IIICh10-13.tex — March 18, 2000 (Preliminary version )

P.J.M. Bongaarts Instituut Lorentz, University of Leiden bongaart@lorentz.leidenuniv.nl

Topics from 20^{th} century physics. An introductory course for students in mathematics

III. QUANTUM THEORY: CHAPTERS 10 - 13

– APPLYING SYMMETRY PRINCIPLES

10. Translation symmetry and linear momentum 10.1. Translation symmetry for a single particle 10.2. Translation symmetry for a system of two particles

11. Rotation symmetry and angular momentum 11.1. The rotation group SO(3) and its Lie algebra so(3) 11.2. Unitary representations in the single particle Hilbert space 11.3. Rotation symmetry for a single particle 11.4. The irreducible unitary representations of so(3) and SO(3) 11.5. The irreducible unitary representations of su(2) and SU(2)

12. The hydrogen atom

13. Spin

1. TRANSLATION SYMMETRY AND LINEAR MOMENTUM

1.1. Translation symmetry for a single particle The translations in space over vectors a⃗ form a 3-dimensional abelian Lie group. It has an obvious unitary representation in the Hilbert space H = L^{2}(R^{3}, dx⃗) of a single particle, given as

(U(a⃗)ψ)(x⃗) = ψ(x⃗ − a⃗),

for all a⃗ in R^{3}. The Hamiltonian of this system has a kinetic energy term

H_{0} = \frac{P⃗^{2}}{2m} = −\frac{h̄^{2}}{2m}Δ,

and possibly a potential energy term V consisting of the operator of multiplication by a potential function V (x⃗). The unitary operators U(a⃗) commute with H_{0} because taking second derivatives and translation over a⃗ commutes. The U(a⃗) do not commute with V because V (x⃗)ψ(x⃗ − a⃗) ≠ V (x⃗ − a⃗)ψ(x⃗ − a⃗).

Taylor expansion gives

(U(a⃗)ψ)(x⃗) = ψ(x⃗ − a⃗) =

= ψ(x⃗) + ∑_{j=1}^{3}(−a_{j})\frac{∂}{∂x_{j}} ψ(x⃗) + \frac{1}{2} ∑_{j,k=1}^{3}(−a_{j})(−a_{k})\frac{∂^{2}}{∂x_{j}∂x_{k}} ψ(x⃗) + …

This suggests that U(a⃗) can be written as

U(a⃗) = e^{a⃗·P⃗′},

with the three commuting antihermitian operators

P′_{j} = −\frac{∂}{∂x_{j}} ,

representing the three basis elements of the 3-dimensional abelian Lie algbra of the group of translations. Instead of the antihermitic P′_{j} we can use the selfadjoint generators

P_{j} = ih̄P′_{j} = \frac{h̄}{i} \frac{∂}{∂x_{j}} ,

which as observables we already know as the components of linear momentum. This gives

U(a⃗) = e^{− \frac{i}{h̄} a⃗·P⃗} .

Conclusion : The group of translations in space is a symmetry group for a single free particle. The conserved quantities associated with this symmetry are the three components of linear momentum . A single particle in a potential has no translation symmetry.

10.2. Translation symmetry for a system of two particles Consider a system of two different particles, with masses m_{1} and m_{2}. Its Hilbert space of states is H = L^{2}(R^{6}, dx⃗dy⃗). The group of 3-dimensional translations has a unitary representation in this H given by

(U(a⃗)ψ)(x⃗, y⃗) = ψ(x⃗ − a⃗, y⃗ − a⃗),

for all a⃗ in R^{3}. The Hamiltonian of this system contains a kinetic enery term

H_{0} = \frac{P⃗_{x}^{2}}{2m_{1}} + \frac{P⃗_{y}^{2}}{2m_{2}} = −\frac{h̄^{2}}{2m_{1}} Δ_{x} − \frac{h̄^{2}}{2m_{2}} Δ_{y},

which clearly commutes with all U(a⃗). An interaction potential of the form V (x⃗, y⃗) = V (x⃗ − y⃗), i.e. depending only on the difference of the positions, also commutes with the U(a⃗). However, an additional potential V_{1}(x⃗) + V_{1}(y⃗), associated with an external force, does not commute with the U(a⃗).

