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In this video, I go over a deep dive into the famous spherical harmonics, which are the solutions to Laplace’s equation in spherical coordinates and are very prevalent in physics, especially in the quantized configurations of atomic orbitals or the electron charge distribution around the atom's nucleus. Laplace's equation is the sum of the 2nd partial derivatives of a function with respect to the x, y, z terms (in rectangular coordinates) and is equated to zero. Converting this equation into spherical coordinates (r, θ, φ) (which I do by first obtaining the corresponding Laplace operator or Laplacian in polar coordinates), and then solving it produces solutions known as solid spherical harmonics. Focusing only on the angular terms θ and φ obtains the spherical harmonics, which, when plotting only the real terms (ignoring the imaginary terms), and distorting radial terms proportional to the magnitude of values on the sphere, we obtain the distinct 3D lobes!
#math #sphericalharmonics #calculus #atomicorbitals #science
Timestamps
- Intro – 0:00
- Topics to cover – 0:47
- Introduction to Spherical Harmonics – 1:40
- Visualization of Real Spherical Harmonics – 25:58
- Atomic Orbitals – 31:24
- Magnetic Fields – 56:40
- Mathematical Review – 1:04:00
- Laplace Operator in Polar Coordinates – 1:14:07
- Laplace Operator in Spherical Coordinates – 2:09:00
- Solid Spherical Harmonics: Solutions to Laplace Equation in Spherical Coordinates – 2:50:38
- Associated Legendre Equation – 3:50:00
- Spherical Harmonics: Solution to the Angular Laplacian – 4:05:39
- Deriving the Real Spherical Harmonics – 4:16:44
- Graphing Real Spherical Harmonics in Desmos Calculator – 5:07:41
- Outro – 5:19:55
View Video Notes Below!
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Topics to Cover
- Introduction to Spherical Harmonics
- Visualization of Real Spherical Harmonics
- Atomic Orbitals
- Magnetic Fields
- Mathematical Review
- Laplace Operator in Polar Coordinates
- Laplace Operator in Spherical Coordinates
- Solid Spherical Harmonics: Solutions to Laplace Equation in Spherical Coordinates
- Associated Legendre Equation
- Spherical Harmonics: Solution to the Angular Laplacian
- Deriving the Real Spherical Harmonics
- Graphing Real Spherical Harmonics in Desmos Calculator
Introduction to Spherical Harmonics
Spherical harmonics are special functions defined on the surface of a sphere, and have many applications in physics.
Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics.
The Laplace Equation in rectangular coordinates (x, y, z) is shown below.
Note that the above equation describes steady-state conditions (no sources or sinks), i.e. where things are in "harmony", hence solutions are called "harmonic functions".
The above equation can be rewritten to obtain the Laplace Equation in spherical coordinates (r, θ, ɸ).
The physics convention. Spherical coordinates (r, θ, ɸ) as commonly used: radial distance r (slant distance to origin), polar angle θ (theta) (angle with respect to positive polar axis), and azimuthal angle ɸ (phi) (angle of rotation from the initial meridian plane).
Solving the above equation using separation of variables obtains the Solid Spherical Harmonics:
Where:
n = 0, 1, 2, 3, … (non-negative integer)
m = -n, -n+1, …, n (integer)
Pnm(t) is the Associated Legendre function with t = cos θ (so t ∈ [−1, 1]):
Pn(t) is the Ordinary Legendre Polynomials (Rodrigues' formula):
Or compacted together:
The general solutions to the Laplace Equation are called Solid Spherical Harmonics as they contain both the radial term (r) and angular terms (θ, ɸ).
If instead we remove the radial term so that we obtain only the angular terms, then such solutions are termed Spherical Harmonics.
Where ∇2 (or Δ) is the Laplace operator or Laplacian:
Multiplying by r2 we get:
or
Where Λ2 is the angular Laplacian (or Laplace-Beltrami operator on the sphere):
Rewriting f:
Solving Y we obtain the complex Spherical Harmonics:
Where N is a normalization constant, usually:
To ensure the following integral (used in Quantum Mechanics for probability) is equal to 1:
or
The real Spherical Harmonics are defined such that the functions are real-valued everywhere, with no complex term i. The usual definition is given below, and is almost always used when displaying Spherical Harmonics visually, since they exclude the complex Phase factor eimφ (which is the rotating about the z-axis).
