Orbital angular momentum and the spherical harmonics

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1 Obital angula momentum and the spheical hamonics Apil 2, 207 Obital angula momentum We compae ou esult on epesentations of otations with ou pevious expeience of angula momentum, defined fo a point paticle as L = x p o, fo a quantum system as the opeato elationship ˆL = ˆx ˆp Notice that since ˆL i = ε ijk ˆx j ˆp k thee is no odeing ambiguity: ˆx j and ˆp k commute as long as j k, and the coss poduct insues this. Computing commutatos of the components of L, we have [ˆLi, ˆL m ] = [ε ijk ˆx j ˆp k, ε mnsˆx n ˆp s ] = ε ijk ε mns [ˆx j ˆp k, ˆx n ˆp s ] = ε ijk ε mns ˆx j [ˆp k, ˆx n ˆp s ] + [ˆx j, ˆx n ˆp s ] ˆp k = ε ijk ε mns ˆx j [ˆp k, ˆx n ] ˆp s + ˆx n [ˆx j, ˆp s ] ˆp k = ε ijk ε mns i δ knˆx j ˆp s + i δ jsˆx n ˆp k = i ε ijk ε mksˆx j ˆp s + ε ijk ε mnj ˆx n ˆp k = i ε ijk ε msk + ε iks ε mjk ˆx j ˆp s Using the Jacobi identity eq.2 in Angula Momentum Notes ε jkm ε inm + ε kim ε jnm + ε ijm ε knm = 0, we ewite ε ijk ε msk + ε iks ε mjk = ε ijk ε msk + ε jmk ε isk = ε mik ε jsk and the commutato becomes [ˆLi, ˆL m ] = i ε ijk ε msk + ε iks ε mjk ˆx j ˆp s = i ε mik ε jsk ˆx j ˆp s = i ε imk ˆLk We see that ˆL m satisfies the fundamental angula momentum commutation elations and must theefoe admit l, m epesentations satisfying ˆL z l, m = m l, m ˆL 2 l, m = l l + 2 l, m

2 along with aising and loweing opeatos, ˆL ±. Howeve, in the case of obital angula momentum, we have an explicit coodinate epesentation fo the opeatos. Fo the z-component, x ˆL 3 α = x ˆx ˆp 2 ˆx 2 ˆp α = i x y y x α x The eigenvalues of ˆL 3 ae given by solving i x y y x α = m x α x but this takes a simple fom in tems of an azimuthal coodinate. Let x = ρ cos ϕ and y = ρ sin ϕ. Then and we ewite the eigenvalue equation as with the immediate solutions ϕ = x ϕ x + y ϕ y = ρ sin ϕ x + ρ cos ϕ y = y x + x y i x α = m x α ϕ x α = e imϕ Single valuedness of the wave function means that we must have e 2πmi = and theefoe only intege values fo m ae allowed. This exposes an essential asymmety between spinos and vectos. We have seen that 3-vectos may be epesented as matices in a complex, 2-dim spino epesentation, but thee does not exist a simila epesentation of spinos using 3-dim coodinates. Having shown that j and m may take both intege and half-intege values, we now see that classical angula momentun is not the whole stoy. While the physical existence of intinsic angula momentum o spin was only discoveed afte the advent of quantum mechanics, its existence is a consequence of the goup-theoetic natue of otations, and could have existed classically. We continue with ou examination of intege j epesentations, and the states l, m of obital angula momentum. 2 Changing to spheical coodinates It is not supising that obital angula momentum is most tanspaently studied in tems of spheical coodinates. Hee we ewite ˆL z, ˆL ± and ˆL 2 in spheical coodinates. The coodinate tansfomation and its invese ae given by = x 2 + y 2 + z 2 x θ = tan 2 + y 2 2 ϕ = tan y x 2