Taylor expansion of (U(a⃗)ψ)(x⃗, y⃗) gives in this case

(U(a⃗)ψ)(x⃗, y⃗) = ψ(x⃗ − a⃗, y⃗ − a⃗) =

= ψ(x⃗, y⃗) + ∑_{j=1}^{3}(−a_{j}) (\frac{∂}{∂x_{j}} + \frac{∂}{∂y_{j}} ) ψ(x⃗, y⃗) +

+ ∑_{j,k=1}^{3}(−a_{j})(−a_{k}) (\frac{∂^{2}}{∂x_{j}∂x_{k}} + 2\frac{∂^{2}}{∂x_{j}∂y_{k}} + \frac{∂^{2}}{∂y_{j}∂y_{k}} ) ψ(x⃗, y⃗) + …,

which suggest that the U(a⃗) can be written as

U(a⃗) = e^{a⃗·P⃗′},

with the three commuting antihermitian operators P′_{j} in this case defined as

P′_{j} = −\frac{∂}{∂x_{j}} − \frac{∂}{∂y_{j}} ,

or as

U(a⃗) = e^{− \frac{i}{h̄} a⃗·P⃗} ,

with three commuting selfadjoint generators

P_{j} = ih̄P′_{j} = \frac{h̄}{i} \frac{∂}{∂x_{j}} + \frac{h̄}{i} \frac{∂}{∂y_{j}} = (P_{x})_{j} + (P_{y})_{j},

the three components of the total linear momentum of the system.

Conclusion : A system of two particles, interacting through a potential depending only on the difference x⃗ − y⃗, has translation symmetry. The conserved quantities associated with this symmetry are the three components of the total linear momentum of the system. Adding a potential connected with external forces breaks this symmetry.

Remark : The examples of this section and the preceding one are meant as very simple demonstrations of the relation between symmetries and conserved quantities. The fact that the momentum P⃗ of a free particle is a conserved quantity follows of course immediately from the form of the Hamiltonian; it does not need the context of Lie groups. The example of a system of two particles is only slightly less trivial as a group theory exercise. However, the general principle involved in the relation between translation symmetry and conservation of momentum is very important in systems where one does not have an obvious suitable notion of total momentum, such as classical and quantum field theory. In such cases it is used to define it.

11. ROTATION SYMMETRY AND ANGULAR MOMENTUM

11.1. The rotation group SO(3) and its Lie algebra so(3) SO(3), the group of (continuous) rotations in 3-dimensional Euclidean space, consists of all 3 × 3 orthogonal matrices with determinant equal to 1. Consider the 1-parameter subgroup of matrices

D_{1}(θ) = ( \begin{matrix}1 & 0 & 0 \\ 0 & cos θ & − sin θ \\ 0 & sin θ & cos θ \end{matrix} ) ,

the rotations over an angle θ around the x_{1}-axis. One has

(\frac{d}{dθ} D_{1}(θ))_{θ=0} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & 1 & 0 \end{matrix} ) = d_{1},

which implies that D_{1}(θ) can be written as D_{1}(θ) = e^{θd_{1}}. Similarly, the subgroups of rotations around the x_{2}- and x_{3}-axis can be written as D_{2}(θ) = e^{θd_{2}} and D_{3}(θ) = e^{θd_{3}}, with

d_{2} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ −1 & 0 & 0 \end{matrix} ) ,

respectively

d_{3} = ( \begin{matrix}0 & −1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

The matrices D_{j}(θ) describe what are called infinitesimal rotations around the x_{j}-axes. They form a basis for the vector space of all antisymmetric 3 × 3 matrices, the Lie algebra so(3) of SO(3), and satisfy the commutation relations

[d_{1}, d_{2}] = d_{3}, [d_{2}, d_{3}] = d_{1}, [d_{3}, d_{1}] = d_{2}.

11.2. Unitary representations in the single particle Hilbert space The group of spatial rotations is represented in H = L^{2}(R^{3}, dx⃗), the single particle state space, by unitary operators as

(U(D)ψ)(x⃗) = ψ(D^{−1}x⃗),

for all D in SO(3) and with (D^{−1}x⃗)_{j} = ∑_{k=1}^{3}(D^{−1})_{jk}x_{k}. According to the general scheme discussed in Appendix D, Section 5.3, the 1-parameter groups of orthogonal matrices D_{j}(θ) = e^{θd_{j}} are represented by the operators

U(D_{j}(θ)) = U(e^{θd_{j}}) = e^{θπ(d_{j})},

with the antihermitian operators π(d_{j}) representing the Lie algebra elements d_{j}, with the same commutation relations as the d_{j}, i.e. [π(d_{1}), π(d_{2})] = π([d_{1}, d_{2}]) = π(d_{3}), etc.

