Calculating spherical harmonics without derivatives
The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural...
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irk-123456789-1570842019-06-20T01:30:17Z Calculating spherical harmonics without derivatives Weitzman, M. Freericks, J.K. The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates. Спосiб отримання сферичних гармонiк приводиться однаково майже в кожному пiдручнику чи на кожному заняттi з квантової механiки. Як виявляється, його трудно вслiдкувати, важко зрозумiти й складно вiдтворити бiльшостi студентiв. В цiй роботi нами показано як обчислити сферичнi гармонiки природнiшим способом з допомогою операторiв та дивовижної тотожностi, вiдомої як тотожнiсть експоненцiйного розплутування операторiв (знаної в квантовiй оптицi, але мало застосовної десь iнакше). Цей новий пiдхiд виникає природнiм чином пiсля введення дiракових позначень, встановлення алгебри оператора кутового моменту та визначення дiї операторiв збiльшення та зменшення кутового моменту на спiльний базис власних функцiй. 2018 Article Calculating spherical harmonics without derivatives / M. Weitzman, J.K. Freericks // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33002: 1–12. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 01.40.gb, 02.20.Qs, 03.65.Fd, 31.15.-p DOI:10.5488/CMP.21.33002 arXiv:1805.12125 http://dspace.nbuv.gov.ua/handle/123456789/157084 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom.
It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In
this work, we show how one can determine spherical harmonics in a more natural way based on operators
and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but
little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed
the angular momentum algebra, and determined the action of the angular momentum raising and lowering
operators on the simultaneous angular momentum eigenstates. |
| format |
Article |
| author |
Weitzman, M. Freericks, J.K. |
| spellingShingle |
Weitzman, M. Freericks, J.K. Calculating spherical harmonics without derivatives Condensed Matter Physics |
| author_facet |
Weitzman, M. Freericks, J.K. |
| author_sort |
Weitzman, M. |
| title |
Calculating spherical harmonics without derivatives |
| title_short |
Calculating spherical harmonics without derivatives |
| title_full |
Calculating spherical harmonics without derivatives |
| title_fullStr |
Calculating spherical harmonics without derivatives |
| title_full_unstemmed |
Calculating spherical harmonics without derivatives |
| title_sort |
calculating spherical harmonics without derivatives |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157084 |
| citation_txt |
Calculating spherical harmonics without derivatives / M. Weitzman, J.K. Freericks // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33002: 1–12. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT weitzmanm calculatingsphericalharmonicswithoutderivatives AT freericksjk calculatingsphericalharmonicswithoutderivatives |
| first_indexed |
2025-07-14T09:24:54Z |
| last_indexed |
2025-07-14T09:24:54Z |
| _version_ |
1837613810585698304 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 3, 33002: 1–12
DOI: 10.5488/CMP.21.33002
http://www.icmp.lviv.ua/journal
Calculating spherical harmonics without derivatives
M. Weitzman1, J.K. Freericks2
1 3515 E. Rochelle Ave., Las Vegas, NV 89121, USA
2 Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057, USA
Received May 30, 2018
The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom.
It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In
this work, we show how one can determine spherical harmonics in a more natural way based on operators
and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but
little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed
the angular momentum algebra, and determined the action of the angular momentum raising and lowering
operators on the simultaneous angular momentum eigenstates (under L̂2 and L̂z ).
Key words: angular momentum, spherical harmonics, operator methods
PACS: 01.40.gb, 02.20.Qs, 03.65.Fd, 31.15.-p
1. Introduction
One of us (JKF), first met Prof. Stasyuk in the early 2000s when visiting Ukraine for a conference
and then subsequently became involved in a long-standing collaboration with Andrij Shvaika, who is
also at the Institute for Condensed Matter Physics in Lviv. It is always a pleasure to meet Prof. Stasyuk
on subsequent trips to Lviv as his friendly smile and enthusiasm for physics is contagious. I know
much of Prof. Stasyuk’s career has been spent working with operator formalisms (especially Hubbard
X-operators) and finding different ways to perform calculations or simplify complex expressions. It is
within this spirit that we present this work, which provides a clear operator-based methodology for
determining spherical harmonics. In spite of some of the lengthy algebraic manipulations, it is the type
of derivation that can fit well into an undergraduate quantum mechanics course. For sure, it can be used
in the graduate curriculum.
The motivation for this work is a book project of one of the authors (JKF) on teaching quantum
mechanics without calculus [1]. This project is a follow-up of a massively open on-line course called
Quantum Mechanics for Everyone, which is running on edX from April 2017–April 2019 [2]. While
for many, the idea of finding spherical harmonics without derivatives might seem sacrilegious, operator
methods allow one to proceed quite far into the quantum curriculum without requiring derivatives or
integrals. Indeed, most of our research work focuses on using these operator methods, so why not
teach them to students from the beginning and provide a more uniform perspective on how to perform
calculations in quantum mechanics? Even quantum experts will find something new in the way we
compute the spherical harmonics.
Our procedure follows a number of steps. First, we introduce the Dirac notation used to define the
spherical harmonics and we analyze the required rotation operators needed to complete the calculation.