Plugging in all the values into the above definition obtains the simplified form below, in which the proportionality symbol ∝ is used to just show the functional shape, rather than the normalization constant.
or
The complete, correctly normalized real spherical harmonics would be something like below.
Where the exact value of the normalization constant is usually:
Thus the full general solution to Laplace's equation in spherical coordinates is a sum (or series) over all possible n and m:
Where:
Anm and Bnm are constants determined by boundry conditions (e.g., values of f. on a sphere or at infinity).
The Spherical Harmonics describe angular patterns that fit the sphere's geometry, and are "natural modes" or "basis functions" for solving problems on or around spheres, such as waves on a balloon or electron clouds in atoms.
Visualization of Real Spherical Harmonics
Visualizations of the first few real spherical harmonics are shown below.
Animation: Real (Laplace) spherical harmonics for (top to bottom) and (left to right)
Note that the spherical harmonics can be visualized by distorting the sphere radially proportional to the magnitude of the values on the sphere, thus obtaining the familiar "lobes".
Alternative picture for the real spherical harmonics Yℓm.
Figure: Blue portions represent regions where the function is positive, and yellow for when it is negative. The distance of the surface from the origin indicates the absolute value of Yℓm(θ,φ) in angular direction (θ,φ).
Atomic Orbitals
Interestingly, spherical harmonics resemble atomic orbitals, which describe the location and wave-like behavior of an electron in an atom (where ℓ is often used instead of n as I do in this video). Note that atomic orbitals are not simply radial distortions as in pure spherical harmonics, but they actually contain the radial term in the solutions.
Atomic orbitals are solutions to the Schrödinger equation, which is shown for the electron in a hydrogen atom (or a hydrogen-like atom) as well as its solution below.
Summary of terms via Grok AI and their physical meaning.
Using separation of variables in spherical coordinates:
Figure: Cross-sections of atomic orbitals of the electron in a hydrogen atom at different energy levels. The probability of finding the electron is given by the color, as shown in the key at upper right (the - / + indicate the phase of the Wave Function).
Figure: 3D views of some hydrogen-like (any atom or ion with a single electron) atomic orbitals showing probability density and phase (g orbitals and higher not shown)
The table below shows the real hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table up to radium.
"One should remember that these orbital 'states', as described here, are merely eigenstates of an electron in its orbit. An actual electron exists in a superposition of states, which is like a weighted average, but with complex number weights."
"The quantum state after a measurement is in an eigenstate corresponding to that measurement and the value measured."
"Numerical or analytic solutions in quantum mechanics can be expressed as pure states. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant pure states are identified by the principal quantum number n, the angular momentum quantum number ℓ, the magnetic quantum number m, and the spin z-component sz."
Instead of the complex orbitals described earlier, it is common, especially in the chemistry literature, to use real atomic orbitals. These real orbitals arise from simple linear combinations of complex orbitals. Real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics.
Animation: Continuously varying superpositions between the p1 and the px orbitals.
Note that the superpositions are the separated lobes (chosen so the complex terms are eliminated) and not the rainbow eigenstates (or "pure" fundamental solutions.
Figure: Atomic orbitals spdf m-eigenstates (right) and superpositions (left)
Fascinating stuff and a bit beyond the scope of this video and my head (for now)!
Magnetic Fields
Likewise, magnetic fields can be expressed as spherical harmonics, as expressed in Lori Gardi aka FractalWoman's paper.
For a magnetic multipole, the scalar potential outside sources (mathematical function describing magnetic fields away from a source of current) can be expressed as:
The magnetic field is then obtained as the negative gradient:
Measuring a single component of B (via a Hall sensor) gives a scalar projection that depends on the orientation, producing lobes and nodes analogous to orbital probability patterns.