Using the Taylor expansion of (U(D_{j}(θ))ψ)(x⃗) one finds the generators π(d_{j}) as differential operators in H = L^{2}(R^{3}, dx⃗). For j = 1 one has

(U(D_{1}(θ))ψ)(x⃗) = ψ(D_{1}^{−1}(θ)x⃗) =
= ψ(e^{−θd_{1}} x⃗) = ψ(x⃗ − θd_{1}x⃗ + …).

After substitution of the explicit form of the matrix d_{1} this becomes

ψ(x_{1}, x_{2} + θx_{3} + …, x_{3} − θx_{2} + …) =

= ψ(x⃗) + θ (x_{3} \frac{∂}{∂x_{2}} − x_{2} \frac{∂}{∂x_{3}} ) ψ(x⃗) + …

Comparison with

U(D_{1}(θ)) = e^{θπ(d_{1})} = 1 + θπ(d_{1}) + …

gives finally

π(d_{1}) = x_{3} \frac{∂}{∂x_{2}} − x_{2} \frac{∂}{∂x_{3}} = −\frac{i}{h̄}(Q_{2}P_{3} − Q_{3}P_{2}) = −\frac{i}{h̄}L_{1},

with L_{1} the selfadjoint operator for the x_{1} component of the quantum mechanical angular momentum, introduced and discussed in 7.1. For j = 1 and j = 3 the same procedure gives

π(d_{2}) = x_{1} \frac{∂}{∂x_{3}} − x_{3} \frac{∂}{∂x_{1}} = −\frac{i}{h̄}(Q_{3}P_{1} − Q_{1}P_{3}) = −\frac{i}{h̄}L_{2},

respectively

π(d_{3}) = x_{2} \frac{∂}{∂x_{1}} − x_{1} \frac{∂}{∂x_{2}} = −\frac{i}{h̄}(Q_{1}P_{2} − Q_{2}P_{1}) = −\frac{i}{h̄}L_{3}.

The commutation relations for the L_{j},

[L_{1}, L_{2}] = ih̄L_{3}, [L_{2}, L_{3}] = ih̄L_{1}, [L_{3}, L_{1}] = ih̄L_{2},

can be calculated using these explicit expressions, but they follow here immediately from the group theoretic relation

[π(d_{j}), π(d_{k})] = π([d_{j}, d_{k}]),

for j, k = 1, 2, 3. We have for the unitary operators

U(D_{j}(θ)) = e^{θπ(d_{j})} = e^{− \frac{i}{h̄} θL_{j}}.

11.3. Rotation symmetry for a single particle The operators U(D), representing the group SO(3) in H = L^{2}(R^{3}, dx⃗) commute with the Hamiltonian H = H_{0} + V of a particle moving in a rotation invariant potential, i.e. a potential of the form V (|x⃗|). (Note that the simplest way to check that the U(D) commute with the kinetic energy part of the Hamiltonian H_{0} is to look at the Fourier transformed picture in which H_{0} acts on the Fourier transformed functions ψ̂(p⃗) as multiplication by a factor −(1/h̄^{2})|p⃗|^{2} and U(D) changes ψ̂(p⃗) into ψ̂(D^{−1}p⃗).

Conclusion : SO(3), the group of rotations in space, is a symmetry group for the system of a single particle moving in a centrally symmetric potential V (|x⃗|). The conserved quantities associated with this symmetry are the three components of angular momentum .

To understand the physical properties of angular momentum in this quantum mechanical system one has to study in the first place the spectrum of the operators L_{j}. Note that because the L_{j} do not commute one cannot look for a common system of eigenvectors; the L_{j} have to be discussed separately. The L_{j} can be transformed into each other by unitary operators U(D). One can for instance easily derive that e^{\frac{i}{h̄} \frac{π}{2} L_{1}}L_{2} e^{− \frac{i}{h̄} \frac{π}{2} L_{1}} is equal to L_{3}. This implies that the three L_{j} have the same spectrum. To find this spectrum, say that of L_{3}, it is of no help that it is a simple expression in operators for position and momentum, operators of which we know the spectrum. In fact the spectrum of L_{3} turns out to be discrete, contrary to that of the position and momentum operators.

