I wanted to write up an introductory description of the Standard Model of particle physics, but the project has gotten out of hand as I want to start from first principles and I forget that from there, the trail is long.
The gauge group of the Standard Model is SU(3)xSU(2)xU(1). These objects, for example, SU(3), are Lie algebras, which are mathematical objects that can be derived from Lie groups. In this post we will discuss a particular example of a Lie group, the rotation group, and its application to differential equations, but we will postpone using the language of Lie groups (and algebras) to the next post, and then eventually gauge principles. For this post, I will discuss only how it is that quantum numbers show up in quatnum mechanics, that is, from solutions to differential equations such as the Schroedinger’s equation, with initial conditions subject to a symmetry.
A group is a set of objects (we will call “elements” ) that have a group relation between them. A group relation is a special sort of associative binary product. For reasons that will become obvious when we discuss Lie groups, we write our binary operation as a multiplication, for example, if A and B are elements of the group, we call their product AB, which doesn’t have to be commutative. There must be a “unit” which when it multiplies all the other group elements leaves them unchanged, and each element of the group has an inverse. The non-zero complex numbers with the binary relation “multiplication” is a good example of a group. Various sets of matrices under multiplication provide more examples of groups. Mathematicians will give examples of various matrices where addition can be used as the “binary operation”, but this would be unnecessarily confusing when we work with Lie algebras. From a physical point of view, very good examples of Lie groups are symmetries of spacetime.
One generally assumes that space is isotropic, that is, that there is no preferred direction. To specify a direction in space we can use a vector in Cartesian coordiantes: (x,y,z). To specify a rotation, we can specify where the unit basis vectors (0,0,1), (0,1,0), and (1,0,0) are sent by the rotation. There is some minor technical confusion based on whether or not we treat space inversions, (x,y,z) -> (-x,-y,-z) in our group, we will not.
Given two rotations, we define the binary operation on them as consecutive rotations. These form a group where the unit is the rotation that doesn’t rotate anything. This is the Rotation group, SO(3). Note that we follow the (sloppy) physics tradition and use the same notation for a Lie group and its associated algebra. Thus SO(3) can mean the algebra of the rotation group as well as the rotation group itself. Mathematicians might use so(3) for the algebra. But we won’t get to the Lie algebra in this post.
Lie groups and Lie algebras can be used to quickly provide information about the solutions to differential equation problems when the problems are set up with initial conditions that satisfy a symmetry. The result of this is that one obtains quantum numbers, and one obtains ways of computing ratios of computations with different solutions (i.e. Clebsch-Gordan coefficients). We will illustrate this process by looking at solutions to Laplace’s equation subject to the rotational symmetry. First, we find the quantum numbers of Laplace’s equation.
Example: Laplace’s Equation and Spherical Harmonics
Laplace’s equation in 3-dimensions in Cartesian coordinates is: Laplace’s equation is linear, that is, given two solutions, their sum is also a solution. The equation is symmetric under rotations (as well as under certain discrete transformations that we will not need to discuss). To explore the behavior of solutions to Laplace’s equation under rotations, we naturally write it in spherical coordinates. The method of solving differential equations known as separation of variables can be used to write general solutions to Laplace’s equation in spherical coordinates. We look for solutions to Laplace’s equation that separate into radial and angular parts, that is, we write . These become three separate differential equations which can be solved. General solutions can then be stitched together by linear superposition of products of the separated solutions.
When one applies this machinery to Laplace’s equation, the solutions to the angular differential equations are known as spherical harmonics. The spherical harmonics can be written as: where are integer constants that satisfy and , and are the associated Legendre functions. If the Laplace equation were used in quantum mechanics, physicists would call the integers “quantum numbers” and the spherical harmonics “states.” To illustrate the relationship, we will use this phrase though it is a little strained. The possible values of , that is, the quantum numbers of the spherical harmonics, can be plotted as follows:
In quantum mechanics as in spherical harmonics, one writes a general solution to the differential equation as a sum over states with different quantum numbers. Since the differential equations are linear, such sums are always solutions. That states with the given quantum number are sufficient for the arbitrary general solution is due to the fact that the spherical harmonics are complete. In addition they are normalized (somewhat arbitrarily) and orthogonal. That is, they form an orthonormal basis for the general solutions to the differential equation.
Of the spherical harmonics, only the first one, with is symmetric under rotation. Some of the other, asymmetric, spherical harmonics can be rotated into each other. Since the differential equation possesed rotational symmetry, such pairs of spherical harmonics must be equivalent in their characteristics.
The spherical harmonics are:
One can take linear combinations of these three orthonormal states to produce linear combinations that are still orthonormal, but are real valued, and that are related to each other by rotations. Labeling the according to the orientation direction, the linear combinations are:
The “2” in the above definitions refers to the energy level. The energy level comes from the radial part of the differential equation (Schroedinger’s) and is not a part of the spherical harmonic. The lowest energy p-wave has an energy (i.e. radial) quantum number of n=2. I’ve labeled the p-waves with 2 for this reason. There are also , etc., p-waves, and since the spherical harmonic does not include the radial wave function, all these waves are equivalent as far as the spherical harmonic goes.
The complex spherical harmonics, , are commonly used in quantum mechanics as they are convenient for the raising and lowering operators (that change ). Chemists need geometric intuition and use the forms of the spherical harmonics in describing atomic orbitals. Higher spherical harmonics are not as simple as these p-wave orbitals, but they all have geometric interpretations.
Since the p-wave orbitals are rotations of each other, their physical characteristics are identical. In terms of quantum mechanics, they share the same quantum numbers (other than of course), and form a triplet. These spherical harmonic triplets can be described geometrically, as related to the directions x, y, and z, or described in raising / lowering operator form, by the quantum number .
One could take the orbital / spherical harmonic, and rotate to any arbitrary direction to produce an orbital oriented in that direction. We could call it the orbital. Such an orbital can be written as a linear combination of , or as a linear combination of . On the other hand, these sets of spherical harmonics are linearly independent. They form an orthogonal basis for this type of spherical harmonic.
For any given , these (or p-wave) spherical harmonics are identical and the physical characteristics of the states are identical. For different , they are different so we cannot assume that the physical characteristics are the same. Since shorter wavelengths are associated in quantum mechanics with higher energies, physicists expect that higher are associated with higher energy particles.
The spherical harmonics are specific to Laplace’s equation. Other differential equations, even if they possess rotational symmetry, will generally have different orthonormal states. Schrödinger’s time-independent equation: , is an inhomogeneous form of the Laplace equation. The general solutions to an inhomogeneous differential equation can be obtained from the general solutions to the homogeneous differential equation (in this case Laplace’s equation), plus any specific solution to the inhomogeneous equation. Because of this relationship, the quantum numbers of the spherical harmonics can appear as quantum numbers in Schroedinger’s equation, if , the potential energy, is a function of radius only. This is true even though, because of the inhomgeneity, the solutions of Schroedinger’s equation are not solutions of Laplace’s equation. This case applies to atomic hydrogen, and the wave structure of the hydrogen atom uses spherical harmonics, and consequently the quantum numbers of the excited states of hydrogen follow the same quantum numbers as Laplace’s equation. Graphing the quantum numbers in 3 dimensions, the quantum numbers look like:
The presence of rotational symmetry, along with the nice geometric form for the p-wave spherical harmonics, might lead one to suppose that all the spherical harmonics might be expressed in forms where the orthonormal functions are equivalent under rotation, but this is not the case.
Next: A more abstract way of finding quantum numbers, Lie groups and Lie algebras, with examples from the rotation group.