Second, we derive the exponential disentangling operator identity, show its relationship to Lie algebras
and Lie groups, and verify later that it works in a different representation than the one it was originally
derived for. Third, we use the angular momentum algebra of raising and lowering operators to derive two
equivalent (but different) formulas for the spherical harmonics. We show how one form can immediately
be put into the standard format. We then analyze the second in more detail to verify that it also yields the
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33002-1
https://doi.org/10.5488/CMP.21.33002
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
M. Weitzman, J.K. Freericks
same result. We end with a discussion of how this approach can be extended to determine the rotation
matrices and provide some commentary about pedagogy.
2. Definition of the spherical harmonics
Wework under the assumption that Dirac’s bra-ket notation has already been introduced. Since this is
simply a notation that emphasizes states and operators and how they inter-relate, it can be introduced early
into the curriculum.We also assume that the notion of a wavefunction as the projection of a quantum state
onto a coordinate-basis eigenfunction is also known. We are working with orbital angular momentum, so
the total angular momentum quantum number l is a nonnegative integer l ∈ N. The angular momentum
operators L̂x , L̂y , and L̂z satisfy the commutation relation
[L̂i, L̂j] = i~εi jk L̂k , (2.1)
with εi jk the completely antisymmetric tensor (Levi-Civita symbol). We also define raising and lowering
operators via L̂+ = L̂x + iL̂y and L̂− = L̂x − iL̂y . These operators then satisfy the following commutation
relations:
[L̂z, L̂±] = ±~L̂± , [L̂+, L̂−] = 2~L̂z . (2.2)
We then start with the angular momentum eigenstates |l,m〉 given by the eigenvalue-eigenvector
relations
L̂2 |l,m〉 = (L̂2
x + L̂2
y + L̂2
z )|l,m〉 = ~
2l(l + 1)|l,m〉 (2.3)
and
L̂z |l,m〉 = ~m|l,m〉. (2.4)
Next, we need to determine the angular coordinate-basis state |θ, φ〉, where θ is the angle of the spherical
coordinate with respect to the z-axis and φ is the azimuthal angle rotating around the z-axis. Using the
angular momentum operators to represent the rotations about the y and z-axes, we can express this state
as
|θ, φ〉 = e−iφ L̂z
~ e−iθ L̂y
~ |θ = 0, φ = 0〉, (2.5)
which expresses the general state as the rotation of the state that starts oriented toward the north pole
along the z-axis — we rotate along the y-axis by the angle θ and along the z-axis by the angle φ to reach
the |θ, φ〉 state. We compute the wavefunction (or the spherical harmonic), by simply taking the hermitian
conjugate of the coordinate ket state (to make it a bra) and overlapping it with the angular-momentum
eigenstate. This gives
Ym
l (θ, φ) = 〈θ, φ|l,m〉 = 〈θ = 0, φ = 0|eiθ L̂y
~ eiφ L̂z
~ |l,m〉, (2.6)
where we use the standard notation Ym
l
(θ, φ) for the spherical harmonic wavefunction. Since the state
|l,m〉 is an eigenstate of L̂z , the first operator can be immediately evaluated as exp(imφ). We introduce
a complete set of states in between the coordinate bra on the left and the remaining rotation operator.
This complete set of states includes only states with the same l eigenvalue, because the L̂2 operator
commutes with the rotation operator (implying that the overlap with any angular-momentum eigenstate
from a different multiplet vanishes). This then yields
Ym
l (θ, φ) = eimφ
l∑
m′=−l
〈θ = 0, φ = 0|l,m′〉〈l,m′ |eiθ L̂y
~ |l,m〉, (2.7)
which experts will recognize as involving Wigner’s rotation matrix in the angular momentum multiplet
corresponding to total angular momentum l.
We next note that the “north pole” state |θ = 0, φ = 0〉 must have zero overlap with any |l,m〉 state that
has m , 0. This is because the matrix element 〈θ = 0, φ = 0|l,m〉 is equal to 〈θ = 0, φ = 0|eiφ′ L̂z/~ |l,m〉,
since the north-pole state is unchanged by any rotation about the z-axis. This implies that we must have
33002-2
Calculating spherical harmonics
m = 0, because that is the only eigenstate in the angular momentum multiplet that also is unchanged by
a rotation along the z-axis.
We pause here for a moment to give an important comment. It is often claimed that orbital angular
momentum must involve an integer eigenvalue for the total angular momentum because this is the only
way to have the wavefunction be continuous as we wind the angle φ by 2π. But such a condition is not
required, since it is only the probability density that must be single-valued, not the wavefunction [3, 4].
Nevertheless, the above argument, that we can only find a wavefunction by expanding in terms of the
m′ = 0 row of the rotation matrix, naturally implies that l must be a nonnegative integer. Otherwise, there
is no m′ = 0 component!
This observation reduces our computation of the spherical harmonics to the final identity
Ym
l (θ, φ) = eimφ 〈θ = 0, φ = 0|l,m′ = 0〉〈l,m′ = 0|eiθ L̂y
~ |l,m〉. (2.8)
Note that the factor 〈θ = 0, φ = 0|l,m′ = 0〉 is just an overall normalization constant. The difficulty
lies in calculating the matrix element of the rotation operator. One can see the challenge by expressing
L̂y = (L̂+ − L̂−)/2i and recognizing that one cannot simply express this as a product of exponentials
of the raising and lowering operators multiplied by a correction term because the correction term from
the Baker-Campbell-Hausdorff expression will never terminate. Instead, we will employ the exponential
disentangling operator identity to determine a simple operator form that can be evaluated. As you will
see below, this powerful identity should become part of the standard quantum mechanics curriculum.