Figure: Any complex magnetic field can be expressed in terms of a sum of weighted spherical harmonics of different orders. Order 0 is a constant value. Order 1 contains simple linear harmonics, here principally directed and labeled in the Cartesian coordinate system as Z, Y, and X respectively. The 2nd Order has five harmonics, called Z², ZX, X²-Y², ZY, and XY. Order 3 has seven harmonics, a few of which are illustrated. As the order increases so do both the number and complexity of the harmonics.
Mathematical Review
Laplace's Equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied it in 1786.
A partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
A multivariable function is a function with more than two arguments, such as the volume of a cone.
A partial derivative is the derivative of a multivariable function with respect to one of those variables, with the others held constant. The partial derivative of a function f(x, y, …) with respect to the x-direction is denoted by any of the following ways:
It can be thought of as the rate of change of the function in the x-direction.
For a function f(x, y, z) of three variables, Laplace's equation is
A function that obeys this equation is called a harmonic function.
The Laplace equation can be written using the Laplace operator or Laplacian (Δ) or the divergence operator (∇ ·) of the gradient (∇f).
The nabla symbol ∇, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.
The gradient is the vector field (or vector function) ∇f whose value at a given point gives the direction and rate of fastest increase.
In coordinate-free terms, the gradient of a function f(r) may be defined by:
where df is the total infinitesimal change in f for an infinitesimal displacement dr, and is seen to be maximal when dr is in the direction of the gradient ∇f.
Figure: The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).
In 3D coordinates, the divergence is just the sum of partial derivatives and uses the dot product notation as a mnemonic device to aid in remembering.
Image: The divergence of different vector fields in 2 dimensions.
Laplace Operator in Polar Coordinates
I will generally follow the derivations shown in videos by Michael Dabkowski from the Mike, the Mathematician YouTube channel for:
- Converting the Laplacian in Polar Coordinates
- Converting the Laplacian into Spherical Coordinates
- Solving the Laplacian to obtain Spherical Harmonics.
I have also used some help from Grok AI for when the above videos are lacking.
To help in obtaining the Laplace Operator or Laplacian in Spherical Coordinates, lets first obtain it for the simpler, but related, polar coordinates.
Let's write out the first partial derivative of f in terms of x and y by using the Chain Rule for Partial Derivatives and its associated Tree Structure.
I will cover partial derivatives more closely in future videos, but the chain rule is similar to that for normal derivatives from my earlier video.
We can solve the first partial derivative of f in terms of x as follows from our polar coordinates formulas.
Recall from my earlier video the derivative of inverse tangent.
Similarly, for the partial derivative of f in terms of y we have:
Now let's find the second partial derivative of f in terms of x:
Likewise, we have the second partial derivative of f in terms of y:
Note that, by symmetry of second derivatives (also called the Schwarz integrability condition or Clairaut's theorem), we have:
Now we can put it all together to obtain the Laplacian in polar coordinates.
Double-checking with Wikipedia, we obtain the correct answer.
Laplace Operator in Spherical Coordinates
Now let's determine the Laplacian in Spherical Coordinates by first converting rectangular coordinates into spherical coordinates.
Recall the distance formula in 3D (and 2D):
We will now apply the Laplacian in Poler Coordinates twice to find the Laplacian in Spherical Coordinates: once for the x-y terms, and once for the z-s terms.
Likewise for the z-s terms:
Adding these together we get:
Thus our Laplacian in spherical coordinates becomes:
Double-checking with Wikipedia, we obtain the correct answer, which lines up with the third variation of the formula!
Solid Spherical Harmonics: Solutions to the Laplace Equation in Spherical Coordinates
Solid spherical harmonics are the solutions to the Laplace Equation in spherical coordinates.
Laplace's Equation is just the Laplace operator or Laplacian set to equal zero.
From the Laplacian in spherical coordinates (r, θ, ɸ), we can write Laplace's Equation as follows:
We can apply separation of variables to obtain.
We now set the equations equal to their corresponding separation constants, to ensure a periodic solution, with the middle function to be set as the difference between the constants, thus ensuring the overall equation sums to zero.