To find the spectrum of L_{3} we introduce spherical coordinates r, ν and θ, by the formulas

x_{1} = r sin ν cos θ
x_{2} = r sin ν sin θ
x_{3} = r cos ν

Writing the wave functions ψ as functions of these coordinates, the inner product of two wave functions ψ_{1} and ψ_{2} becomes

(ψ_{1}, ψ_{2}) = ∫_{0}^{∞} ∫_{0}^{2π} ∫_{0}^{π} \overline{ψ_{1}(r, ν, θ)}ψ_{2}(r, ν, θ)r^{2} sin ν dνdrdθ.

It is clear that the operators U(D) act only on the angle variables ν and θ; the 1-parameter group U(D_{3}(θ)) acts only on θ. We have

(U(D_{3}(θ′)ψ)(r, ν, θ) = (e^{− \frac{i}{h̄} θ′L_{3}}ψ) (r, ν, θ) = ψ(r, ν, θ − θ′),

which implies

L_{3}ψ = \frac{h̄}{i} \frac{∂ψ}{∂θ} .

The general solution of the eigenvalue/eigenvector equation

\frac{h̄}{i} \frac{∂ψ}{∂θ} = m_{3}ψ

has the form

ψ_{m_{3}}(r, ν, θ) = ϕ(r, ν) e^{im_{3}θ},

for a real eigenvalue m_{3}h̄ and an arbitrary function ϕ(r, ν) for which the integral ∫_{0}^{∞} ∫_{0}^{2π} |ϕ(r, ν)|^{2}r^{2} sin ν drdν is convergent. The 1-parameter group of unitary operators e^{− \frac{i}{h̄} θ′L_{3}}, representing the rotations along the x_{3}-axis over an angle θ′, act on these eigenstates as

(e^{− \frac{i}{h̄} θ′L_{3}}ψ_{m_{3}}) (r, ν, θ) = ψ_{m_{3}}(r, ν, θ − θ′) = e^{−im_{3}θ′}ψ_{m_{3}}(r, ν, θ).

Because a rotation over an angle θ′ is the same as a rotation over θ′ + 2π, the number m_{3} has to be one of the discrete values m_{3} = 0, ±1, ±2, . . ..

This result can be slightly generalized to the statement that the component of the angular momentum in any given direction is ‘quantized’ in the sense that it can only assume the discrete values m_{3}h̄, m_{3} ∈ Z. The eigenvalues clearly have infinite degeneration, due to the presence of the functions ϕ(r, ν). To discuss this and other physically interesting features of the situation one needs to know more about the way the representation {U(D)}_{D∈SO(3)} can be written in terms of irreducible representations. Mathematical material for this and for a further study of eigenstates, etc., will be supplied in the next section. This will be useful for our treatment of the hydrogen atom in Chapter 12 and for the notion of spin in Chapter 13.

11.4. The irreducible unitary representations of so(3) and S)(3) The group SO(3) is compact. It is known that the irreducible unitary representations of compact Lie groups, together with the corresponding Lie algebra representations, are all finite dimensional.

Assume that we have an irreducible unitary representation of the Lie algebra so(3) in a finite dimensionial Hilbert space K. Such a representation is generated by three selfadjoint operators A_{1}, A_{2} and A_{3} in K, satisfying the commutation relations

[A_{1}, A_{2}] = iA_{3}, [A_{2}, A_{3}] = iA_{1}, [A_{3}, A_{1}] = iA_{2}.

Define the operators A^{2} = (A_{1})^{2} + (A_{2})^{2} + (A_{3})^{2} and A_{±} = A_{1} ± iA_{2}. The operator A^{2} is positive and commutes with the three A_{j}, as one easily verifies. Schur’s lemma states that in a finite dimensional space a linear operator that commutes with all the members of an irreducible system of operators is necessarily a multiple of the unit operator. Therefore A^{2}ψ = λψ, for a fixed positive constant λ and for all ψ in K. For the A_{±} one has A_{−}^{∗} = A_{+} and the commutation relations

[A_{+}, A_{−}] = 2A_{3}, [A_{3}, A_{±}] = ±A_{±}.