3. Exponential disentangling operator identity
In a seminal paper byArecchi, Courtens, Gilmore, and Thomas from 1972 [5], many different relation-
ships between harmonic oscillator coherent states and angular momentum coherent states are discussed.
In particular, the exponential disentangling operator identity is fully derived in their appendix A.
Before we get into the details of the derivation, we want to first briefly describe what a Lie algebra and
a Lie group are. We will not give the most general mathematical definition, but rather describe a working
definition that is appropriate for quantum mechanics. A Lie algebra consists of a set of operators that are
closed under commutation relations. An example is the angular momentum operators. One key property
is that the Lie algebra satisfies the Jacobi identity, which we know every commutator satisfies. We form
the Lie group from exponentiating scalars multiplied by the Lie algebra operators in arbitrary linear
combinations. The product of any two Lie group elements can then be expressed as the exponential of
a linear combination of the Lie algebra operators due to the Baker-Campbell-Hausdorf theorem and the
fact that the Lie algebra closes under commutation. Hence, the Lie group is a mathematical group that has
multiplication, a unique inverse, etc. For physics, we often focus on finite-dimensional representations
of Lie groups and Lie algebras (Ado’s theorem [6] guarantees that such representations exist with finite-
dimensional matrices). In this case, a faithful representation is a representation that is injective (or
one-to-one) so that every element of the Lie group is mapped to a different element of the representation.
Faithful representations also have a trivial null space. The theory of representations then tells us that any
identity written in terms of exponentials of the Lie algebra operators that one can derive in a faithful
representation will hold both at the general operator level and for all other faithful representations (see for
example [7]). This follows, essentially, from the definition of a faithful representation, due to the one-to-
one nature of the mapping. We use this important fact in the derivation of the exponential disentangling
operator identity.
We employ the simplest representation of angular momentum, namely the results for l = 1/2 with
2 × 2 matrices. The representation of the angular momentum matrices is
L̂x =
~
2
(
0 1
1 0
)
=
~
2
σx , L̂y =
~
2
(
0 −i
i 0
)
=
~
2
σy , L̂z =
~
2
(
1 0
0 −1
)
=
~
2
σz , (3.1)
where we continue to use L̂ to represent the angular momentum, even though this case is for spin one-half.
33002-3
M. Weitzman, J.K. Freericks
We also find that
L̂2 = ~2
(
3
4 0
0 3
4
)
, L̂+ =
~
2
(
0 2
0 0
)
=
~
2
σ+ , L̂− =
~
2
(
0 0
2 0
)
=
~
2
σ− . (3.2)
The Pauli matrices σi satisfy σ2
i = I, so we can immediately derive the fundamental exponential identity
exp (iα®n · ®σ) = cosα I + i sinα ®n · ®σ, (3.3)
where ®n is a unit vector and ®n · ®σ =
∑3
i=1 niσi . The derivation proceeds simply by expanding the
exponential in a power series and using the fact that the Pauli matrices square to the identity. Note that
this identity works in two ways: first, it shows us how to go from the exponential form of a Lie group
element to its 2×2matrix form and second, it shows us how to convert any 2×2matrix into an exponential
Lie group form.
We are now ready to derive the exponential disentangling operator identity. We have for spin one-half
eiθ L̂y
~ = ei θ2 σy = cos
(
θ
2
)
I + i sin
(
θ
2
)
σy =
(
cos
(
θ
2
)
sin
(
θ
2
)
− sin
(
θ
2
)
cos
(
θ
2
) )
. (3.4)
We wish to factorize this in the following form
eiθ L̂y
~ =
(
1 0
a 1
) (
b 0
0 c
) (
1 d
0 1
)
=
(
b bd
ab abd + c
)
, (3.5)
which immediately tells us that a = − tan(θ/2), b = cos(θ/2), c = sec(θ/2), and d = tan(θ/2). This
allows us to re-express the factorization in terms of exponentials of matrices via
eiθ L̂y
~ = e−
1
2 tan( θ2 )σ−eln[cos( θ2 )]σz e
1
2 tan( θ2 )σ+ (3.6)
and we do not need to worry about the absolute value of the logarithm because 0 6 θ 6 π. This latter
form follows because σ± are nilpotent matrices, which satisfy σ2
± = 0 or
eασ+ =
(
1 2α
0 1
)
and eασ− =
(
1 0
2α 1
)
. (3.7)
We now re-express this identity in terms of the Lie algebra, and claim it holds for all representations.
Namely, we have
eiθ L̂y
~ = e− tan( θ2 )
L̂−
~ eln[cos2( θ2 )]
L̂z
~ etan( θ2 )
L̂+
~ (3.8)
as the first form for the exponential disentangling operator identity. Note that the above strategy works
with all Lie algebras/groups and can be employed to determinemore complicated identities than the above
one. It seems that much of the confusion about these identities arises from the fact that they were derived
for a particular representation and then the claim is that they hold for all representations. While this must
be true simply from the definition of a faithful representation [7], the result can appear “magical”.
