# 5. Spherically symmetric potentials

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1 TFY4215/FY1006 Tillegg 5 1 TILLEGG 5 5. Spherically symmetric potentials Chapter 5 of FY1006/TFY4215 Spherically symmetric potentials is covered by sections 5.1 and in Hemmer s bok, together with the present Tillegg 5. References will be given also to the relevant sections in B&J. In the courses FY2045/TFY4250, this chapter is part of the background, that you should study on your own. 5.1 Isotropic harmonic oscillator (Hemmer 5.1, B&J 7.1) The three-dimensional harmonic oscillator is discussed in section 5.1 in Hemmer and in 7.1 in B&J. Here we want to stress some important points. Point no 1: The energy eigenfunctions can be written on product form Even for an anisotropic oscillator, with different spring constants for the three Cartesian directions, V = 1k 2 xx 2 + 1k 2 yy 2 + 1k 2 zz mω2 xx mω2 yy mω2 zz 2, (T5.1) the Hamiltonian can be written as a sum of three independent (commuting) terms: Ĥ = Ĥ(x) + Ĥ(y) + Ĥ(z), with Ĥ(x) h2 2m x mω2 xx 2, etc (T5.2) If we denote the well-know eigenfunctions of Ĥ(x) by ψ nx (x): Ĥ (x) ψ nx (x) = hω x (n x )ψ n x (x) E (x) ψ nx (x), n x = 0, 1, 2,, it is easily seen that the product states 2 ψ nxn yn z (x, y, z) = ψ nx (x)ψ ny (y)ψ nz (z) (T5.3) are energy eigenstates (of Ĥ) with the energy eigenvalues E nxnyn z = E (x) + E (y) + E (z) = hω x (n x + 1) + hω 2 y(n y + 1) + hω 2 z(n z + 1 ). (T5.4) 2 These are at the same time eigenfunctions of Ĥ(x) etc, that is, simultaneous eigenfunctions of the operator set Ĥ, Ĥ(x), Ĥ(y), Ĥ(z), which all commute.

2 TFY4215/FY1006 Tillegg 5 2 Exercise: Another example of such product states is encountered for the threedimensional box, for which Ĥ(x) = ( h 2 /2m) 2 /x 2, Ĥ(y), Ĥ(z) and Ĥ = Ĥ (x) + Ĥ(y) + Ĥ(z) all commute. Here too we can construct eigenstates in the form of product states: ψ nxnyn z (x, y, z) = 23 sin n xπx sin n yπy sin n zπz. L x L y L z L x L y L z Find the energy eigenvalues expressed in terms of the independent quantum numbers n x, n y and n x. In the example with the three-dimensional oscillator, we can call the x-, y- and z- directions the quantization directions. Since the three quantum numbers n x, n y and n z are independent, we also say that the motions in the x- y and z-directions represent three independent degrees of freedom. If we now consider the special case where the three spring constants are equal, k x = k y = k z = k, there emerges a symmetry in this system; the potential becomes spherically symmetric and the force becomes a central force: V = 1 2 kx ky kz2 = 1 2 kr2 1 2 mω2 r 2, = (T5.5) F(r) = V (r) = mω 2 r. (T5.6) Point no 2: The spherical symmetry leads to degenerate energy levels The energy now depends only on the sum of the three quantum numbers: E = hω(n x + n y + n z + 3/2) hω(n ) E N; (N = 0, 1, 2, ). As shown in the book, the number of states for level number N [the (degree of) degeneracy] is g N = 1 (N + 1)(N + 2). 2 The ground state is (as always) non-degenerate; for the lowest energy (N = 0, E = 3 2 hω) we have only one state, ψ 000 (x, y, z) (000) = ψ 0 (x)ψ 0 (y)ψ 0 (z) = C 3 0 e mωr2 /2 h. (T5.7) For the first excited level (N = 1, E = 5 2 hω) we have three states, ψ 100 (100) = ψ 1 (x)ψ 0 (y)ψ 0 (z) = C 2 0C 1 x e mωr2 /2 h, ψ 010 (010) = ψ 0 (x)ψ 1 (y)ψ 0 (z) = C 2 0C 1 y e mωr2 /2 h, (T5.8) ψ 001 (001) = ψ 0 (x)ψ 0 (y)ψ 1 (z) = C 2 0C 1 z e mωr2 /2 h.

3 TFY4215/FY1006 Tillegg 5 3 While the ground state ψ 000 is spherically symmetric, we see that these three excited states are not; as an example, the state ψ 001 (001) is rotationally symmetric with respect to the z-axis and antisymmetric with respect to the xy-plane. The state (100) behaves the same way with respect to the x-axis, and so on. Point no 3 is that when a degenarate level has several eigenfunctions with the same eigenvalue (here 5 E = 2 hω), then these eigenfunctions are not unique. This is because we are free to use linear combinations of these eigenfunctions. Such linear combinations are alse eigenfunctions with the same energy. They are therefore equally relevant. For the oscillator we can illustrate this point in several ways: (i) Since the potential is spherically symmetric, we can just as well use another set of coordinate axes than the ones above, e.g. x, y, z (= z): By repeating the above process, with x, y and z as quantization directions, we then find for N = 1 the following three simultaneeous eigenstates of the operators Ĥ, Ĥ(x ), Ĥ(y ), Ĥ(z ) : (100) ψ 1 (x )ψ 0 (y )ψ 0 (z ) = C 2 0C 1 x e mωr2 /2 h, (010) ψ 0 (x )ψ 1 (y )ψ 0 (z ) = C 2 0C 1 y e mωr2 /2 h, (001) ψ 0 (x )ψ 0 (y )ψ 1 (z ) = C 2 0C 1 z e mωr2 /2 h. Here there is a lot of moral to be gained: With V = V (r) there is nothing particular about the original axis system x, y, z. The new choice, and the resulting eigenfunctions are just as relevant. The new eigenfunctions are linear combinations of the old ones. Thus with the concrete choice of x, y, z above, which corresponds to we have for example x = x cos α + y sin α, y = x( sin α) + y cos α, z = z, (100) = (100) cos α + (010) sin α, (010) = (100)( sin α) + (010) cos α (001) = (001). Here, the state (100) has the same form seen from the x -axis as (100) seen from the x-axis.

