Bound States in Three Dimensions – Orbital Angular Momentum

22.101 Applied Nuclear Physics Lecture 4 Bound States in Three Dimensions – Orbital Angular Momentum ________________________________________________________________________ References -R. L. Liboff, Introductory Quantum Mechanics (Holden Day, New York, 1980). L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955). P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953). ________________________________________________________________________ We will now extend the bound-state calculation to three-dimensional systems. The problem we want to solve is essentially the same as before, except that we wish to determine the bound-state energy levels and corresponding wave functions for a particle in a three-dimensional spherical well potential. Although this is a three-dimensional potential, we can take advantage of its symmetry in angular space and reduce the calculation to an equation still involving only one variable, the radial distance between the particle position and the origin. In other words, the spherical potential is still a function of one variable, V (r) = −Vo r < ro (4.1) =0 otherwise Here r is the radial position of the particle relative to the origin. Any potential that is a function only of r, the magnitude of the position r and not the position vector itself, is called a central-force potential. As we will see, this form of the potential makes the solution of the Schrödinger wave equation particularly simple. For a system where the potential or interaction energy has no angular dependence, one can reformulate the problem by factorizing the wave function into a component that involves only the radial coordinate and another component that involves only the angular coordinates. The wave equation is then reduced to a system of uncoupled one-dimensional equations, each describing a radial component of the wave function. As to the justification for using a 1 central-force potential for our discussion, this will depend on which properties of the nucleus we wish to study. We again begin with the time-independent wave equation ⎡ h2 2 ⎤ ∇ + V (r)⎥ψ (r) = Eψ (r) ⎢− ⎣ 2m ⎦ (4.2) Since the potential function has spherical symmetry, it is natural for us to carry out the analysis in the spherical coordinate system rather than the Cartesian system. A position vector r then is specified by the radial coordinate r and two angular coordinates, θ and ϕ , the polar and azimuthal angles respectively, see Fig. 1. In this coordinate system Fig. 1. The spherical coordinate system. A point in space is located by the radial coordinate r, and polar and azimuthal angles θ and ϕ . the Laplacian operator ∇ 2 is of the form ∇ 2 = Dr2 + 1 ⎡ − L2 ⎤ ⎢ ⎥ r2 ⎣ h2 ⎦ (4.3) where Dr2 is an operator involving only the radial coordinate, 2 Dr2 = 1 ∂ ⎡ 2 ∂⎤ r r 2 ∂r ⎢⎣ ∂r ⎥⎦ (4.4) and the operator L2 involves only the angular coordinates, − ∂ ⎤ 1 ∂ ⎡ 1 ∂2 L2 = sin + θ ∂θ ⎥⎦ sin 2 θ ∂ϕ 2 h 2 sin θ ∂θ ⎢⎣ (4.5) In terms of these operators the wave equation (4.2) becomes ⎡ h2 2 ⎤ L2 D − + + V (r)⎥ψ (rθϕ ) = Eψ (rθϕ ) r ⎢ 2 2mr ⎣ 2m ⎦ (4.6) For any potential V(r) the angular variation of ψ is always determined by the operator L2/2mr2. Therefore one can study the operator L2 separately and then use its properties to simplify the solution of (4.6). This needs to be done only once, since the angular variation is independent of whatever form one takes for V(r). It turns out that L2 is very well known (it is the square of L which is the angular momentum operator); it is the operator that describes the angular motion of a free particle in three-dimensional space. We first summarize the basic properties of L2 before discussing any physical interpretation. It can be shown that the eigenfunction of L2 are the spherical harmonics functions, Ylm (θ , ϕ ) , L2Ylm (θ , ϕ ) = h 2 l(l + 1)Ylm (θ , ϕ ) (4.7) where ⎡ 2l + 1 (l − m)! ⎤ Y (θ , ϕ ) = ⎢ ⎥ ⎢⎣ 4π (l + m )!