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260 7 QUANTUM THEORY
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Figure 7.11
E7F.14(b) �e angle in question is that between the z-axis and the vector representing the
angular momentum. �e projection of the vector onto the z-axis is m lħ, and
the length of the vector is ħ
√
l(l + 1). �erefore the angle θ that the vector
makes to the z-axis is given by cos θ = m l /
√
l(l + 1).
When l is very large, the number of projections onto the z axis, (2l +1), is also
very large implying that the angular momentum vector can take any direction.
In addition, for l >> 1, the vector representing the state with m l = l makes an
angle with the z-axis given by cos θ = l/
√
l(l + 1) ≈ l/l = 1.�us, in this limit,
θ = 0 and the vector may point along the z-axis. Both of these results �t in with
the correspondence principle.
Solutions to problems
P7F.2 (a) A function ψ is an eigenfunction of an operator Ω̂ if Ω̂ψ = ωψ where ω is
a constant called the eigenvalue.
(i) l̂z(eiϕ) = (ħ/i)d/dϕ(eiϕ) = (ħ/i) × ieiϕ = ħeiϕ . Hence eiϕ is an
eigenfunction of the operator (ħ/i)d/dϕ, eigenvalue ħ .
(ii) l̂z(e−2iϕ) = (ħ/i)d/dϕ(e−2iϕ) = (ħ/i) × −2ie−2iϕ = −2ħe−2iϕ . Hence
e−2iϕ is an eigenfunction of the operator (ħ/i)d/dϕ, eigenvalue −2ħ .