The second identity simply reverses the order of the raising and lowering operator factors. Hence, we
factorize
eiθ L̂y
~ =
(
1 a′
0 1
) (
b′ 0
0 c′
) (
1 0
d ′ 1
)
=
(
b′ + a′c′d ′ a′c′
c′d ′ c′
)
, (3.9)
to find a′ = tan(θ/2), b′ = sec(θ/2), c′ = cos(θ/2), and d ′ = − tan(θ/2). This then yields our second
exponential disentangling operator identity:
eiθ L̂y
~ = etan( θ2 )
L̂+
~ e− ln[cos2( θ2 )]
L̂z
~ e− tan( θ2 )
L̂−
~ . (3.10)
Although these two identities are simple to derive, they end up being quite powerful to use in a number
of different applications. They should be better known outside the realm of quantum optics. Perhaps
this will occur if others use them to teach about spherical harmonics. Pressing forward, we use them to
complete our spherical harmonics derivation next.
33002-4
Calculating spherical harmonics
4. Derivation of the spherical harmonics from the angular momentum
algebra
Wenowhave all of the elements needed to determine the spherical harmonics. Because the exponential
disentangling operator identity has two forms, we have two ways we can proceed. We begin by working
with the identity that has the raising operator to the right and the lowering to the left. Then, we find
Ym
l (θ, φ) = eimφ 〈θ = 0, φ = 0|l,m′ = 0〉〈l,m′ = 0|eiθ L̂y
~ |l,m〉 (4.1)
becomes
Ym
l (θ, φ) = eimφ 〈θ = 0, φ = 0|l,m′ = 0〉〈l,m′ = 0|e− tan( θ2 )
L̂−
~ eln[cos2( θ2 )]
L̂z
~ etan( θ2 )
L̂+
~ |l,m〉. (4.2)
We can immediately analyze the strategy for completing the calculation, which is illustrated schemat-
ically in figure 1. Because the highest-weight state cannot be raised — i.e., L̂+ |l,m = l〉 = 0 — the
exponential of the raising operator acting on the state on the right will involve exactly l −m+ 1 terms in a
polynomial in L̂+. A similar result is true for the exponential of the L̂− term. Furthermore, the Lie algebra
for the angular momentum operators immediately tells us what each term is from repeatedly using the
raising and lowering operator matrix elements
L̂+ |l,m〉 = ~
√
(l − m)(l + m + 1)|l,m + 1〉 and L̂− |l,m〉 = ~
√
(l + m)(l − m + 1)|l,m − 1〉, (4.3)
which we assume have already been derived prior to working out the spherical harmonics wavefunctions.
Hence, we simply employ equation (4.3) multiple times to determine each term in the series. For example,
Figure 1. Schematic figure indicating the raising and lowering operators for the two forms of the expo-
nential disentangling operator identity in equations (3.8) and (3.10) (left and right, respectively). Since
we must end at m = 0, the first form requires us to raise states (arrows moving up on the right) that start
with m < 0 to at least m = 0 (if raised higher, they must be lowered down to m = 0 indicated by the
arrows going down on the left). Of course, there is no constraint for states that start with m > 0. Similarly
for the second form, we must lower an m > 0 state to at least m = 0 (arrows moving down on the left)
with no constraint for the states that start with m 6 0. We only illustrate the ladder operators action
with the arrows — we also need to attach the L̂z contributions along the way. The exponentials produce
polynomials in the raising and lowering operators that involve all possible number of steps up and down
(or vice versa) that start from the initial m value and end at m = 0.
33002-5
M. Weitzman, J.K. Freericks
we immediately find
etan( θ2 )
L̂+
~ |l,m〉 =
l−m∑
n=0
(
tan θ
2
)n
n!
[
n∏
r=1
√
(l − m − r + 1)(l + m + r)
]
|l,m + n〉. (4.4)
Operating the middle term next yields
eln[cos2( θ2 )]
L̂z
~ etan( θ2 )
L̂+
~ |l,m〉 =
l−m∑
n=0
(
tan θ
2
)n
n!
[
n∏
r=1
√
(l − m − r + 1)(l + m + r)
] (
cos
θ
2
)2m+2n
|l,m + n〉.
(4.5)
The last step is to operate the final factor with the lowering operator on this state. Because we take the
overlap with 〈l,m′ = 0|, we only need to include the term that applies the L̂− operator m + n times. We
also need to have m + n > 0, otherwise, we cannot lower to the state with m′ = 0. This then gives us
〈l,m′ = 0|eiθ L̂y
~ |l,m〉 =
l−m∑
n=max(0,−m)
(
− tan θ
2
)m+n
(m + n)!
(
tan θ
2
)n
n!
[
n∏
r=1
√
(l − m − r + 1)(l + m + r)
]
×
(
cos
θ
2
)2m+2n
[
m+n∏
s=1
√
(l + m + n − s + 1)(l − m − n + s)
]
, (4.6)
where we used the fact that 〈l,m = 0|l,m = 0〉 = 1. This final result can be simplified. Start with
the trigonometric factors. Using the half-angle identities sin θ = 2 sin(θ/2) cos(θ/2) and 1 − cos θ =
2 sin2(θ/2) then gives(
− tan
θ
2
)m+n (
tan
θ
2
)n (
cos
θ
2
)2m+2n
=
(
−
1
2
)m+n
(sin θ)m (1 − cos θ)n . (4.7)
The two product terms become√
(l − m)!(l + m + n)!(l + m + n)!l!
(l − m − n)!(l + m)!l!(l − m − n)!
=
(l + m + n)!
(l − m − n)!
√
(l − m)!
(l + m)!
