The spherical harmonics
Summary
TLDRIn this video, Professor Erin Das explores spherical harmonics, essential eigenfunctions in quantum mechanics, particularly relevant to the hydrogen atom. The discussion covers their mathematical forms, visualization techniques, and their role in describing orbital angular momentum. Using visual representations on the unit sphere, the video explains the angular dependence of spherical harmonics and how they vary with quantum numbers l and m. The content is supported by Python code available through a linked Jupyter notebook, encouraging viewers to interact with and generate their own visualizations. This video is ideal for anyone interested in the mathematical beauty and application of spherical harmonics in quantum systems.
Takeaways
- 📚 Spherical harmonics are the eigenfunctions of orbital angular momentum in quantum mechanics, crucial in many problems, especially in the hydrogen atom.
- 🔢 These harmonics depend on two quantum numbers: l (angular momentum) and m (magnetic quantum number).
- 📈 Spherical harmonics are often visualized on the unit sphere, where positive values are red, negative values are blue, and zero values are white.
- 🧮 The Y00 spherical harmonic is a constant function, representing a uniform red sphere, while Ylm functions show more complex patterns depending on l and m.
- 🎨 The visualization of spherical harmonics helps in understanding their angular dependence, with each harmonic represented by color-coded spheres based on their values.
- 🔄 The values of spherical harmonics can change with the angles θ and φ, producing varying patterns across different harmonics as l and m increase.
- 💻 The script refers to a linked Jupyter notebook containing Python code that can generate spherical harmonics, allowing viewers to visualize them themselves.
- 🔬 Spherical harmonics play a central role in quantum mechanics, especially in understanding the angular part of wavefunctions in the hydrogen atom.
- 🔍 The script also explains how spherical harmonics are related to associated Legendre polynomials and their use in different mathematical forms.
- 🔧 An alternative way to visualize spherical harmonics is by plotting the magnitude of the function, which produces different, often used, visual representations.
Q & A
What are spherical harmonics and why are they important in quantum mechanics?
-Spherical harmonics are the eigenfunctions of orbital angular momentum in quantum mechanics. They are important because they describe the angular part of wavefunctions in systems such as the hydrogen atom, making them crucial for solving problems involving spherical symmetry.
What is the general form of a spherical harmonic function?
-The general form of a spherical harmonic function \( Y_{l}^{m}(θ, φ) \) consists of a pre-factor, a phase factor, and a term involving associated Legendre polynomials, depending on the angular variables \( θ \) (colatitude) and \( φ \) (azimuthal angle).
What are the quantum numbers associated with spherical harmonics?
-Spherical harmonics are labeled by two quantum numbers: \( l \), which is associated with the magnitude of orbital angular momentum, and \( m \), which describes the z-component of angular momentum. The values of \( l \) are non-negative integers, and for a given \( l \), \( m \) ranges from \( -l \) to \( l \).
How do spherical harmonics relate to the hydrogen atom?
-In the hydrogen atom, spherical harmonics describe the angular part of the electron's wavefunction. This is due to the spherical symmetry of the atom, making spherical harmonics essential for solving the Schrödinger equation in spherical coordinates.
What do the colors in spherical harmonic visualizations represent?
-In spherical harmonic visualizations, colors represent the value of the function at each angular position on the unit sphere. Red indicates positive values, blue indicates negative values, and white corresponds to zero. These visualizations help show the angular dependence of the function.
How are spherical coordinates used in the context of spherical harmonics?
-Spherical coordinates are used to describe the position of a point in space using three numbers: \( r \) (distance from the origin), \( θ \) (angle from the z-axis), and \( φ \) (angle in the horizontal plane from the x-axis). Spherical harmonics depend only on \( θ \) and \( φ \), making them ideal for problems with spherical symmetry.
What are associated Legendre polynomials and their role in spherical harmonics?
-Associated Legendre polynomials, denoted as \( P_{l}^{m}(\cos θ) \), appear in the mathematical expression of spherical harmonics. They define the angular dependence of spherical harmonics in terms of the colatitude angle \( θ \).
What does the spherical harmonic \( Y_{0}^{0} \) look like?
-The spherical harmonic \( Y_{0}^{0} \) is a constant and has no angular dependence. It is represented by a uniformly red sphere because the function has the same value at all points on the unit sphere, corresponding to a value of \( 1 / \sqrt{4π} \).
How do the real and imaginary parts of spherical harmonics differ in their visual representation?
-The real and imaginary parts of spherical harmonics can be represented separately. For example, for \( Y_{1}^{-1} \), the real part exhibits a cosine dependence, while the imaginary part exhibits a sine dependence. These two parts are 90 degrees out of phase and show different patterns in the color plot.
What are the common methods for visualizing spherical harmonics, and how do they differ?
