Further Quantum Theory, MT2014, Problem sheet 1
1. For the harmonic oscillator Hamiltonian H =
a± = P ± imωX. Show that
1
~ω
1
~ω
a+ a− +
=
a− a+ −
,
2m
2
2m
2
Show further that
H=
[a− , a+ ] = 2mω~ ,
P2
2m
+ 12 mω 2 X 2 , introduce the operators
and
||a± ψ|| = 2mEψ (H) ± m~ω .
[H, a± ] = ±~ωa± .
Suppose that there exists a state ψ0 so that a− ψ0 = 0. Show that ψn = (a+ )n ψ0 is
an energy eigenstate with En = (n + 21 )~ω. Comment briefly on the existence and
uniqueness of these energy eigenstates.
2. The spherical harmonics Ylm (θ, φ), when multiplied by rl are polynomials of degree
l in xi that satisfy the laplacian ∇2 (rl Ylm ) = 0. Set
x± = x1 ± ix2 = r sin θe±iφ ,
x3 = r cos θ ,
and define
1
∂± =
2
∂
∂
∓i
∂x1
∂x2
,
so that ∂± x± = 1 , and ∂± x∓ = 0.
Show that the components of L = x ∧ P = −i~x ∧ ∇ in the above basis become
L± = L1 ± iL2 = −~x±
∂
± 2~x3 ∂∓ ,
∂x3
L3 = ~(x+ ∂+ − x− ∂− ) .
Show that
||L− Ylm ||2 = (l(l + 1) − m2 + m)~2 ||Ylm ||2 .
Show that rl Yl±l is a constant multiple of xl± and use the raising and lowering operators to find the normalized Ylm for l = 0, 1, 2.
3. The Pauli spin matrices are defined by
0 −i
0 1
,
,
σ2 =
σ1 =
i 0
1 0
σ3 =
1 0
,
0 −1
and σ · a = σ1 a1 + σ2 a2 + σ3 a3 . Show that (σ · a)(σ · b) = a.b + iσ · (a ∧ b). Deduce
that 21 ~σ1 , 12 ~σ2 , 21 ~σ3 satisfy the angular momentum commutation relations, and
that
(σ.a)(σ · b) + (σ · b)(σ · a) = 2a · b.
Deduce also that the eigenvalues of σ · a are ±|a|.
4. Suppose that A and B are vector operators, in the sense that
X
X
[Ji , Aj ] = i~
ǫijk Ak ,
[Ji , Bj ] = i~
ǫijk Bk
k
k
Show that the wedge product A ∧ B is a vector operator in the same sense.
1

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