PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 9
Nov. 9, 2004
Due: Nov. 17, 2004
Reading: Griffiths Chapter 4
1. Ladder Operators. A simultaneous eigenfunction of the anuglar momentum
operators L2 and Lz can be labelled as ψ` m . In the standard sign convention,
the raising angular momentum ladder operator satisfies the sign convention
L+ ψ` m = h̄(`(` + 1) − m(m + 1))1/2 ψ` m+1 .
(1)
The square root factor on the right hand side preserves the normalization.
(a) Use the expression for L− L+ in notes and the known results of operating
on ψ` m with L2 and Lz to derive the square root in equation (1).
(b) Use L− operatoing on (1) to find the normalizing constant c in the analogous equation
L− ψ` m = c ψ` m−1 .
2. Angular momentum states of 2 particles. Read Griffiths 4.4.3 before
doing this problem. Consider a two-particle system, with angular momentum
observables L(a) and L(b). The complete set of observables for this system is
L(a)2 , L(b)2 , Lz (a), and Lz (b),
(2)
φ(`a , ma , `b , mb ),
(3)
L2 , Lz , L(a)2 , and L(b)2 ,
(4)
ψ(`, m, `a , `b ).
(5)
with eigenfunctions
or
with eigenfunctions
In this problem, L2 is the square of total angular momentum operator associated
with the two particles, L = L(a) + L(b), and Lz is its z component.
(a) By using the equations Lz = Lz (a) + Lz (b), and L± = L± (a) + L± (b), show
that the state with `a = ma = 2 and `b = mb = 1 is an eigenfunction of
the set of observables in Eq. (4) with ` = m = 3, i.e. show that
φ(2, 2, 1, 1) = ψ(3, 3, 2, 1)
1
(6)
(b) Using what you learned about the ladder operators in the previous problem, find an expression for ψ(3, 2, 2, 1) as a linear combination of the
φ(2, ma , 1, mb ).
(c) Find an expression for the state ψ(2, 2, 2, 1) as a linear combination of the
φ(2, ma , 1, mb ). One approach is to look for that linear combination of the
φ functions with the correct total m value that is orthogonal to the wave
function from part (b). The other is to observe that the desired linear
combination must be annihilated by L+ .
(d) Figure out how to use the Clebsch-Gordon table on p. 188 of Griffiths to
check that your results (a-c) are correct (Hint: you need to use the table
labelled 2×1). Then use the table to express ψ(1/2, 1/2, 1, 1/2) as a linear
combination of φ(1, ma , 1/2, mb ). It’s easier than calculating as in (a-c)
once you figure it out! (Hint: Griffiths sometimes uses a notation where
he suppresses `a , `b , i.e. he doesn’t write them out explicitly since they
occur in both φ and ψ above).
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