂i ∂r + 1 )rShow that for its expectation value to be real: 〈ψ| ˆp r |ψ〉 = 〈ψ| ˆp r |ψ〉 ∗ , the radial wavefunction must satisfy the condition u(0) = 0. Suggestion: Express the expectation value inposition space in spherical coordinates and integrate by parts.Homework Assignment 9 29 March 2006

29 March 2006 Homework Assignment 9Problem 5 – Townsend 10.7potential wellShow that there are no allowed energies E < −V 0 for theV ={−V 0 r < a0 r > aby explicitly solving the Schrödinger equation and attempting to satisfy all the appropriateboundary conditions.Peter N. Saeta 2 Physics 116