-There are two common methods for visualizing spherical harmonics. One plots the function on a unit sphere, using colors to represent the value at different angles. Another method uses the magnitude of the function to define a radial distance, effectively stretching the surface based on the function's value. Both methods highlight different aspects of the function's angular dependence.
Outlines
🔬 Introduction to Spherical Harmonics in Quantum Mechanics
Professor Erin Das introduces spherical harmonics, which are eigenfunctions of orbital angular momentum in quantum mechanics. The discussion centers on their role, particularly in the hydrogen atom. The video will explore the mathematical forms of spherical harmonics, visualize them, and provide Python code to generate these plots.
🌈 Visualizing Spherical Harmonics on the Unit Sphere
The visualization of spherical harmonics is explained, where the color represents function values on the unit sphere. Red indicates large positive values, blue indicates large negative values, and white represents zero. The angular coordinates, theta and phi, help describe points on the sphere, and rotation helps in better understanding these visualizations.
📐 Real and Imaginary Parts of Spherical Harmonics
The focus shifts to the real and imaginary parts of spherical harmonics, starting with the case of l=1, m=-1. The behavior of the spherical harmonics is broken down by analyzing how the cosine and sine functions change across different angles, giving insight into the patterns of positive and negative values across the surface.
🎨 Exploring Various Quantum States: L=1 and Beyond
Different spherical harmonics for quantum states with l=1 and varying m are explored, including their mathematical derivation and visual representation. The constant nature of some functions across specific angles and their color-coded depiction is discussed. This provides deeper insights into how different quantum states manifest geometrically.
🔢 Understanding Higher-Order Spherical Harmonics
The video explores spherical harmonics with l=2, presenting five possible states (m = -2 to 2) and their corresponding mathematical and visual representations. The complex patterns of these harmonics are visualized in both real and imaginary parts, showcasing their beauty and symmetry in quantum mechanical systems.
💻 Generate Your Own Spherical Harmonics with Python
A detailed comparison is provided between traditional spherical harmonics plots and an alternative approach where the magnitude of the function defines a radial distance. The rotation of these plots helps viewers understand the full angular dependence. The video concludes by encouraging viewers to use Python code to create their own spherical harmonics.
Mindmap
Keywords
💡spherical harmonics
💡orbital angular momentum
💡eigenfunctions
💡hydrogen atom
💡Legendre polynomials
💡quantum numbers
💡visualization
💡theta and phi
💡Jupyter notebook
💡real and imaginary parts
Highlights
Introduction to spherical harmonics as eigenfunctions of orbital angular momentum, relevant in quantum mechanics.
Spherical harmonics play a central role in understanding orbital angular momentum in the hydrogen atom.
Discussion of the mathematical form and visualization of spherical harmonics using Python code linked in the description.
Quantum numbers L and M govern the eigenvalues of orbital angular momentum, with L being non-negative integers and M ranging from -L to L.
Spherical harmonics depend on angular variables theta and phi, and are represented on a unit sphere with color-coded values.
Theta and phi in spherical coordinates describe angular positions, with theta ranging from 0 to pi and phi from 0 to 2pi.
Visualization of spherical harmonics shows red for large positive values, blue for large negative values, and white for zero values.
For L = 0 and M = 0, the spherical harmonic Y_00 is a constant value and visualized as a solid red sphere.
For L = 1 and M = -1, Y_1_-1 shows a cosine phi dependence, creating alternating red, white, blue sections on the unit sphere.
Real and imaginary parts of spherical harmonics are plotted separately, showcasing their full angular dependence.
At L = 1 and M = 0, Y_1_0 depends only on theta and creates a red-white-blue pattern from the north to south poles.
Alternative visualization method involves using radial distance to define the magnitude of the spherical harmonic function, rather than color.
Complex spherical harmonics for L = 2 and L = 3 are visualized through rotations, revealing intricate angular dependencies.
Spherical harmonics are essential in quantum systems, with practical applications in hydrogen atom studies.
Encouragement to experiment with the provided Python code to generate and visualize spherical harmonics independently.