4 TFY4215/FY1006 Tillegg 5 4 The change of basis does not alter the degeneracy (which here is equal to 3). The eigenfunctions above are automatically orthogonal [cf (T2.34)]. (The orthogonality is in general important; cf the discussion of orbitals below.) (ii) Another way to obtain a set of orthogonal eigenfunctions for a symmetric potential is to look for simultaneous eigenfunctions of Ĥ, L 2 and one of the components of L, e.g. L z. Such eigenfunctions exist because the three operators mutually commute, as we shall see. Cf section 2.3.c, where we found that L 2 and L z commute. Since these two operators contain derivatives only with respect to the angles θ and φ, it follows that they also commute with the Hamiltonian, when the potential is spherically symmetric. The three-dimensional isotropic oscillator is very special in that the potential is both spherically symmetric and can be separated in terms of the independent (Cartesian) contributions 1 2 mω2 x 2 etc. In a short while we shall attack the hydrogen atom (the Coulomb problem). Then again we have a spherically symmetric potential. However, this potential can not be separated into Cartesian terms. So in that case we are forced to look for simultaneous eigenfunctions of the operators Ĥ, L 2 and one of the components of L, e.g. L z. This must be point no 4: The angular momentum is an important physical observable for particles moving in a spherically symmetric potential, like that of the hydrogen atom. We shall therefore proceed to study angular momentum in the next section. 5.2 Angular momentum and spherically symmetric potentials (Hemmer 5.4, B&J 6.1 and 6.3) 5.2.a Angular-momentum operators All information about a state sits in the wave function, combined with the operators for energy, momentum etc. The momentum operator p = ( h/i) essentially asks for the gradient of ψ, that is, how fast and in which direction ψ changes, the operator K = ( h 2 /2m) 2 for the kinetic energy checks div grad ψ = 2 ψ, etc. Similarly, the orbital angular-momentum operator L = r p = r ( h/i) asks about the variation of ψ perpendicularly to r. Thus it is the variation in the angular direction which determines the angular momentum. Then we understand immediately that the ground state of the hydrogen atom, ψ = (πa 3 0) 1/2 e r/a 0, is a state with zero angular momentum. This is because here the gradient points in the radial direction, so that The moral is that L ψ = r h ψ = 0. i A spherically symmetric wave function ψ(r) corresponds to a state with zero angular momentum. (T5.9)

5 TFY4215/FY1006 Tillegg 5 5 Another exaample is the ground state (T5.7) of the isotropic oscillator, ψ 000 (000) = C0e 3 mωr2 /2 h. This conclusion ( Lψ(r) = 0) also follows if we express the operators in spherical coordinates. As shown in section 5.2.g below, the gradient operator in spherical coordinates is = ê r r + ê 1 θ r θ + ê φ 1 r sin θ φ. It then follows that L contains derivatives with respect to the angles: L = r h i ( = h i ê φ θ ê θ 1 sin θ (T5.10) ), (T5.11) φ since ê r ê r = 0, ê r ê θ = ê φ, ê r ê φ = ê θ. The same holds for the Cartesian components, even if it isn t obvious in the expressions L z = x p y y p x, etc. Thus, expressed in spherical coordinates the Cartesian components are L x = h i L y = h i ( ( sin φ θ cos φ θ ) cot θ cos φ, φ ) cot θ sin φ, φ (T5.12) L z = h i φ.

6 TFY4215/FY1006 Tillegg 5 6 And then it is perhaps not surprising that the same holds also for the square of the angularmomentum operator, L 2 = h 2 ( 2 θ 2 + cot θ θ + 1 sin 2 θ 2 ). (T5.13) φ 2 Here we note that the spherical coordinates result in a particularly simple form for L z b Compatible observables in spherically symmetric potentials Spherically symmetrice potentials V (r) play a central role in atomic physics, and for a particle moving in such a central potential the angular momentum is an important observable, together with the energy. For such systems the angular momentum and the energy are compatible observables they can have sharp values simultaneously, because the angularmomentum operators commute with the Hamiltonian for a spherically symmetric potential, Ĥ = K + V (r) = h2 2m 2 + V (r). By comparing the last part of the Laplace operator in spherical coordinates, 2 = 2 r r r + 1 ( 2 r 2 θ + cot θ 2 θ + 1 sin 2 θ 2 ), (T5.14) φ 2 with the above expression for L 2 we see that the Laplace operator can be written as 2 = 2 r r r L 2 h 2 r 2. (T5.15) Using this expression we find a practical expression for the Hamiltonian for a particle moving in a spherically symmetric potential: ( Ĥ = h2 2 2m r r ) + r L 2 2mr 2 + V (r) K r + K L + V (r). (T5.16) Here we note that the angular part of K is K L = L 2 /(2mr 2 ). Since L 2 contains only derivatives with respect to angles, we realize that the above Hamiltonian commutes with L 2 : [Ĥ, L 2 ] = 0 (V = V (r)). The moral is that the size L = L 2 x + L 2 y + L 2 z of the angular momentum can have a sharp value together with the energy. 1 In the figure above, the angle θ between r and the z-axis is the so-called polar angle. The angle φ between the rẑ-plane and the xz-plane is the so-called asimuthal angle. In mathematics, you have possibly used a different notation.