Write A_{+}A_{−} = (A_{1} +iA_{2})(A_{1} −iA_{2}) = (A_{1})^{2} +(A_{2})^{2} −i[A_{1}, A_{2}] = A^{2} −(A_{3})^{2} + A_{3}, etc., and a similar expression for A_{−}A_{+}. This gives the expressions

A_{+}A_{−} = A^{2} − A_{3}(A_{3} − 1)
A_{−}A_{+} = A^{2} − A_{3}(A_{3} + 1)

LEMMA 1: Let ϕ_{α} be an eigenvector of A_{3} with eigenvalue α. For α(α+1) ≠ λ one has A_{+}ϕ_{α} ≠ 0, A_{+}ϕ_{α} an eigenvector of A_{3} with eigenvalue α + 1, and A_{−}A_{+}ϕ_{α} = (λ − α(α + 1))ϕ_{α}. For α(α + 1) = 0 one has A_{+}ϕ_{α} = 0

Proof : Follows easily from the property A_{+}^{∗} = A_{−} and two identities:

1. A_{3}(A_{+}ϕ_{α}) = ([A_{3}, A_{+}] + A_{+}A_{3})ϕ_{α} = (A_{+} + A_{+}A_{3})ϕ_{α} = (α + 1)A_{+}ϕ_{α}.

2. A_{−}A_{+}ϕ_{α} = (A^{2} − A_{3}(A_{3} + 1))ϕ_{α} = (λ − α(α + 1))ϕ_{α}.

LEMMA 2: Let ϕ_{α} be an eigenvector of A_{3} with eigenvalue α. For α(α−1) ≠ λ one has A_{−}ϕ_{α} ≠ 0, A_{−}ϕ_{α} an eigenvector of A_{3} with eigenvalue α − 1, and A_{+}A_{−}ϕ_{α} = (λ − α(α − 1))ϕ_{α}. For α(α − 1) = 0 one has A_{−}ϕ_{α} = 0

Proof : Follows easily from the property A_{+}^{∗} = A_{−} and two identities:

1. A_{3}(A_{−}ϕ_{α}) = ([A_{3}, A_{−}] + A_{−}A_{3})ϕ_{α} = (−A_{−} + A_{−}A_{3})ϕ_{α} = (α − 1)A_{−}ϕ_{α}.

2. A_{+}A_{−}ϕ_{α} = (A^{2} − A_{3}(A_{3} − 1))ϕ_{α} = (λ − α(α − 1))ϕ_{α}.

The operator A_{3} has at least one eigenvector ϕ_{α} with eigenvalue α. Because of the two lemmas there is a sequence of eigenvectors of A_{3}

…, (A_{−})^{2}ϕ_{α}, A_{−}ϕ_{α}, ϕ_{α}, A_{+}ϕ_{α}, (A_{+})^{2}ϕ_{α}, …

with eigenvalues …, α − 2, α − 1, α, α + 1, α + 2, . . .. Because K is finite dimensional this sequence is finite; it starts with an eigenvector ϕ_{α_{1}} with the eigenvalue α_{1} and ends with an eigenvector ϕ_{α_{2}} with eigenvalue α_{2}, with the difference α_{2} −α_{1} a nonnegative integer p. One has A_{−}ϕ_{α_{1}} = 0 and A_{+}ϕ_{α_{2}} = 0, which implies α_{1}(α_{1}−1) = λ and α_{2}(α_{2}+1) = λ. Combining these two relations with α_{2} − α_{1} = p and p ≥ 0 leads to α_{1} = −\frac{1}{2}p, α_{2} = +\frac{1}{2}p and λ = \frac{1}{2}p(\frac{1}{2}p + 1). The linear subspace spanned by the sequence of p + 1 eigenvectors of A_{3}, say ϕ_{α_{1}}, …, ϕ_{α_{2}}, is invariant under the action of A_{−}, A_{+} and A_{3}, and therefore under A_{1}, A_{2} and A_{3}. These three operators were assumed to generate an irreducible representation of so(3), so the invariant subspace must be K itself.

Introducing j = \frac{1}{2}p, the result obtained in this manner can be stated as follows: the irreducible representation that we assumed to be given is characterized by a number j taking one of the values j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, . . .. The representation space K is (2j + 1)-dimensional, spanned by eigenvectors ϕ_{m} of A_{3} with eigenvalues m = −j, −j +1, …, +j −1, +j. The operator A^{2} multiplies all vectors in K by the constant j(j + 1).

The vectors ϕ_{m} are orthogonal but not normalized. We are free to choose one eigenvector, say the one with the lowest eigenvalue m = −j, to be of unit lenght. We call it ϕ_{−j}. The length of the next eigenvector is then given by

||A_{+}ϕ_{−j}||^{2} = (A_{+}ϕ_{−j}, A_{+}ϕ_{−j}) = (ϕ_{−j}, A_{−}A_{+}ϕ_{−j}) =
= (ϕ_{−j}, (A^{2} − A_{3}(A_{3} + 1))ϕ_{−j}) = 2l(ϕ_{−j}, ϕ_{−j}) = 2j.