3 and x = cos ϕ y = sin ϕ z = cos θ We also need the deivative opeatos, x i. Using the chain ule, we have x = x + θ x θ + ϕ x ϕ = y y + θ y θ + ϕ y ϕ = z z + θ z θ + ϕ z ϕ We would like to wite the ight hand sides of these equations in spheical coodinates. We may find the patials by witing the total diffeentials of, θ and ϕ. Stating with the diffeential of, d = x dx + y dy + z dz we ead off the patial deivatives, x = x = cos ϕ y = y = sin ϕ z = z = cos θ Next, fo θ, we take the diffeential of tan θ, x2 + y tan θ = 2 z cos 2 θ dθ = x x2 + y 2 z dx + y x2 + y 2 z dy x2 + y 2 z 2 dz Then, with the diffeential of θ becomes dθ = cos 2 θ = cos θ cos ϕdx + x2 + y 2 = x z y z = = cos ϕ cos θ sin ϕ cos θ cos ϕ dx + cos θ cos θ sin ϕdy dz sin ϕ dy cos θ 2 cos 2 θ dz 3

4 and ead off the patials θ x = cos θ cos ϕ θ y = cos θ sin ϕ θ = z Finally, we compute the diffeential of tan ϕ = y x, and use cos2 ϕ = and once again ead off the patials x2 x 2 +y 2 cos 2 ϕ dϕ = y x 2 dx + x dy dϕ = cos 2 sin ϕ ϕ 2 sin 2 θ cos 2 ϕ dx + cos2 ϕ cos ϕ dy = sin ϕ cos ϕ dx + dy ϕ x ϕ y ϕ z = sin ϕ = cos ϕ = 0 Now, etuning to the chain ule expansions, eqs., we substitute to find x = x y z + xz x2 + y 2 2 θ y x 2 + y 2 ϕ = cos ϕ + cos θ cos ϕ θ sin ϕ ϕ = y + yz x2 + y 2 2 θ + x x 2 + y 2 ϕ = sin ϕ + cos θ sin ϕ θ + = z x2 + y 2 2 θ = cos θ θ In the next section, we substitute to find the obital angula momentum opeatos in angula coodinates. Finally, it is easy to find the Laplacian in spheical coodinates using the techniques of diffeential geomety. Using the metic in spheical coodinates g ij = 2 2 sin 2 θ and the divegence theoem, the esult is immediate: 2 = D i D i cos ϕ ϕ 2 4

5 = g x i = 2 = 2 gg ij x j [ θ 2 + θ θ + θ ϕ 2 sin 2 θ ] ϕ 2 2 sin 2 θ 2 ϕ Obital angula momentum opeatos in spheical coodiates Caying out the coodinate substitutions, fo ˆL 3 we have i x y y = i cos ϕ sin ϕ x + cos θ sin ϕ θ + +i sin ϕ cos ϕ + cos θ cos ϕ θ = i ϕ as found above. Fo the aising opeato, we have while the loweing opeato is ˆL + = z x + iz y x + iy z = cos θ + cos θ cos ϕ + cos θ cos ϕ θ i sin ϕ + i cos θ sin ϕ θ + i cos ϕ + i sin ϕ cos θ θ sin ϕ ϕ cos ϕ ϕ cos ϕ sin ϕ = cos ϕ + i sin ϕ e iϕ cos θ + cos2 θe iϕ θ + eiϕ sin 2 θ θ +i cos ϕ + i sin ϕ cos θ ϕ = e iϕ θ + icos θ ϕ ˆL = z x + iz y + x iy z = cos θ + cos θ cos ϕ + cos θ cos ϕ θ i sin ϕ + i cos θ sin ϕ θ + i + cos ϕ i sin ϕ cos θ θ = e iϕ cos θ e iϕ cos θ e iϕ cos 2 θ θ e iϕ sin 2 θ cos θ + ie iϕ θ ϕ 5 sin ϕ ϕ cos ϕ ϕ ϕ ϕ