7 TFY4215/FY1006 Tillegg 5 7 But what about the direction of L? Well, here we must remember (cf 2.3.c in Tillegg 2) that the operator components L x, L y and L z do not commute, but satisfy the so-called angular-momentum algebra [ L x, L y ] = i h L z, [ L y, L z ] = i h L x, (T5.17) [ L z, L x ] = i h L y. Then, according to the rules in section 4.1, there does not exist simultaneous eigenfunctions of L x, L y and L z, and hence of L. Thus the observables L x, L y and L z can not be sharp simultaneously; they are not compatible. The only exception to this rule is when the angular momentum is zero, as we saw above. Apart from this exception, we can state that the angular-momentum operator L = r p (contrary to the momentum operator p) has no eigenfunctions. Thus it is impossible for the particle to have a well-defined angularmomentum vector L, with a sharply defined direction. On the other hand, we have seen in Tillegg 2 that L x, L y and L z all commute with the square of the angular-momentum operator, L 2 = (r p) (r p) = L 2 x + L 2 y + L 2 z : [ L 2, L i ] = 0, i = x, y, z. (T5.18) Therefore it is possible to find simultaneous eigenfunctions of L 2 and one of the components of L (e.g. L x, or L y, or L z, or for that matter any component ˆn L). It is customary to choose to work with simultaneous eigenfunctions of L 2 and L z, because L z as stated has a particularly simple form in spherical coordinates. Then it only remains to note that L z (as well as L 2 ) only contains an angular derivative and therefore commutes with Ĥ = K + V (r). Thus, for spherically symmetric potentials the energy E, the size L of the angular momentum and its z-component L z are compatible observables. This means for example that it is possible to find energy eigenfunctions for the isotropic oscillator which are simultaneously eigenfunctions of L 2 and L z. Since these operators depend only on the angles, such eigenfunctions can be written as products of a radial function R(r) and an angular function Y (θ, φ): ψ(r, θ, φ) = R(r)Y (θ, φ). As an example, let us consider one of the three states (T5.8) for the first excited level of the isotropic oscillator. With z = r cos θ we can write ψ 001 as ψ 001 = RY, with R = C 2 0C 1 r e mωr2 /2 h and Y = cos θ. In this angular function there is no variation in the φ-direction, so that the partial derivative of Y = cos θ with respect to φ is equal to zero. Thus, L z Y = 0, that is, Y and hence ψ 001 are eigenfunctions of L z with eigenvalue zero. It is also easy to show that the variation in the θ-direction is such that Y is an eigenfunction of L 2 with eigenvalue 2 h 2.

8 TFY4215/FY1006 Tillegg 5 8 A small exercise: Show this using (T5.13). We can conclude that in the oscillator state ψ 001 the energy is 5 hω/2, while L is h 2 and L z is equal to zero. Note that all these quantities have sharp values (zero uncertainty). Another exercise: Since x = r sin θ cos φ and y = r sin θ sin φ, the two other states in (T5.8) can be written as ψ 100 = R sin θ cos φ and ψ 010 = R sin θ sin φ, with the same radial function R as above. Show that also these two functions are eigenfunctions of L 2 with the eigenvalue 2 h 2. None of them are eigenstates of L z. Show that linear combinations (ψ 100 ± iψ 010 )/ 2 = R/ 2 sin θ e ±iφ are eigenfunctions of L z, with the eigenvalues ± h. This is an example of the general fact that it is possible to find simultaneous eigenstates of Ĥ, L 2 and L z for a spherically symmetric potential. (Cf rule B in section 4.1 of Tillegg 4.) 5.2.c Quantization of angular momentum. The spherical harmonics The example above also illustrates that the angular momentum is quantized. It turns out that the simultaneous eigenfunctions and the corresponding eigenvalues of the angular-momentum operators L 2 and L z are given by: L 2 Y lm (θ, φ) = h 2 l(l + 1) Y lm (θ, φ), l = 0, 1, 2, L z Y lm (θ, φ) = hm Y lm (θ, φ), m = 0, ±1, ±2,, ±l. (T5.19) Such an eigenfunction Y lm corresponds to a state where both the size L = L 2 and the z-component of the angular momentum have sharp values, while L x and L y have to be uncertain, because they are not compatible with L z. (Then also the direction of L is uncertain.) We note that the possible eigenvalues are quantized. The size is given in terms of the so-called orbital angular-momentum quantum number l, which must be a nonnegative integer. The z-component L z is proportional to the so-called azimuthal quantum number m, also called the magnetic quantum number, which is an integer. Note that for a given quantum number l, the maximal m is l. This limitation is obviously connected with the fact that L z can not be larger than L. The eigenfunctions Y lm are known as the spherical harmonics. We shall soon discover that they can be written on the form Y lm (θ, φ) = Θ lm (θ) e imφ, where the dependence on θ and φ is separated. The quantization of angular momentum, as expressed by the quantum numbers l and m being integers, simply stated is a cosequence of the fact that the angular space is compact, so that the angular functions must bite themselves in the tail, so to speak. This is seen most clearly when one tries to solve

9 TFY4215/FY1006 Tillegg d the eigenvalue equation L z Y = h i φ Y = hm Y The partial differentiation with respect to φ means that θ is to be kept fixed. This eigenvalue equation therefore takes the form dy/dφ = imy, or which can be integrated to dy Y = im dφ, ln Y = imφ + ln Θ(θ), = Y (θ, φ) = Θ(θ) e imφ, where the integration constant ln Θ(θ) is independent of φ, but other than that can be an arbitrary function of θ. The requirement that Y (θ, φ) be continuous then implies that the quantum number m must be an integer; Y (θ, 2π) = Y (θ, 0) = e im 2π = 1; = m = 0, ±1, ±2,. Thus L z is quantized; L z = 0, ± h, ±2 h,. Note that a non-integer m would imply a wave function ψ(r, θ, φ) = R(r)Θ(θ) e imφ with a discontinuity in a half-plane corresponding to φ = 0 (y = 0, x > 0). 5.2.e The eigenvalue equation L 2 Y = h 2 l(l + 1) Y is more of a mathematical challenge. It is this equation that determines the quantized values of l, and the corresponding functions Θ lm (θ). This is explained in detail In Hemmer s book and also in B&J. The main points of the derivation are repeated here. Inserting Y = Θ(θ) exp(imφ) we note that 2 /φ 2 gives a factor (im) 2 = m 2. As in Hemmer, we introduce the abbreviation cos θ x (NB! x is not the coordinate x), so that Re-writing slightly, d 2 f df + cot θ dθ2 dθ sin θ = 1 x 2 and dx = sin θ dθ. = 1 sin θ [ = d dx ( d dθ sin θ df ) dθ ] (1 x 2 ) df dx = 1 sin θ ( d sin 2 θ dθ sin θ ) df dθ = (1 x 2 ) d2 f df 2x dx2 dx, we then find that the eigenvalue equation L 2 Y = h 2 l(l + 1)Y can be written as [ (1 x 2 ) d2 dx 2x d ] + l(l + 1) m2 Θ(x) = 0. 2 dx 1 x 2 (T5.20) Here we know that m is an integer, while the quantum number l is now regarded to be unknown, together with the possible solutions Θ lm (θ) of this eigenvalue equation. For the case m = 0 this is a standard equation in applied mathematics. It is called the Legendre equation: [ (1 x 2 ) d2 dx 2x d ] + l(l + 1) P 2 l (x) = 0. dx (T5.21)