. (4.8)
Using these two results, then yields
〈l,m′ = 0|eiθ L̂y
~ |l,m〉 =
l−m∑
n=max(0,−m)
(
−
1
2
)m+n
(sin θ)m (1 − cos θ)n
(l + m + n)!
(m + n)!n!(l − m − n)!
√
(l − m)!
(l + m)!
.
(4.9)
This is multiplied by exp(imφ)〈θ = 0, φ = 0|l,m′ = 0〉 to finally yield the spherical harmonic.
However, we do need to put this result into a standard form, which we can directly do for m > 0. In
this case, we employ the definition of the associated Legendre function via [8]
Pm
l (cos θ) =
1
2mm!
(l + m)!
(l − m)!
(sin θ)m
l−m∑
n=0
(−1)n
(
l − m
n
) (
l + m + n
n
)
(
m + n
n
) (
1 − cos θ
2
)n
, (4.10)
where the combinatorial symbol satisfies (
m
n
)
=
m!
(m − n)!n!
. (4.11)
After evaluating the combinatorial factors, one can immediately see that
Pm
l (cos θ) =
1
2m
(sin θ)m
l−m∑
n=0
(−1)n
(l + m + n)!
(l − m − n)!(m + n)!n!
(
1 − cos θ
2
)n
(4.12)
33002-6
Calculating spherical harmonics
and hence
Ym
l (θ, φ) = 〈θ = 0, φ = 0|l,m′ = 0〉(−1)m
√
(l − m)!
(l + m)!
Pm
l (cos θ)eimφ, (4.13)
for m > 0. If we proceed in the standard fashion, the case m < 0 follows from the general relation
Y−|m |
l
(θ, φ) = (−1) |m |[Y |m |
l
(θ, φ)]∗. The term 〈θ = 0, φ = 0|l,m′ = 0〉 is a normalization factor, which
can be shown to be equal to
√
(2l + 1)/4π by normalizing the wavefunction.
If we instead determine Ym
l
directly from the formula derived above for m < 0 in equation (4.9), we
see that it appears to be a new formula, as it does not seem to have any simple relationship with the
conventional formula. In particular, we examine the case l = 1 and m = −1. Here, we have
Y−1
1 (θ, φ) =
√
3
4π
e−iφ
[
1 − cos θ
sin θ
√
2 −
1
2
(1 − cos θ)2
sin θ
√
2
]
=
√
3
8π
e−iφ 1 − cos2 θ
sin θ
=
√
3
8π
e−iφ sin θ,
(4.14)
which is the correct result, but requires a nontrivial cancellation of sin θ in the numerator and denominator.
This brings to bear the question of whether the result for m < 0 is a new result, or the one that reduces to
the known result after using the special properties of the associated Legendre functions. We discuss this
in detail below and show that it is not a new identity.
But first, we derive a similar result for the other form of the exponential disentangling operator
identity. Here, the raising operator appears on the left and the lowering operator on the right. Otherwise,
the procedure is precisely the same. We simply report the final result analogous to equation (4.9):
〈l,m′ = 0|eiθ L̂y
~ |l,m〉 =
l+m∑
n=max(0,m)
(−1)m
(
−
1
2
)n−m
(sin θ)−m (1 − cos θ)n
×
(l − m + n)!
(n − m)!n!(l + m − n)!
√
(l + m)!
(l − m)!
. (4.15)
So the result for the second form of the exponential disentangling operator identity is the same as the
first form with m → −m and an additional phase factor of (−1)m. Note how, in this case, the results
for m 6 0 are the conventional results, while those for m > 0 appear to be new formulas. Note further,
that the symmetry of m → −m and the extra factor of (−1)m, are exactly what is needed to verify the
aforementioned identity, that Ym
l
(θ, φ) = (−1)mY−m∗
l
(θ, φ), which is usually derived by taking a complex
conjugate of the differential equation. Of course, the final evaluation of the results derived from the two
different forms of the exponential disentangling operator identity must be equal to each other for the
corresponding m values.
We now return to equation (4.9) for the case m < 0. That this is equivalent to the conventional
spherical harmonics formula is essentially shown in appendix D of [5] where their equations (D5) and
(D6) are shown to be equivalent. But the form that those equations take, being the full rotation matrices,
is not the most direct way to show that the new forms of the identities actually reduce to the conventional
ones. We know that this equivalence is highly nontrivial, given the example we worked out for l = 1
above. Hence, we do not recommend going through the details of the following derivation with students,
as it is likely to be too much algebra for them to be able to follow all of the steps. But for completeness,
we do include all of those details here.
It turns out that the strategy we use is to start with the first form of the exponential disentangling
operator identity for m < 0, apply the methodology described in [5] to arrive at an intermediate form,
similar to their equation (D6), and then apply the methodology in reverse, to show that the intermediate
result also agrees with the first form of the identity with m > 0 [up to an additional factor of (−1)m],
which then shows us the equivalence of the negative m and positive m results. We assume m < 0,
so m = −|m|. We begin with equation (4.9) and rewrite the trigonometric functions in terms of their
half-angle identities, to yield
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l+ |m |∑
n= |m |
(−1)−|m |+n
(
sin
θ
2
)2n−|m | (
cos
θ
2
)−|m |
33002-7
M. Weitzman, J.K. Freericks
×
(l − |m| + n)!
(−|m| + n)!n!(l + |m| − n)!
√
(l + |m|)!
(l − |m|)!