Transcripts
hi everyone this is professor erin das
science and today i want to discuss
spherical harmonics in another one of
our videos on rigorous quantum mechanics
spherical harmonics are the
eigenfunctions of orbital angular
momentum in quantum mechanics as such
they feature in many problems but most
importantly they feature in the hydrogen
atom today we're going to dissect the
mathematical form of the first few
spherical harmonics and we're also going
to visualize them
and i hope you will agree that they look
rather cool we also have a link in the
description to a jupiter notebook which
has some python code to generate the
spherical harmonics so i encourage you
to take a look and see what it does so
let's go
the main reason why we study spherical
harmonics is that they play a central
role in the theory of orbital angular
momentum in quantum mechanics
this is the eigenvalue equation for the
squared angular momentum
and this is the eigenvalue equation for
the z component of orbital angular
momentum
the l squared eigenvalues are labeled by
the quantum number l and it can only
take non-negative integer values
and the lz eigenvalues are labeled by
the quantum number m
and for a given l
m can only take one of the values minus
l minus l plus 1 and so on all the way
to l
however today's focus is not on the
eigenvalues but instead on the
eigenstates that are shared by both l
squared and lz here and here
these eigenstates are the so-called
spherical harmonics
in the video on orbital angular momentum
eigenfunctions we determine an explicit
expression for these eigenfunctions
we can write the spherical harmonic ylm
of theta phi as equal to this long
pre-factor
multiplied by a phase factor
and then this other long expression
involving sines and
cosines as you may suspect we can
rewrite spherical harmonics in a number
of alternative but equivalent ways
a very common form is that in which we
dump all the terms involving sines and
cosines into a function
and then we can write the spherical
harmonic ylm as equal to this long
pre-factor
multiplied by the phase factor and
everything multiplied by a function plm
of the cosine
these functions p here are called the
associated legendre polynomials
they're actually a rather interesting
family of mathematical functions but
today we are not going to explore them
any further so if you're interested in
learning more about them i encourage you
to check them out elsewhere
today we will explore the first few
spherical harmonics and to do so we're
going to use this expression up here
we're going to both write down their
mathematical form explicitly and we're
also going to visualize them
the visualization of spherical harmonics
is something that you will often
encounter because spherical harmonics
feature in a variety of quantum systems
most importantly the hydrogen atom
to understand how we will visualize them
we first need to note that the spherical
harmonics depend on the two angular
variables theta and phi
so let's draw a set of coordinate axes
let's consider a general point here at a
position given by the vector r from the
origin
and in spherical coordinates we describe
the position of this point with a set of
three numbers
the first is the distance between the
origin and the point which is the
magnitude of the vector r and we call it
the scalar r
the second is the angle between the
vector r and the third axis and we call
it theta and the third is built by first
projecting the vector r onto the
horizontal plane
and then measuring its angle with
respect to the first axis and we call it
phi
as r is the length of a vector it can
only be zero or positive
the angle theta runs from zero to pi and
the angle five from zero to two pi
the spherical harmonics only depend on
the spherical coordinates and what we
will do to represent them is to plot
them on the unit sphere
now this here is an example of how we
will plot the spherical harmonics
this happens to be the real part of the
spherical harmonic y one minus one but
for now all that matters is how we are
representing it
we're not going to plot the coordinate
axes in general but for now uh they're
these three here and they should help
you as a reference
at every angular position on the unit
sphere the color shows the value of the
function such that the red regions show
the angular positions where the function
takes large positive values in this case
this region here which corresponds to
theta equals pi over 2 and phi equals 0
has large and positive values
and the blue regions show the angular
positions where the function takes large
negative values and in this case this is
hidden at the other side of the sphere
with the most negative value centered
around theta equals pi over 2
and phi equals pi
for intermediate values we plot the
function with a color scale that goes
from positive red to negative blue
through zero which is white
in this particular example both the
north pole here corresponding to theta
equals zero and the south pole heated
behind here corresponding to theta
equals pi are both white
which means that the function is zero at
those positions and in fact if the
function is zero along a whole great
circle that unites the two poles for
values of phi equals pi over two here
and phi equals three pi over two on the
other side
as i said for clarity we're not going to
show the axes and to understand these
plots better we will often rotate the
coordinate axes like this
allowing us to see the value of the
spherical harmonic in all angular
directions
right so with this let's get started
with our exploration of the spherical
harmonics
the spherical harmonics are labeled by
the quantum numbers l and n
as l is associated with the magnitude of
orbital angular momentum we will use
this as the main quantum number from
which to build the spherical harmonics
so let's start with l equals zero
this implies that m is also zero
the corresponding spherical harmonic is
therefore y zero zero and so if we start
with the pre-factor here we can write it
down inserting l equals 0 and n equals
0.
most terms are now trivial and the
pre-factor reduces to 1 over square root
of 4 pi
let's next look at the face factor and
using m equals 0 it becomes this
which is trivially equal to 1.