10 TFY4215/FY1006 Tillegg 5 10 As explained in the book, this equation can be solved using the power series expansion method. With Θ = a n x n, 2xΘ = 2x a n nx n 1 and n=0 n=0 Θ = a n n(n 1)x n 2 = a n+2 (n + 2)(n + 1)x n n=2 n=0 we find that { a n+2 (n + 2)(n + 1) a n [n(n 1) + 2n l(l + 1)] } x n = 0. n=0 This equation is satisfied for all x only if the coefficient of each power of x vanishes, that is, if all the expressions {...} are eaual to zero. This gives the recursion relation a n+2 a n = n(n + 1) l(l + 1) (n + 1)(n + 2) = (n l)(n + l + 1) (n + 1)(n + 2), n = 0, 1, 2,. Using this relation, we can express a 2, a 4 etc in terms of a 0 and, in the same manner, a 3, a 5 etc in terms of a 1. The solution thus is the sum of two infinite power series, with respectively even and odd powers of x: [ Θ = a l (l + 1) 2! x 2 + l( l + 2) (l + 1)(l + 3) 4! x 4 + [ ( l + 1) (l + 2) +a 1 x + x 3 ( l + 1)( l + 3) (l + 2)(l + 4) + x 5 + 3! 5! Here, a 0 and a 1 are two arbitrary constants. From the recursion formula we note that a n+2 = n ( 1 + O(n 2 ) ). a n n + 2 This means that both series will for large n behave as n 1 n xn (where n is even or odd). Both series will therefore diverge for x = ±1, corresponding to θ = 0 and π. In order to obtain a finite and acceptable solution (eigenfunctions) we must therefore demand that both series terminate, giving a polynomial solution. From the formula above (and from the recursion relation) we see that the series with even powers becomes a polynomial of degree l if l is a non-negative even number, l = 0, 2, 4,. In these cases, the series with odd powers does not terminate. We must therefore set the coefficient a 1 equal to zero for even l. For odd l we must similarly set a 0 equal to zero, and the solution becomes a polynomial of degree l with only odd powers of x. This is how the quantization of L enters the picture, with integer orbital angular-momentum quantum numbers l = 0, 1, 2,. ] ].

11 TFY4215/FY1006 Tillegg 5 11 These polynomial solutions are uniquely determined by the recursion relation, apart from an arbitrary multiplicative constant (a 0 or a 1 ). When the solutions are normalized so that their value for x = 1 (θ = 0) becomes 1, they are called the Legendre polynomials, and they are denoted by P l (x) = P l (cos θ). Thus P l (1) = 1 and P l ( x) = ( 1) l P l (x), (T5.22) where the last relation states that the parity is ( 1) l (because the powers of x are either even or odd). Explicitly the first few of these polynomials are P 0 (x) = 1, (med x = cos θ :) P 1 (x) = x = cos θ, P 2 (x) = 1 2 (3x2 1) = 1 2 (3 cos2 θ 1), P 3 (x) = 1 2 (5x3 3x) = 1 2 (5 cos3 θ 3 cos θ), osv. A useful formula for the Legendre polynomials is So, for m = 0 we have P l (x) = 1 2 l l! d l dx l (x2 1) l. (Rodrigues formula) (T5.23) Θ l0 (θ) P l (cos θ). See by the way the diagramm with P 0, P 1 as functions of θ page 95 in Hemmer. A small exercise: a. In section 5.2.b we found that Y = cos θ is an eigenfunction of L 2 and L z with the eigenvalues h 2 l(l + 1) = 2 h 2 and hm = 0, that is, a solution with the quantum numbers l = 1 and m = 0. Check that Rodrigues formula for Θ l0 (θ) P l (cos θ) gives the same result. b. Use (T5.24) and (T5.23) to find the spherical harmonic Y 20, apart from the normalization. One more: Check that Rodrigues formula gives the above result for the Legendre polynomial P 3 (x). Use the above recursion relation, together with the normalization convention (T5.22), to show that P 4 (x) = 1 8 ( 35 x 4 30 x ). (T5.24) For the case m 0 we see that the differential equation (T5.20) depends only on m 2, that si on m. This means that we can set Θ m l = Θ m l, and consider only the case m > 0. As explained in the book, one then finds that the acceptable solutions of this differential equation are the so-called associated Legendre functions; Θ lm (θ) = P m l (x) (1 x 2 ) m/2 ( ) m d P l (x), (T5.25) dx where x = cos θ and (1 x 2 ) m/2 = sin m θ. You can find the proof in the book, or in the attachment at the end of this Tillegg.

12 TFY4215/FY1006 Tillegg 5 12 Since the highest power in P l is x l, the differentiation here gives a polynomial of degree l m, and zero for m > l. 2 Thus we can conclude that the differential equation gives solutions for integer l which are m, and the other way around: For a given orbital angular-momentum quantum number l the magnetic quantum number m is limited to m = 0, ±1,, ±l. The results for the normalized spherical harmonics can be written as Y lm (θ, φ) = δ 2l + 1 m 4π (l m )! (l + m )! P m l (cos θ)e imφ, (T5.26) where δ m = { 1 for m 0, ( 1) m for m > 0 is the most commonly used sign convention. In the table below you will find the explicit results for l 3. Spherical harmonics l m Y lm (θ, φ) s 0 0 Y 00 = 1 4π p 1 0 Y 10 = 3 4π cos θ ±1 Y 1,±1 = 3 8π sin θ e±iφ d 2 0 Y 20 = 5 16π (3 cos2 θ 1) ±1 Y 2,±1 = 15 8π ±2 Y 2,±2 = 15 32π sin θ cos θ e±iφ sin2 θ e ±2iφ f 3 0 Y 30 = 7 16π (5 cos3 θ 3 cos θ) ±1 Y 3,±1 = 21 64π ±2 Y 3,±2 = π ±3 Y 3,±3 = 35 64π sin θ (5 cos2 θ 1) e ±iφ sin2 θ cos θ e ±2iφ sin3 θ e ±3iφ 2 Note also that the powers of x in this polynomial are even/odd when l m is even/odd. This means that P m l ( x) = ( 1) l m P m (x). l