. (4.16)
We then shift n→ n + |m| to find
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l∑
n=0
(−1)n
(
sin
θ
2
)2n+ |m | (
cos
θ
2
)−|m |
(l + n)!
n!(n + |m|)!(l − n)!
√
(l + |m|)!
(l − |m|)!
.
(4.17)
The next step uses the van der Monde identity, given by
min(r,m)∑
k=0
(
m
k
) (
n
r − k
)
=
(
n + m
r
)
, (4.18)
where m, n, r , and k are all nonnegative integers, and we will assume that n > r . The first form we use
the identity in, after some rearranging of terms, is
(l + n)!
n!(l − |m|)!
=
min(n,l−|m |)∑
k=0
l!(n + |m|)!
k!(l − |m| − k)!(n − k)!(|m| + k)!
. (4.19)
Substituting this in, then gives
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l∑
n=0
min(n,l−|m |)∑
k=0
(−1)n
(
sin
θ
2
)2n+ |m | (
cos
θ
2
)−|m |
×
l!
√
(l + |m|)!(l − |m|)!
k!(l − |m| − k)!(n − k)!(|m| + k)!(l − n)!
. (4.20)
Switching the order of the k and n sums yields
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
k=0
l∑
n=k
(−1)n
(
sin
θ
2
)2n+ |m | (
cos
θ
2
)−|m |
×
l!
√
(l + |m|)!(l − |m|)!
k!(l − |m| − k)!(n − k)!(|m| + k)!(l − n)!
. (4.21)
Now, we shift n→ n + k to find
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
k=0
(−1)k
(
sin
θ
2
)2k+ |m | (
cos
θ
2
)−|m | l!
√
(l + |m|)!(l − |m|)!
k!(l − |m| − k)!(|m| + k)!
×
l−k∑
n=0
(−1)n
(
sin
θ
2
)2n 1
n!(l − k − n)!
. (4.22)
The sum over n can now be done, yielding a [1 − sin2(θ/2)] = cos2(θ/2) raised to the l − k power and a
factorial term, as follows
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
k=0
(−1)k
(
sin
θ
2
)2k+ |m | (
cos
θ
2
)2l−2k−|m | l!
√
(l + |m|)!(l − |m|)!
k!(l − |m| − k)!(|m| + k)!(l − k)!
.
(4.23)
This is the middle point of our derivation and corresponds to the formula (D6) from [5]. The rest of
the derivation proceeds in a similar fashion backwards, reversing all of the steps, but making changes
along the way to produce the m > 0 result.
33002-8
Calculating spherical harmonics
Okay, here we go. We start by factoring out a term [cos2(θ/2)]l−|m |−k = [1 − sin2(θ/2)]l−|m |−k and
then expanding the term in the binomial expansion to give
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
k=0
(−1)k
(
sin
θ
2
)2k+ |m | (
cos
θ
2
) |m | l!
√
(l + |m|)!(l − |m|)!
k!(l − |m| − k)!(|m| + k)!(l − k)!
×
l−|m |−k∑
n=0
(−1)n
(
sin2 θ
2
)n
(l − |m| − k)!
n!(l − |m| − k − n)!
. (4.24)
Now, we shift n→ n − k:
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
k=0
(
sin
θ
2
)2k+ |m | (
cos
θ
2
) |m | l!
√
(l + |m|)!(l − |m|)!
k!(|m| + k)!(l − k)!
×
l−|m |∑
n=k
(−1)n
(
sin2 θ
2
)n−k 1
(n − k)!(l − |m| − n)!
. (4.25)
Now, we switch the order of the k and n sums to give
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
n=0
n∑
k=0
(−1)n
(
sin
θ
2
)2n+ |m | (
cos
θ
2
) |m |
×
l!
√
(l + |m|)!(l − |m|)!
k!(|m| + k)!(l − k)!(n − k)!(l − |m| − n)!
. (4.26)
We next shift k → k − |m|:
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
n=0
n+ |m |∑
k= |m |
(−1)n
(
sin
θ
2
)2n+ |m | (
cos
θ
2
) |m |
×
l!
√
(l + |m|)!(l − |m|)!
(k − |m|)!k!(l − k + |m|)!(n − k + |m|)!(l − |m| − n)!
. (4.27)
As before, we now introduce the van der Monde identity in the form
(l + n + |m|)!
n!(l + |m|)!
=
n+ |m |∑
k= |m |
l!(n + |m|)!
(k − |m|)!k!(l − k + |m|)!(n − k + |m|)!
(4.28)
and find
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 =
l−|m |∑
n=0
(−1)n
(
sin
θ
2
)2n+ |m | (
cos
θ
2
) |m |
(l + n + |m|)!
(n + |m|)!n!(l − |m| − n)!
√
(l − |m|)!
(l + |m|)!
.
(4.29)
Finally, using the half-angle formulas, again gives our final result
〈l,m′ = 0|eiθ L̂y
~ |l,−|m|〉 = (−1) |m |
l−|m |∑
n=0
(
−
1
2
) |m |+n
(sin θ) |m | (1 − cos θ)n
×
(l + n + |m|)!
(n + |m|)!n!(l − |m| − n)!
√
(l − |m|)!
(l + |m|)!
. (4.30)
Hence, we have shown that the result for m = −|m| is the same as the result for m = |m| with an additional
factor of (−1) |m | . This is exactly what is needed to show Ym
l
(θ, φ) = (−1)mY−m∗
l
(θ, φ).