The first constant -m2 (where m is an integer) is selected such that we have a periodic solution:
Note that the general solution allows for writing every possible solution as a linear combination of two basis functions.
This complex exponential function are equivalent to trigonometric equations via Euler's formula, as shown in my earlier video.
Thus the periodicity for our solution is:
The selection of the negative sign (-) in -m2 allows for our specific complex solution and the squared ensures we get m to be integer values and thus our solution is periodic about 2π or 360 degrees.
Let's examine the second separated function and its chosen constant n(n+1) (where n is a positive integer), which yields the Euler-Cauchy equation, which we can solve via substitution.
Again, note the general solution is the linear combination of the two basis functions.
Recall my earlier video on the Quadratic Formula.
Let's now solve the middle term in the Laplace equation, this term is key to Spherical Harmonics.
Associated Legendre Equation
This is the Associated Legendre Equation:
or
where ℓ and m are integers and refer to the degree and order (not a power) of the associated Legendre polynomial, respectively.
The above equation has nonzero solutions that are non-singular (or regular and means P remains finite, bounded, continuous, and differentiable) on [-1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values (replacing m with - m).
The solution is given in terms of the Legendre polynomials Pℓ(x) (m ≥ 0).
Note the factor (-1)m is known as the Condon-Shortley phase but some authors omit this phase factor (which just rotates / shifts the phase or angle).
Where Pℓ(x) is the solution of the Legendre equation:
By Rodrigues' formula we can obtain Pℓ(x) as:
Then Pmℓ(x) can be expressed as:
The derivation of the Legendre polynomials are outside the scope of this video but I may cover them in a future video, see Mike, the Mathematician's playlist for now.
Now plugging all of this together, we obtain our solution for the Solid Spherical Harmonics:
Spherical Harmonics: Solution to the Angular Laplacian
The solid spherical harmonics includes the radial terms r, while the spherical harmonics, in general, typically are just in terms of the angular terms θ and φ.
We can combine the radial terms as a function Y(θ,φ):
This Y is typically considered as the solution to the angular Laplacian (or Laplace-Beltrami operator on the sphere), which can be denoted as Λ2 (or denoted as Δs2).
Y is a solution to the above equation involving the angular Laplacian:
Deriving the Real Spherical Harmonics
The real Spherical Harmonics are defined such that the functions are real-valued everywhere, with no complex term i, with the typical definition shown as follows:
Note that the complex spherical harmonics are chosen using the pure complex exponentials (the basis solutions), rather than the linear combinations, and the Yn-m term is defined via the following complex conjugate form (flip the sign of the complex rotation angle term) because this makes important formulas and operations in quantum mechanics cleaner, simpler, and more elegant.
And using the proportional forms (ignoring the Normalization constant discussed earlier), we obtain.
Note the complex conjugation in the above definition.
For the Yn-m term, we use Pnm instead of Pn-m because the Associated Legendre equation depends on m2, and not m.
Note that this exact relation is obtained from Rodrigues' formula for the Associated Legendre Function in which we have the following identity, which I may derive in a future video.
For m > 0, the definition is:
For m = 0, the definition is:
Note that Pn0 = Pn and is just the ordinary Legendre polynomial.
For m < 0, the definition is:
Thus, putting it all together, we obtain the real spherical harmonics in proportional form, excluding the normalization constant.
Graphing Real Spherical Harmonics in Desmos Calculator
Here is an amazing Real Spherical Harmonics graph in Desmos Calculator that someone made!
Note the normalization constants used, as well as the same formulas we obtained above!
Note the Associated Legendre Polynomials and its giant derivative function.
Since the Spherical Harmonics are graphed on the sphere, the radial disance needs to be stretched proportional to the magnitude of its value on the sphere. This gives the distinctive lobes.
Note the relationships between the principle, orbital angular momentum, and magnetic quantum numbers. The principle sets the maximum orbital angular momentum which then sets positive and negative limits of the magnetic.
Here are the Real Spherical Harmonics for various values of the quantum numbers.
Fascinating stuff!!
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