13 TFY4215/FY1006 Tillegg 5 13 As you probably have observed, it takes quite a lot of work to obtain these results for the angular functions. If one finds it difficult to grasp all the details of the derivation, some consolation may be found by checking explicitly the eigenvalues of the functions given in the table. The normalization can be checked by integrating Y lm (θ, φ) 2 over all angles. 5.2.f Some points worth noticing (i) All the functions Y lm contain the factors exp(imφ) and sin m θ together with the polynomial of degree l m in x cos θ. (ii) The spherical harmonics are parity eigenstates. The parity operation P means space inversion, that is, reflection with respect to the origin (see the figure p 5): This corresponds to P : r r, i.e. r r, θ π θ, (T5.27) φ φ + π. x = cos θ cos θ = x, P m Thus, the parity of Y lm simply is ( 1) l : P l (cos θ) P l ( cos θ) = ( 1) l P l (cos θ), l (cos θ) ( 1) l m P m l (cos θ), e imφ ( 1) m e imφ. P Y lm (θ, φ) = Y lm (π θ, φ + π) = ( 1) l Y lm (θ, φ). (T5.28) These symmetry properties are important e.g. when we want to study hybrid orbitals. (iii) The functions Y lm are not only normalized, but also orthogonal, because they are eigenfunctions with different eigenvalues [remember the rule (T2.34) in Tillegg 2]: Y l m Y lmdω = δ l lδ m m (orthonormality). (T5.29) (iv) The fact that l can not be smaller than m is of course connected with the obvious fact that L = h l(l + 1) can not be smaller than L z = h m. As an example, let us consider l = 2, which corresponds to L = h h. As illustrated in the figure, the allowed values of L z then are 0, ± h and ±2 h. Each of these values corresponds to a well-defined angle α between L and the z-axis, given by cos α = L z L = m l(l + 1) = m 6, (m = 2, 1, 0, 1, 2).

14 TFY4215/FY1006 Tillegg 5 14 In this case we find the smallest angle for m = 2, α = arccos(2/ 6) = In this state Y 22, also L 2 x + L 2 y = L 2 L 2 z = 6 h 2 (2 h) 2 = 2 h 2 is sharply defined. The observables L x and L y, on the other hand, are not compatible with L z, and must therefore be uncertain. It is easy to show that the expectation values of both these observables are equal to zero: L x Y22 = L y Y22 = 0. From the symmetry we can also understand that the expectation values of L 2 x and L 2 y must be equal, and since the sum of them is 2 h 2, we thus have that L 2 x = L 2 Y y = h Y 22 It follows that the uncertainties in the state Y 22 are L x = L y = h. This situation can be illustrated (to a certain extent) by the sketch on the right in the figure above: L and L z are sharp. So is L 2 x + L 2 y (the radius of the circle). L x and L y are uncertain (with zero averages), and then also the direction of L is uncertain. In a way this means that L is confined to the surface of a cone. 4 (v) The spherical harmonics Y lm (θ, φ) constitute a complete set; an arbitrary function of θ and φ can be expanded as: g(θ, φ) = l l=0 m= l c lm Y lm (θ, φ). (T5.30) If the function depends only on θ, one gets contributions only for m = 0, that is from Y l0 P l (cos θ) : g(θ) = c l P l (cos θ). (T5.31) l=0 Because of this completeness, the spherical harmonics are relevant under many circumstances, not only in quantum mechnics. 3 Note that the angle α is not allowed to be zero, because that would correspond to a vector L with a well-defined direction, and that as we know is impossibble, because L x, L y and L z are not compatible, that is, they can not have sharp values simultaneously. 4 However, here we must be careful. This illustration may lead us to believe that the direction of L is an observable. That is not the case, because L x, L y and L z are not compatible.

15 TFY4215/FY1006 Tillegg g Attachment on spherical coordinates etc From this figure we can read out the connections between the spherical coordinates r, θ, φ and the Cartesian coordinates x, y, z: 5 x = r sin θ cos φ r = x 2 + y 2 + z 2 y = r sin θ sin φ cos θ = z/ x 2 + y 2 + z 2 z = r cos θ tan φ = y/x (0 r <, 0 θ π, 0 φ < 2π). From the figure it is also easy to see that infinitesimal increments dr, dθ, dφ of r, θ, φ correspond to an infinitesimal change of r given by Integration in spherical coordinates dr = ê r dr + ê θ rdθ + ê φ r sin θ dφ. Because the three unit vectors ê r, ê θ and ê φ are orthonormal, we can figure out that the volume element can be written as d 3 r dv = dr r dθ r sin θ dφ = r 2 dr sin θdθ dφ. Integration over the entire space is accomplished by integrating r from 0 to, θ from 0 to π, and φ from 0 to 2π. For a spherically symmetric integrand this simplifies to 2π π f(r)d 3 r = dφ sin θ dθ f(r) r 2 dr [sin θ dθ = d(cos θ)] 0 = 2π = 0 cos θ=1 cos θ= 1 0 d(cos θ) f(r) 4πr 2 dr. 0 0 f(r)r 2 dr Note that 4πr 2 dr is the volume of a spherical shell with radius r and thickness dr. 5 In the figure above, the angle θ between r and the z-axis is the so-called polar angle. The angle φ between the rẑ-plane and the xz-plane is the so-called asimuthal angle. From mathematics you are probably used to a different notation.

16 TFY4215/FY1006 Tillegg 5 16 The gradient operator in spherical coordinates For a function f(r), the infinitesimal change dr (given above) corresponds to an infinitesimal change of the function, given by the scalar product of dr and the gradient of the function: Comparing with df = f(r + dr) f(r) = f dr = {ê r ( f) r + ê θ ( f) θ + ê φ ( f) φ } {ê r dr + ê θ rdθ + ê φ r sin θ dφ} = ( f) r dr + ( f) θ r dθ + ( f) φ r sin θ dφ. df = f r f f dr + dθ + θ φ dφ we then find that the components of the gradient of f in spherical coordinates are given by ( f) r = f r, ( f) θ = 1 r f θ, ( f) φ = 1 r sin θ f φ. Thus the gradient operator is (see also Rottmann) = ê r r + ê 1 θ r θ + ê φ 1 r sin θ φ. The orbital angular-momentum operator in spherical coordinater then is L = r p = r h i = h ( ê φ i θ ê θ 1 sin θ ), φ since ê r ê r = 0, ê r ê θ = ê φ, ê r ê φ = ê θ. Here it is important to notice that the orbital angular-momentum operator L contains derivatives only with respect to the angles θ and φ. This implies among other things that a spherically symmetric function f(r) is an eigenfunction of the angular-momentum operator with eigenvalue zero. Then the function f(r) is also an eigenfunction of L 2 L L with zero eigenvalue. Thus a wave function that depends only on r (and not on the angles) describes a state with zero angular momentum. L x, L y and L z in spherical coordinates By careful consideration of the figure one finds that ê r = ê x sin θ cos φ + ê y sin θ sin φ + ê z cos θ, ê θ = (ê x cos φ + ê y sin φ) cos θ ê z sin θ, ê φ = ê x sin φ + ê y cos φ. Using these formulae and the above expression for L it is easy to see that the Cartesian components of the orbital angular-momentum operator can be expressed in spherical coordinates