This derivation was tortuously long, but we could not find any shortcut to do it faster. It is possible
that some of the identities for the Jacobi polynomials could be employed to show this directly, but we
have not been able to sort that out here.
33002-9
M. Weitzman, J.K. Freericks
5. Pedagogical aspects
Historically, there are five ways that spherical harmonics can be derived. The first way, and the oldest,
is from the works of Laplace [9] and Legendre [10] who determined the spherical harmonics by solving
the differential equation. The only change, when one goes to quantum mechanics, is that we now also
have the normalization condition. The second way is via harmonic polynomials, a method first developed
by Kramers [11] and most recently popularized by Weinberg [12]. The third, is to use the highest weight
state |l, l〉 and apply the lowering operator to it to generate all of the different states [13–15]. One works
in the coordinate representation and uses differential operators for L̂−. The fourth is to determine the
rotation matrices as Wigner originally did [16–18] and to take the m′ = 0,m matrix element, which
also leads to the spherical harmonics. The final approach is to work with simple harmonic oscillator
wavefunctions and the angular momentum representation discovered by Schwinger [19]. This approach
also allows one to determine the spherical harmonics from the rotation matrices [20] in the same fashion
as above.
Our approach, while closely related to methods four and five, is decidedly different. We focus on
the meaning of the wavefunction as an overlap of the two eigenstates and develop the required rotation
operators directly. The exponential disentangling operator identity is then naturally employed to get the
final result. It does require a bit of operator algebra, but nothing that is too complicated. We feel it is a
useful method six for deriving spherical harmonics.
Depending on the ordering of material in a course, this approach can fit in such a way that it could be
the primary methodology for deriving spherical harmonics. In courses that begin with a discussion of the
Stern-Gerlach experiment and spin, one would naturally discuss Pauli matrices and angular momentum
early in the course. Once the angular momentum algebra and eigenstates are determined, then this
approach fits well as a method to finish the spherical harmonics discussion. The advantage is that it
focuses directly on reinforcing operator methods and that it also helps use the algebra in a meaningful
way to derive a nontrivial result.
We end this section with a discussion of some typical reinforcing exercises one can assign when
teaching this material. First, one can develop the angular momentummatrices for l = 1 and show how the
identity L̂3
α = L̂α (α = x, y, or z) allows one to also compute the exponential of linear combinations of
the angular momentum operators. Then, one can check the exponential disentangling operator identities
for the faithful l = 1 representation, which helps galvanize the general group theory result. This exercise
works in the following way. We begin with the standard 3× 3 representations for the angular momentum:
L̂x =
~
√
2
©«
0 1 0
1 0 1
0 1 0
ª®¬ , L̂y =
~
√
2
©«
0 −i 0
i 0 −i
0 i 0
ª®¬ , and L̂z = ~
©«
1 0 0
0 0 0
0 0 −1
ª®¬ . (5.1)
Then, using the fact that L̂3
i = ~
2 L̂i , allows us to derive that
eiθ L̂y
~ = I + i sin(θ)
L̂y
~
+ (cos θ − 1)
(
L̂y
~
)2
=
©«
1+cos θ
2
sin θ√
2
1−cos θ
2
− sin θ√
2
cos θ sin θ√
2
1−cos θ
2 − sin θ√
2
1+cos θ
2
ª®®¬ . (5.2)
Similarly, we have
L̂+ = ~
©«
0
√
2 0
0 0
√
2
0 0 0
ª®¬ and L̂− = ~
©«
0 0 0
√
2 0 0
0
√
2 0
ª®¬ (5.3)
so that
e− tan( θ2 )
L̂−
~ =
©«
1 0 0
−
√
2 tan
(
θ
2
)
1 0
tan2 (
θ
2
)
−
√
2 tan
(
θ
2
)
1
ª®¬ , (5.4)
33002-10
Calculating spherical harmonics
eln[cos2( θ2 )]
L̂z
~ =
©«
cos2 (
θ
2
)
0 0
0 1 0
0 0 sec2 (
θ
2
) ª®¬ , (5.5)
and
etan( θ2 )
L̂+
~ =
©«
1
√
2 tan
(
θ
2
)
tan2 (
θ
2
)
0 1
√
2 tan
(
θ
2
)
0 0 1
ª®¬ . (5.6)
Putting this together in the first form of the exponential disentangling operator identity, then gives
e− tan( θ2 )
L̂−
~ eln[cos2( θ2 )]
L̂z
~ etan( θ2 )
L̂+
~ =
©«
1+cos θ
2
1√
2
sin θ 1−cos θ
2
− 1√
2
sin θ cos θ 1√
2
sin θ
1−cos θ
2 − 1√
2
sin θ 1+cos θ
2
ª®®¬ , (5.7)
after using the half-angle formulas to convert all trigonometric functions to functions of θ. This verifies
the exponential disentangling operator identity for the l = 1 multiplet. One can similarly derive the
second form of the identity directly as well.
Second, one can ask students to explicitly extract the low l spherical harmonics from the two different
forms for the identity. This approach works well for l = 1 and l = 2. Since this expansion is easy to
perform, we do not provide further details here. Finally, one could discuss how parity arises by taking
θ → π − θ and φ → φ + π; determining how Ym
l
(θ, φ) changes under the parity transformation; and
concluding that the parity operator acting on |l,m〉 returns (−1)l times the state. Once again, details are
straightforward and are omitted.