We choose therefore the second normalized eigenvector as

ϕ_{−j+1} = (2j)^{−1/2}A_{+}ϕ_{−j}.

Continuing this process we obtain an orthonormal basis of 2j + 1 eigenvectors of A_{3}, denoted as

ϕ_{m} = C_{m}(A_{+})^{m+l}ϕ_{−j},

for m = −j, −j+1, …, +j. The normalization constants C_{m} can be calculated; they are determined up to a phase factor which we fix by requiring that the C_{m} are positive. One next calculates the matrix elements of the operators A_{−} and A_{+} with respect to the ϕ_{m} as

(ϕ_{m}, A_{−}ϕ_{m′}) = (j(j + 1) − m(m + 1))^{1/2} δ_{m+1,m′}
(ϕ_{m}, A_{+}ϕ_{m′}) = (j(j + 1) − m′(m′ + 1))^{1/2} δ_{m,m′+1}.

From this we obtain the matrix elements of A_{1} and A_{2}. This means that from our initial assumption that an irreducible representation of so(3) was given, we have finally derived the explicit matrix form of such a representation, in fact the explicit form of all possible finite dimensional irreducible representations of so(3).

The irreducible representations of the Lie algebra so(3) are characterized by a number j, integer or half-integer, i.e. taking the values j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, . . .. Can all these representations be exponentiated to representations of the group SO(3)? The answer follows immediately from the form of the representation of the 1-parameter subgroup D_{3}(θ) = e^{θd_{3}}, as defined in 11.1. One should have

U(e^{θd_{3}})ϕ_{m} = e^{− \frac{i}{h̄} θL_{3}}ϕ_{m} = e^{−iθA_{3}}ϕ_{m} = e^{−iθm}ϕ_{m}.

Rotation over an angle θ is the same as rotation over an angle θ + 2π. This excludes the half integer values of m, and consequently the half-integer values of j. We can state therefore the following general result for the representations of SO(3): An irreducible unitary representation of the group SO(3), necessarily finite dimensional, is characterized by an integer usually denoted as l, which takes one of the values l = 0, 1, 2, 3, . . .. The dimension of such a representation is 2l + 1; it is spanned by vectors ϕ_{m_{3}}, eigenvectors of L_{3} with eigenvalues h̄m_{3}, for m_{3} = −l, −l + 1, …, +l.

11.5. The irreducible unitary representations of su(2) and SU(2) Consider the Lie algebra su(2), which consists of all 2×2 antihermitian matrices. The three matrices

d̂_{1} = ( \begin{matrix}0 & −\frac{i}{2} \\ −\frac{i}{2} & 0 \end{matrix} ) , d̂_{2} = ( \begin{matrix}0 & −\frac{1}{2} \\ \frac{1}{2} & 0 \end{matrix} ) , d̂_{3} = ( \begin{matrix}−\frac{i}{2} & 0 \\ 0 & \frac{i}{2} \end{matrix} )

form a basis of su(2). Their commutation relations are

[d̂_{1}, d̂_{2}] = d̂_{3}, [d̂_{2}, d̂_{3}] = d̂_{1}, [d̂_{3}, d̂_{1}] = d̂_{2}.

This shows that su(2) is isomorphic to so(3). The representations of so(3) that we have found can therefore also be seen as representations of su(2). It is not hard to check that as such they can be exponentiated to representations of the group SU(2) for all values of j = 0, \frac{1}{2}, 2, \frac{3}{2}, 3, . . .. The map su(2) → so(3) is the Lie algebra isomorphism associated with a two-to-one Lie group homomorphism from SU(2) onto SO(3). This homomorphism can be worked out explicitly, but this will not be done here. A general element of SU(3) can be written as

( \begin{matrix}z_{1} & −z_{2} \\ \overline{z_{2}} & \overline{z_{1}} \end{matrix} ) ,

with z_{1} and z_{2} two complex numbers satisfying |z_{1}|^{2} + |z_{2}|^{2} = 1. Writing z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2}, one realizes that this means that SU(2) as a manifold is diffeomorphic to the 3-sphere S^{3}, and is therefore simply connected. SU(2) is in fact the unique simply connected covering group of SO(3).