17 TFY4215/FY1006 Tillegg 5 17 as L x = h ( sin φ ) cot θ cos φ, i θ φ L y = h ( cos φ ) cot θ sin φ, i θ φ L z = h i φ. You should pay attentian particularly to the last one. L 2 in spherical coordinates. The Laplace operator By squaring the expression for L in spherical coordinates we have L 2 = L L = h i ( ê φ θ ê θ 1 sin θ ) h ( ê φ φ i θ ê θ 1 sin θ ). φ Here it is important to understand that ê θ depends both on θ and φ; it can be shown that ê θ θ = ê r, ê θ φ = ê φ cos θ, while ê φ is independent of θ: ê φ θ = 0, ê φ φ = ê r sin θ ê θ cos θ. Inserting into the expression above, one then arrives at the expression for L 2 in spherical coordinates, ( L 2 = h 2 2 θ + cot θ 2 θ ) sin 2. θ φ 2 By comparing with the Laplaceian in spherical coordinates, 2 = 2 r r r + 1 ( 2 r 2 θ + cot θ 2 θ + 1 sin 2 θ we see that the angular part of 2 can be expressed in terms of L 2 : 2 = 2 r r r L 2 h 2 r 2. 2 ), φ 2 Using this expression we finally arrive at a practical expression for the Hamiltonian for a spherically symmetric potential: ( Ĥ = h2 2 2m r r ) + r L 2 2mr 2 + V (r).

18 TFY4215/FY1006 Tillegg Rigid rotator (Hemmer 5.5, B&J 6.4) 5.3.a What is a rigid rotator Let us consider a particle of mass µ which is forced to move on the (two-dimensional) surface of a sphere with radius r 0. We can imagine that the particle is attached to one end of a weightless rod (the other end being attached to the origin). This is called a rigid rotator. Such a rotator can be used to model the rotational motion of a two-atomic molecule. Then µ stands for the reduced mass of the two atoms, 6 µ = m 1m 2 m 1 + m 2, The radius r = r 0 is the (equilibrium) distance between the two nuclei, and the angles θ and φ specify the orientation of the vector from m 2 to m 1. For the rigid rotator there is no radial motion, only rotation. The particle with mass µ is moving perpendicularly to r; so that p = p ( r). With L = r p = r 0 p, the (kinetic) energy then is given classically by the expression E = K = p2 2µ = L2 2µr 2 0 L2 2I (I = µr 2 0 = moment of inertia). The corresponding Hamiltonian is Ĥ = K = L 2 2µr 2 0 = L 2 2I. (T5.32) (Here we note that there is no potential energy, and no radial contribution to the kinetic energy. Thus the Hamiltonian in (T5.16) in this case is reduced to Ĥ = K L = L 2 /(2µr 2 0).) 5.3.b Energy eigenvalues and eigenfunctions From the results in section 5.2.c it then follows that the energy levels of the rotator are E l = l(l + 1) h2 2I. (T5.33) 6 By considering the two-atomic molecule as a rigid rotator we are neglecting among other things the vibrational motion (Cf the discussion in Tillegg 3.)

19 TFY4215/FY1006 Tillegg 5 19 Each of these levels is degenerate, with g l = 2l + 1 independent states. As energy eigenfunctions we can then use ψ lm = Y lm (θ, φ), m = 0, ±1, ±2,, ±l. (T5.34) Note that Y (θ, φ) 2 now gives the probability distribution for the direction ˆr r r = r 1 r 2 r 1 r 2 from nucleus 2 to nucleus 1. 7 Note also that the normalization condition is 2π π dφ sin θ dθ Y (θ, φ) 2 Y (θ, φ) 2 dω = c Radiative transitions. Rotational spectra When such a molecule exchanges energy with its surroundings, it can take place e.g. via absorption or emission of photons. It can be shown that transitions between states of the type (T5.34) almost always respect the selection rule l = ±1, meaning that l very rarely changes by more than one unit (±1). exchanged photons have energies given by (T5.35) This implies that the E l E l E l 1 = h2 I l, l = 1, 2,... (T5.36) These energies typically are of the order of ev, and correspond to photons in the infrared frequency range. We note that the emitted photons give a frequency spectrum with equidistant lines, a so-called rotational spectrum. Experiments agree well with this simple theory, when one measures the emission spectra from hot two-atomic gases at moderate temperatures. (See Hemmer p 99.) This agreement may be somewhat surprising because, when treating the two atoms as point particles with a constant distance r 0, we have in fact simplified in a seemingly rather brutal way. Among other things, we have neglected the following effects: (1)Vibration. The distance between the two nuclei actually is not constant, but may vary slightly about the (classical) equilibrium distance r 0. For small deviations this corresponds to a one-dimensional harmonic oscillation, as discussed in Tillegg 3. The distance hω between the energy levels of such an oscillator typically is of the order of 0.1 ev, which is significantly larger than the energy amounts ( ev ) which are needed to excite the rotational degrees of freedom. In statistical mechanics one learns that the probability of finding a molecule in an excited vibrational state is negligible when the temperature is so low that k B T << hω, that is, when T << hω k B 0.1eV ev/k 103 K. 7 From (T5.28)we see that the probability density Y lm (θ, φ) 2 is always symmetric with respect to inversion about the origin. This means among other things that the probability of finding the particle on e.g. the upper half-sphere is 50 percent when the rotator is in a state described by one of the functions Y lm.