⎥⎦ m l 1/ 2 Plm (cos θ )e imϕ (4.8) and Plm ( µ ) = (1 − µ 2 ) m / 2 d l + m ( µ 2 −1) l l l+m 2 l! dµ (4.9) 3 with µ = cos θ . The function Plm ( µ ) is called the associated Legendre polynomials, which are in turn expressible in terms of Legendre polynomials Pl ( µ ) , P ( µ ) = (1 − µ ) m l m /2 d dµ m m Pl ( µ ) (4.10) with Po(x) = 1, P1(x) = x, P2(x) = (3x2 – 1)/2, P3(x) = (5x3-3x)/2, etc. Special functions like Ylm and Plm are quite extensively discussed in standard texts [see, for example, Schiff, p.70] and reference books on mathematical functions [More and Feshbach, p. 1264]. For our purposes it is sufficient to regard them as well known and tabulated quantities like sines and cosines, and whenever the need arises we will invoke their special properties as given in the mathematical handbooks. It is clear from (4.7) that Ylm (θ , ϕ ) is an eigenfunction of L2 with corresponding eigenvalue l(l + 1)h 2 . Since the angular momentum of the particle, like its energy, is quantized, the index l can take on only positive integral values or zero, l = 0, 1, 2, 3, … Similarly, the index m can have integral values from - l to l , m = - l , - l +1, …, -1, 0, 1, …, l -1, l For a given l , there can be 2 l +1 values of m. The significance of m can be seen from the property of Lz, the projection of the orbital angular momentum vector L along a certain direction in space (in the absence of any external field, this choice is up to the observer). Following convention we will choose this direction to be along the z-axis of our coordinate system, in which case the operator Lz has the representation, L z = −ih∂ / ∂ϕ , and its eignefunctions are also Ylm (θ , ϕ ) , with eigenvalues mh . The indices l and m are called quantum numbers. Since the angular space is two-dimensional 4 (corresponding to two degrees of freedom), it is to be expected that two quantum numbers will emerge from our analysis. By the same token we should expect three quantum numbers in our description of three-dimensional systems. We should regard the particle as existing in various states which are specified by a unique set of quantum numbers, each one is associated with a certain orbital angular momentum which has a definite magnitude and orientation with respect to our chosen direction along the z-axis. The particular angular momentum state is described by the function Ylm (θ , ϕ ) with l known as the orbital angular momentum quantum number, and m the magnetic quantum number. It is useful to keep in mind that Ylm (θ , ϕ ) is actually a rather simple function for low order indices. For example, the first four spherical harmonics are: Y00 = 1/ 4π , Y1−1 = 3 / 8π e −iϕ sin θ , Y10 = 3 / 4π cos θ , Y11 = 3 / 8π e iϕ sin θ Two other properties of the spherical harmonics are worth mentioning. First is that {Ylm (θ , ϕ ) }, with l = 0, 1, 2, … and − l ≤ m ≤ l , is a complete set of functions in the space of 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ 2π in the sense that any arbitrary function of θ and ϕ can be represented by an expansion in these functions. Another property is orthonormality, π 2π 0 0 m* m' ∫ sin θdθ ∫ dϕYl (θ , ϕ )Yl' (θ , ϕ ) = δ ll'δ mm' (4.11) where δ ll' denotes the Kronecker delta function; it is unity when the two subscripts are equal, otherwise the function is zero. Returning to the wave equation (4.6) we look for a solution as an expansion of the wave function in spherical harmonics series, ψ (rθϕ ) = ∑ Rl (r)Ylm (θ , ϕ ) (4.12) l,m 5 Because of (4.7) the L2 operator in (4.6) can be replaced by the factor l(l + 1)h 2 . In view of (4.11) we can eliminate the angular part of the problem by multiplying the wave equation by the complex conjugate of a spherical harmonic and integrating over all solid angles (recall an element of solid angle is sin θdθdϕ ), obtaining ⎡ h 2 2 l(l + 1)h 2 ⎤ Dr + V (r) + ⎢− ⎥ Rl (r) = ERl (r) 2mr 2 ⎣ 2m ⎦ (4.13) This is an equation in one variable, the radial coordinate r, although we are treating a