6. Summary
In this work, we show how one can derive spherical harmonics without using any derivatives and argue
that this approach brings in a number of important pedagogical advantages to the conventional treatment of
orbital angular momentum (in particular, by introducing the powerful exponential disentangling operator
identity). Obviously, one can use this approach to directly calculate the rotation matrices, and this has
been known since the 1970s [5, 21]. But the connection to use these results for spherical harmonics does
not appear to have been made by anyone else, yet.
We strongly feel that much of the beauty of quantum mechanics lies in how one can derive so many
fundamental results from the different operator identities. We now see that this approach extends to
deriving spherical harmonics as well. In our view, the more one can obtain results via operator methods,
the better (just ask students to compare learning the differential equation approach for the simple harmonic
oscillator versus the raising and lowering operator approach). In addition, this methodology teaches many
of the strategies we employ later in research problems and it allows students to become facile in working
with operators right from the start.
We end this work with a final comment to Prof. Stasyuk. We hope that he agrees with us about the
importance of introducing operator methods such as these early into the curriculum. We feel he will
appreciate this approach given his rich history of work with different operator-based formalisms. It is our
honour to include this work in a volume dedicated to his 80th birthday.
Acknowledgements
This work was supported by the National Science Foundation under grant number PHY-1620555. In
addition, J.K.F. was also supported by the McDevitt bequest at Georgetown University.
33002-11
M. Weitzman, J.K. Freericks
References
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URL https://www.edx.org/course/quantum-mechanics-everyone-georgetownx-phyx-008-01x.
3. Green H.S., Matrix Mechanics, P. Noordhoff Ltd., Groningen, Netherlands, 1965.
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doi:10.1103/PhysRevA.6.2211.
6. Ado I.D., Izv. Fiz.-Mat. Obsch. (Kazan’), 1935, 7, 1–43 (in Russian).
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1995.
8. Aubert G., AIP Adv., 2013, 3, 062121, doi:10.1063/1.4811853.
9. Laplace P.S., Théorie des Attractions des Sphéroides et de la Figure des Planeètes, Mémoires de l’Académie
Royale des Sciences, 1782, 113–196.
10. Legendre A.-M., Recherches sur l’Attraction des Sphéroides Homogènes, Mémoires de Mathématiques et de
Physique, présentés à l’Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées, 1785, X,
411–435.
11. Kramers H.A., Quantum Mechanics, North Holland, Amsterdam, 1957.
12. Weinberg S., Lectures on Quantum Mechanics, Cambridge University Press, Cambridge, 2013.
13. Condon E.U., Shortley G.H., Theory of Atomic Spectra, Cambridge University Press, Cambridge, 1935.
14. Landau L.D., Lifshitz E.M., Quantum Mechanics: Nonrelativistic Theory, Pergamon Press, Oxford, 1965.
15. Cohen-Tannoudji C., Diu B., LaLoë F., Quantum Mechanics, Hermann, Paris, 1977.
16. Edmonds A.R., Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, 1957.
17. Wigner E.P., Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press,
London, 1959.
18. Gottfried K., Yan T.-M., Quantum Mechanics: Fundamentals, 2nd Edn., Springer-Verlag, New York, 2003.
19. Schwinger J., On Angular Momentum, Dover Publications, Mineola, New York, 2015.
20. Sakurai J.J., Napolitano J.J., Modern Quantum Mechanics, 2nd Edn., Pearson, Edinburgh, 2014.
21. Santiago M.A.M., Vaidya A.N., J. Phys. A: Math. Gen., 1976, 9, 897–904, doi:10.1088/0305-4470/9/6/010.
Розрахунок сферичних гармонiк без похiдних
М. Вайцман1, Дж.К. Фрiрiкс2
1 просп. Е. Рошелл 3515, Лас-Вегас, Невада 89121, США
2 Фiзичний факультет, Джорджтаунський унiверситет,
вул. 37 & О NW, Вашинґтон, округ Колумбiя 20057, США
Спосiб отримання сферичних гармонiк приводиться однаково майже в кожному пiдручнику чи на ко-
жному заняттi з квантової механiки. Як виявляється, його трудно вслiдкувати, важко зрозумiти й складно
вiдтворити бiльшостi студентiв. В цiй роботi нами показано як обчислити сферичнi гармонiки природнi-
шим способом з допомогою операторiв та дивовижної тотожностi, вiдомої як тотожнiсть експоненцiйно-
го розплутування операторiв (знаної в квантовiй оптицi, але мало застосовної десь iнакше). Цей новий
пiдхiд виникає природнiм чином пiсля введення дiракових позначень, встановлення алгебри оператора
кутового моменту та визначення дiї операторiв збiльшення та зменшення кутового моменту на спiльний
базис власних функцiй (як для L̂2 так i L̂z ).
Ключовi слова: кутовий момент, сферичнi гармонiки, операторнi методи
33002-12
https://www.edx.org/course/quantum-mechanics-everyone-georgetownx-phyx-008-01x
https://doi.org/10.1103/PhysRevA.6.2211
https://doi.org/10.1063/1.4811853
https://doi.org/10.1088/0305-4470/9/6/010
Introduction
Definition of the spherical harmonics
Exponential disentangling operator identity
Derivation of the spherical harmonics from the angular momentum algebra
Pedagogical aspects
Summary
|