19.7 Homework
- A particle is in the stateψ=R(r)
(√
1
3 Y^21 +i√
1
3 Y^20 −√
1
3 Y^22)
. Find the expected values of
L^2 ,Lz,Lx, andLy.
2. A particle is in the stateψ=R(r)
(√
1
3 Y^11 +i√
2
3 Y^10)
. If a measurement of thexcomponent
of angular momentum is made, what are the possilbe outcomes and what are the probabilities
of each?
3. Calculate the matrix elements〈Yℓm 1 |Lx|Yℓm 2 〉and〈Yℓm 1 |L^2 x|Yℓm 2 〉
4. The Hamiltonian for a rotor with axial symmetry isH=
L^2 x+L^2 y
2 I 1 +
L^2 z
2 I 2 where theIare constant
moments of inertia. Determine and plot the eigenvalues ofHfor dumbbell-like case that
I 1 >> I 2.- Prove that〈L^2 x〉=〈L^2 y〉= 0 is only possible forℓ= 0.
- Write the spherical harmonics forℓ≤2 in terms of the Cartesian coordinatesx,y, andz.
- A particle in a spherically symmetric potential has the wave-functionψ(x,y,z) =C(xy+yz+
zx)e−αr
2. A measurement ofL^2 is made. What are the possible results and the probabilities
of each? If the measurement ofL^2 yields 6 ̄h^2 , what are the possible measured values ofLz
and what are the corresponding probabilities?
8. The deuteron, a bound state of a proton and neutron withℓ= 0, has a binding energy of -2.18
MeV. Assume that the potential is a spherical well with potential of−V 0 forr < 2 .8 Fermis
and zero potential outside. Find the approximate value ofV 0 using numerical techniques.