three-dimensional problem. We can make this equation look like a one-dimensional problem by transforming the dependent variable Rl . Define the radial function u l (r) = rRl (r) (4.14) Inserting this into (4.13) we get ⎤ h 2 d 2 u l (r) ⎡ l(l + 1)h 2 − + + V (r) ⎢ ⎥u l (r) = Eu l (r) 2 2m dr 2 ⎣ 2mr ⎦ (4.15) We will call (4.15) the radial wave equation. It is the basic starting point of threedimensional problems involving a particle interacting with a central potential field. We observe that (4.15) is actually a system of uncoupled equations, one for each fixed value of the orbital angular momentum quantum number l . With reference to the wave equation in one dimension, the extra term involving l(l + 1) in (4.15) represents the contribution to the potential field due to the centrifugal motion of the particle. The 1/r2 dependence makes the effect particularly important near the origin; in other words, centrigfugal motion gives rise to a barrier which tends to keep the particle away from the origin. This effect is of course absent in the case of l = 0, a state of zero orbital angular momentum, as one would expect. The first few l states usually are the only ones of interest in our discussion (because they tend to have the lowest energies); they are given special spectroscopic designations, 6 notation: s, p, d, f, g, h, … l = 0, 1, 2, 3, 4, 5, … where the first four letters stand for ‘sharp’, ‘principal’, ‘diffuse’, and ‘fundamental’ respectively. After f the letters are assigned in alphabetical order, as in h, i, j, … The wave function describing the state of orbital angular momentum l is often called the l th partial wave, ψ l (rθϕ ) = Rl (r)Ylm (θϕ ) (4.16) Notice that in the case of s-wave the wave function is spherically symmetric since Y00 is independent of θ and ϕ . Interpretation of Orbital Angular Momentum In classical mechanics, the angular momentum of a particle in motion is defined as the vector product, L = r × p , where r is the particle position and p its linear momentum. L is directed along the axis of rotation (right-hand rule), as shown in Fig. 2. Fig. 2. Angular momentum of a particle at position r moving with linear momentum p (classical definition). L is called an axial or pseudovector in contrast to r and p, which are polar vectors. Under inversion, r → −r , and p → − p , but L → L . Quantum mechanically, L2 is an operator with eigenvalues and eigenfunctions given in (4.7). Thus the magnitude of L is h l(l + 1) , with l = 0, 1, 2, …being the orbital angular momentum quantum number. We can specify the magnitude and one Cartesian component (usually called the z­ 7 component) of L by specifying l and m, an example is shown in Fig. 3. What about the x- and y-components? They are undetermined, in that they cannot be observed Fig. 3. The l(l + 1) = 5 projections along the z-axis of an orbital angular momentum with l = 2. Magnitude of L is 6h . simultaneously with the observation of L2 and Lz. Another useful interpretation is to look at the energy conservation equation in terms of radial and tangential motions. By this we mean that the total energy can be written as E= L2 1 1 m(v r2 + vt2 ) + V = mv r2 + +V 2 2 2mr 2 (4.17) where the decomposition into radial and tangential velocities is depicted in Fig. 4. Eq.(4.17) can be compared with the radial wave equation (4.15). Fig. 4. Decomposing the velocity vector of a particle at position r into radial and tangential components. 8 Thus far we have confined our discussions of the wave equation to its solution in spherical coordinates. There are situations where it will be more appropriate to work in another coordinate system. As a simple example of a bound-state problem, we can consider the system of a free particle contained in a cubical box of dimension L along each side. In this case it is clearly more convenient to write the wave equation in Cartesian coordinates, − h2 ⎡ ∂2 ∂2 ∂2 ⎤ + + ⎥ψ ( xyz ) = Eψ ( xyz ) ⎢ 2m ⎣ ∂x 2 ∂y 2 ∂z 2 ⎦ (4.17) 0