20 TFY4215/FY1006 Tillegg 5 20 (k B = ev/k is Boltzmann s constant). Thus at low temperatures there are practically no excitations of the vibrational degree of freedom. This way the system becomes effectively two-dimensional, as a rotator, with r 0 equal to the equilibrium distance between the two nuclei. 8 (2) Rotation about the axis through the two nuclei. A rigid body actually has three rotational degrees of freedom. Two of these were taken into account in the point-like rigid rotator treated above. The third degree of freedom lies in the possibility of a rotation about the axis through the two nuclei. However, this degree of freedom is not excited at low temperatures. The reason is that the moments of inertia of the electrons with respect to this axis are very small (a factor 10 4 smaller than for the rigid rotator above). The moments of inertia of the nuclei with respect to this axis are even smaller. Based on this we can state that the excitation energy for the electronic degree of freedom will be of the order of electronvolts. (Cf the hydrogen atom, for which it takes 10.2 ev to excite the electron from the ground state to the rotational 2p-state.) This is the reason why the simple rigid-rotator model above works so well at low temperatures. At higher temperatures, one finds spectra corresponding to excitation of both rotational and vibrational degrees of freedom. 5.3.d Comments (i) Physically, we can prepare states with well-defined L 2 and L z for the rotator by measuring these observables. According to the measurement postulate, we must then necessarily get a pair of eigenvalues h 2 l(l + 1) and hm, and after the measurement the rotator will be left in the correspopnding state Y lm. (ii) We should keep in mind that the rotator (as well as the hydrogen atom or any other system described by a spherically symmetric potential V (r)) has an underlying spherical symmetry: There is no preferred direction. Thus, physically there is nothing particular about the z-axis. Instead of L z we can therefore just as well measure any other component of L together with L, e.g. L x, or for that matter an arbitrary component L ˆn. From the symmetry we realize that the simultaneous eigenstates in which the system is left after such a measurement will look the same seen from the ˆn-axis as Y lm seen from the ẑ-axis. Examples are the angular functions p x 3 4π x r and p y 3 4π which have the same appearance seen from respectively the ˆx-axis and the ŷ-axis as 3 z 3 p z 4π r = 4π cos θ = Y 10(θ, φ) has seen from the ẑ-axis. More generally, 3 pˆn 4π ˆn r 3 r = 4π cos θ nr y r, (T5.37) (T5.38) 8 Note that at room temperature, T 300K, we have k B T (= 0.025eV ), which is much larger than the amount needed to excite the rotaional degree of freedom ( ev ). In statistical mechanics one learns that the average rotational energy under such circumstances is equal to k B T. This implies, e.g., that air molecules at room temperature will have average orbital angular-momentum quantum numbers of the order of l 10.

21 TFY4215/FY1006 Tillegg 5 21 has the same appearance seen from the ˆn-axis. (iii) All these states are p-states (l = 1), and from (T5.30) it then follows that they must be linear combinations of the set Y 1m, m = 0, ±1. This holds both for and it also holds for p x = 1 2 (Y 11 Y 1, 1 ) and p y = i 2 (Y 11 + Y 1, 1 ), pˆn 3 4π ˆn r r = n xp x + n y p y + n z p z =. Moral : The choice of the ẑ-axis as quantization axis and the resulting set of states Y lm are not unique. However, this is no problem; all possible angular functions that might be of interest can be expanded in terms of the set Y lm, due to the completeness of this set. 9 (iv) In chemistry one learns that directed bonds and hybridized orbitals are important for the structure of molecules. Angular functions are an important part of the discussion of such orbitals. (v) Because the probability distributions Y lm (θ, φ) 2 are independent of the asimuthal angle φ, that is, are rotationally symmetric with respect to the ẑ-axis, they can be illustrated by polar diagrams, where the size of Y lm 2 as a function of θ is marked along a line in the direction θ. See p 286 in B&J, which shows such polar diagrams for l = 0, 1, 2. To visualize the rotational symmetry we can imagine that these polar diagrams are rotated around the z-axis. We then obtain three-dimensional polar diagrams, where each of the curves in the book creates a two-dimensional surface. For Y 00 2, e.g., this surface is a sphere with radius 1/4π. The figure below shows sketches of such surfaces for Y 00 2, p z 2 = Y 10 2 = cos 2 θ 3/4π, together with Y 20 2 = (3 cos 2 θ 1) 2 5/16π. By rotating the sketch for p z 2 by 90 degrees with respect to the y-axis (or the x-axis) you can obtain the corresponding surfac for p x 2 (or p y 2 ). For the real functions p x, p y and p z it can even be interesting to make polar diagrams for the functions themselves (not only for the absolute squares as above). Since these functions take both positive and negative values, we choose to show their absolute values p x, p y and p z. The figure below shows the polar diagram for 4π/3 p z = cos θ (which simply consists of two circles touching each other), together with the corresponding two-dimensional surfaces, which are rotationally symmetric with respect to respectively the x-, y- and z axes. The signs indicate where the functions p x etc are positive or negative. These signs at the same time tells us that these functions have parity 1; they are antisymmetric with respect to the origin. 9 By superposing many such angular functions with sufficiently large quantum numbers, it is possible to describe a wave packet with small extension in angular space. This is relevant if we want to describe a rotator with a reasonably well-defined direction θ, φ at a given point in time.

22 TFY4215/FY1006 Tillegg 5 22 The p-state pˆn mentione above is of the same type, with the unit vector ˆn as symmetry axis. Angular functions of this type are relevant in the discussion of hybridized orbitals. (vi) The degeneracy in this problem, the fact that we have g l = 2l + 1 states with the same energy, is caused by the spherical symmetry. We have a free rotator, where no forces act along the spherical surface. Thus no particular direction is favoured. This degeneracy, where the energy levels are independent of the quantum number m, is common for all spherically symmetric potentials. 5.4 Motion in spherically symmetric potential. Radial equation (Hemmer 5.6, B&J 7.2, Griffiths 4.1.) When we allow also for motion in the radial direction, in a spherically symmetric potential V (r), we can return to the Hamiltonian (T5.16), on the form 10 ( Ĥ = K + V (r) = h2 2 2m r r ) + r L 2 2mr 2 + V (r). (T5.39) Here, we note that the first term on the right is the radial part K r of the operator K for the kinetic energy. Note that the angular part, K L = L 2 /(2mr 2 ), has the same form as for the rotator. Since the angular differentiations in L 2 and L commute both with /r and with the r-dependent factors, we note that Ĥ, L 2 and L z are a set of commuting operators. The simultaneous eigenfunctions of these operators must be proportional to both e imφ and Θ lm (θ), that is, to Y lm = Θ lm (θ)e imφ : ψ Elm (r, θ, φ) = R El (r)y lm (θ, φ) = R El (r)θ lm (θ)e imφ. (T5.40) Here we have succeeded in factorizing the dependence on the three variables r, θ and φ. You should also notice that the angular functions are the same, no matter which potential V (r) we are considering. 10 Here, we take the risk of using the old symbol m for the mass, even if the same m is also used for the magnetic quantum number.