6 Collecting the esults so fa, we have = e iϕ θ icos θ ϕ ˆL 3 = i ϕ ˆL + = e iϕ θ + icos θ ϕ ˆL = e iϕ θ + icos θ ϕ Execise: Find the fom of ˆL x and ˆL y fom eqs.5 and 6, togethe with the definitions Ĵ ± Ĵ ± iĵ2. Execise: Confim the fom of the Laplacian opeato by diect substitution into ˆL 2 = ˆL 2 x + ˆL 2 y + ˆL 2 z Now, since ˆL + ˆL = ˆL 2 ˆL ˆL 3 the magnitude squaed of the total angula momentum is L 2 = ˆL + ˆL + L 2 3 L 3 = e iϕ θ + icos θ = 2 cos θ = 2 cos θ = 2 cos θ = 2 = 2 θ + θ icos2 sin 2 θ e iϕ ϕ θ + icos θ ϕ + θ 2 θ 2 i ϕ cos2 θ sin 2 θ θ + icos θ ϕ 2 ϕ θ 2 θ 2 2 ϕ 2 cos2 θ 2 sin 2 θ ϕ 2 2 cos θ + θ θ 2 + sin 2 θ + 2 θ θ sin 2 θ ϕ 2 2 ϕ ϕ 2 + i 2 ϕ ϕ θ + icos θ ϕ 2 + i 2 ϕ ϕ 2 2 ϕ 2 + i 2 ϕ This last equation establishes the elationship between the spheical hamonics and the angula momentum states, because the Laplace equation in spheical coodinates is 2 = θ θ 2 sin 2 θ ϕ 2 = ˆL 2 2 6

7 and we know that sepaation of vaiables leads to geneal solution of the Laplace equation, f, θ, ϕ with the angula solution given in tems of spheical hamonics, f, θ, ϕ = l l=0 m= l A l Y l m θ, ϕ The spheical hamonics satisfy the sepaated angula eigenvalue equation, + 2 θ θ sin 2 θ ϕ 2 Y l m θ, ϕ = l l + Y l m θ, ϕ fo intege l and m = l, l +,... + l. Expessing this in tems of ˆL 2, ˆL 2 ψ = l l + 2 ψ we see that ψ = l, m and theefoe identify the spheical hamonics as the intege spin eigenstates of angula momentum in a coodinate basis, These descibe only intege j states. 4 Spheical hamonics Y l m θ, ϕ = θ, ϕ l, m We can now use the quantum fomalism to find the spheical hamonics, Y l m θ, ϕ = θ, ϕ l, m. Fo any state α, we know the effect of ˆL z is given by eq.4, so θ, ϕ ˆL z α = i θ, ϕ α ϕ Since the eigenstates satisfy ˆL z l, m = m l, m in geneal, placing this equation in a coodinate basis it becomes i θ, ϕ l, m = m θ, ϕ l, m ϕ This is tivially integated to give θ, ϕ l, m = e imϕ θ, ϕ l Futhemoe, we know that the aising opeato will anihilate the state with the highest value of m, ˆL + l, m = l = 0 Again choosing a coodinate basis, ˆL + is given by eq.5 so this tanslates to a diffeential equation, 0 = θ, ϕ ˆL + l, l = e iϕ θ + icos θ = e iϕ θ + icos θ = e il+ϕ θ Setting θ, ϕ l = f l θ, we ewite this as θ, ϕ l, m = l ϕ e ilϕ θ, ϕ l ϕ cos θ θ, ϕ l l θ, ϕ l 0 = f l θ l cos θf l 7

8 This is solved by f l = sin l θ, so we have, fo m = l Y l l θ, ϕ = A ll e ilϕ sin l θ Now we can find all othe Y l m θ, ϕ by acting with the loweing opeato, θ, ϕ ˆL l, m = l l + m m θ, ϕ l, m Inseting the coodinate expession, eq.6, fo θ, ϕ ˆL l, m and solving fo the next lowe state, we have θ, ϕ l, m = theeby defining all Y l m θ, ϕ ecusively. e iϕ l l + m m θ + icos θ e iϕ e imϕ = l l + m m θ + mcos θ ϕ θ, ϕ l, m θ l Execise: Find the Y m θ, ϕ fo all allowed m. 8