24 TFY4215/FY1006 Tillegg 5 24 The figure shows the effective potentials for l = 1 and l = 2, for the case that the potential itself is a spherical well. In general we must have some kind of well in order for bound states to exist. As illustrated in the figure, the effective well becomes smaller the larger the angular momentum (l) is. From these sketches, it should be obvious that the ground state for a potential V (r) will in general be a so-called s-state, that is, a state with l = 0, because the effective well for l = 0 is deepest. Examples are the ground states of the hydrogen atom and of the three-dimensional harmonic oscillator. For a given value of l, that is, for a given effective potential Veff(r), l we can expect to find several radial functions u(r) corresponding to bound states, provided that the effective well is sufficiently big. Of these states, the one with the lowest energy will have zero nodes (no zero), that is, it will have minimal curvature and minimal radial kinetic energy. For the other states, the energy will increase with the number of nodes, which is called the radial quantum number (n r ). For a given l, we thus find a set of states with energies E nrl such that E 0l < E 1l < E 2l < E 3l < E 4l. The moral is that the energy eigenvalues for a spherically symmetric potential normally depends both on l and the radial quantum number n r, which gives the number of zeros in the radial function, but not on the magnetic quantum number m, which does not enter the radial equation. This means that the degeneracy of such an energy level is equal to the number of m-values, g l = 2l + 1. This degeneracy is characteristic for the spherical symmetry. For the Coulomb potential ( 1/r), something special happens. For this potential, the energies turn out to depend only on the sum of the quantum numbers l and n r. This means that several l-values give the same energy, so that the degree of degeneracy becomes larger. More about that later. If V (r) approaches zero for large r (as the centrifugal term does), the bound states must have negative energies, E < 0. Then u (2mE/ h 2 ) u κ 2 u for large r, and it follows that the asymptotic behaviour for large r is given by u e ±κr. Here only the exponentially

25 TFY4215/FY1006 Tillegg 5 25 decreasing solution is acceptable as an eigenfunction. Thus we have 11 u(r) e κr, κ 2mE/ h 2 (r ). (T5.43) For positive energies, E > 0, one finds a continuous spectrum of unbound states. For large r we then have u (2mE/ h 2 ) u k 2 u, and it follows that the asymptotic behaviour for large r is given by u(r) sin(kr + α l ). Such solutions are relevant when one considers scattering on a potential V (r). It is also possible to find the general behaviour of the radial functions u l (r) for small r, without specifying the potential. Provided that the potential V (r) is less singular than 1/r 2 (that is, approaches infinity slower than 1/r 2 ), we note that (for l > 0) the centrifugalb term will dominate over the other terms in the differential equation for small r, so that d 2 u l(l + 1) u (r 0). dr2 r 2 By setting u(r) = c 0 r s + c 1 r s+1 +, we can then determine the exponent s in the dominant power for small r. We find the two solutions s = l + 1 and s = l. The last one gives a non-normalizable wave function, which is not acceptable. (The integral ψ 2 d 3 r = 0 [u(r)]2 dr does not exist.) Thus the dominant power in u(r) for small r is given by u l (r) c 0 r l+1 dvs R l (r) c 0 r l (r 0). (T5.44) This argument, which holds for bound as well as unbound states, is valid for l 1. However, it can be shown that the result holds also for l = 0 : 12 u l=0 (r) c 0 r 1 = c 0 r, that is, R l=0 c 0 (r 0). The factor r l+1 implies that the wave function for small r becomes more suppressed the larger l is. This can also be understood qualitatively from the diagram above: We see that the effective potentials make the region for small r more and more forbidden the larger l is. 5.5 The Coulomb potential (Hemmer 5.7, B&J 7.5, Griffiths 4.2.) 11 For the ground state of the Coulomb potential, ψ = (πa 3 0) 1/2 exp( r/a 0 ), the exact result is u r exp( κr). It turns out that the solutions u l for the Coulomb potential in general are given by the exponential function multiplied by a polynomial in which the dominat power for large r is r n, where n is the principal quantum number. Thus, asymptotically we have u l (r) C r n e κr (r ). 12 The ground state ψ = (πa 3 0) 1/2 exp( r/a 0 ) of the hydrogen atom provides an example of this behaviour.

26 TFY4215/FY1006 Tillegg a Hydrogenlike system reduced to one-particle problem The Coulomb potential V (r) = Ze2 1 4πɛ 0 r describes the interaction between two particles forming a hydrogenlike system. One of these particles usually is a nucleus with charge Ze, while the other one often (but not always) is an electron with charge e. Here the masses of the two particles are denoted respectively by M (the nuclar mass) and m 1 (the electron mass). As shown in an exercise, the relative motion of the two particles is described by the time-independent Schrödinger equation h2 2m 2 ψ(r) + V (r) ψ(r) = E ψ(r), where r is the relative coordinate (the distance vector from M to m 1 ) and m = m 1M M + m 1 = m m 1 /M is the reduced mass of this two-particle system. Formally, this Schrödinger equation describes a one-particle system, where a (fictitious) particle with the reduced mass m is moving in a static Coulomb potential. Thus, as far as the relative motion of the two particles is concerned, the system is effectively reduced to a one-particle system. (In the exercise it is shown that the motion of the center of gravity of the system is described by the Schrödinger equation for a free particle.) The central example is the hydrogen atom, which corresponds to Z = 1, m 1 = m e, M = m p and hence a reduced mass 13 m = m e 1 + m e /m p m e. The fact that m lies so close to m e simply means that no big mistake is made by neglecting the motion of the proton from the beginning, calculating as if the proton were at rest (cf an earlier exercise). However, as we have just seen, it is very easy to improve on this approximation, calculating with the reduced mass. This means e.g. that the binding energies are reduced by 0.05 %. In an introductory course in chemical physics, the hydrogen atom is a very important topic for several reasons: This system is the first testing ground for any theory which aims to explain the structure of atoms and molecules 13 Other examples are the ions He + (Z = 2), Li ++ (Z = 3) etc, which all are one-electron systems.

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