finally we can look at this final term
and writing it out with l equals 0 and m
equals zero
we see that it also trivially becomes
one
so overall the spherical harmonic y zero
zero is equal to the constant one over
square root of four pi
this here is a plot of the y zero zero
spherical harmonic it is trivially a
solid uniformly red sphere
the color is uniform because the
function is a constant which means that
it has the same value in all directions
and it is red because we are using red
to depict positive values
also note that the function is purely
real so this single diagram is all that
we need to depict the y0 spherical
harmonic
let's now consider l equals one
for this value of l there are three
possible values of m
minus one zero and one so let's start
with l equals one and m equals minus one
so that we have the spherical harmonic y
one minus one
if we start with a pre-factor here we
can write it down inserting l equals one
and m equals minus one
this part here is equal to minus one
over two
and the argument of the square root here
simplifies to three-eighths of pi
together the pre-factor turns into this
expression
let's next look at the face factor
using m equals minus 1 it becomes this
and finally we can look at this final
term and writing it out with l equals 1
and m equals minus 1 we get this
expression
at this stage it is convenient to
rewrite this sine squared in terms of a
cosine squared using the standard
trigonometric relation
putting everything together we get this
new expression
now looking at the second derivative
term here we get minus two so that the
full term becomes minus two sine theta
overall the spherical harmonic y one
minus one is given by this pre-factor
this phase factor and all multiplied by
sine theta and we can also separate this
expression into its real part and its
imaginary part
these plots show the y one minus one
spherical harmonic
the top diagram shows the real part
while the bottom diagram shows the
imaginary part
so let's start with the real part which
is given by this term up here
remember that the angle along the
horizontal plane is measured by 5
and that 5 runs from 0 to 2 pi
looking at the real part of the
spherical harmonic up here we see that
the phi dependence is fully captured by
this cosine term
let's therefore see how this cosine phi
term changes as we travel along phi and
to do so we will consider a fixed theta
and specifically we will look at the
horizontal plane
at phi equals to zero the cosine is
equal to one
this gives a positive maximum value
which is represented by the red color
here
when phi grows to pi over two the cosine
becomes zero
this corresponds to this point here on
the horizontal plane where the color
plot turns white
when phi goes to pi the cosine becomes
-1 and this gives a negative value which
is represented by the blue color hidden
at the back of the diagram
moving to phi equals 3 pi over 2 the
cosine becomes 0 again
and the function vanishes again
represented by the white color at this
edge
and completing the loop to phi equals to
2 pi we get that cosine is again 1 and
we are back to the maximum rate value
here
so this is it for the phi dependence the
real part of the y one wins one
spherical harmonic exhibits a full
cosine oscillation from positive to
negative and back to positive and
pictorially we get red white blue white
and back to red
so let's make some room
remember that the angle from the
vertical plane is measured by theta
and that theta runs from 0 to pi
looking at the real part of the
spherical harmonic up here we see that
the theta dependence is fully captured
by this sine term
so let's therefore see how this sine
theta term changes as we travel along
theta
for simplicity let's start at the fixed
phi equals zero
and then theta varies along this great
circle from the north to the south poles
at the north pole for theta equals zero
sine theta is also zero
this means that at the north pole here
we have the color white as theta grows
to pi over two sine theta becomes one
and we get the maximum positive value in
red here
and then as we continue all the way to
the south pole at theta equals pi
sine theta becomes zero again
and we get a white south pole down here
although it is hidden behind the sphere
from this view
we of course have the same theta
dependence at different values of phi
but for example if we now travel along
this great circle corresponding to phi
equal pi over 2
we get a white line throughout because
although there is a sine dependent along
theta it is multiplied by this cosine
phi up here which is equal to zero along
this great circle
let's make some room again
if we next look at the imaginary part it
is given by this second term up here
we see that the theta angle has the same
sine dependence as the real part
while the phi angle is now given by a
sine rather than a cosine
this means that the real and imaginary
parts are offset by 90 degrees as it's
clear from the figures
we can finally make these diagrams
rotate to appreciate the full angular
dependence
it's not the easiest to get your head
around this type of plot so i recommend
that you take your time until it becomes
completely completely clear
okay let's now look at l equals one and
m equals zero
we have the spherical harmonic y one
zero and i won't go over the derivation
in detail this time as it is analogous
to what we've just done for the y one
minus one case but i really do encourage
you to try it out as it is really good
practice
we get the pre-factor square root of
three over four pi
times the cosine of theta
and overall y 1 0 is purely real
this here is a plot of the y 1 0
spherical harmonic and as it is purely
real we only need one diagram
we see that y 1 0 does not depend on the
angular variable phi which implies that
the spherical harmonic looks the same in
all directions within the horizontal
plane as we can clearly see in the
diagram y10 does depend on theta through
this cosine function
at the north pole where theta is zero
the cosine takes the maximum value hence
the red color
then on the horizontal plane
corresponding to theta equals pi over
two the cosine vanishes and this is
indicated by the white band
finally at the south pole where theta is
pi the cosine takes its maximum negative
value and we get the blue color
again as there is no phi dependence
these results are true along any phi
direction and again do spend as long as
you need to really make sure that this
plot makes sense
finally let's look at l equals one and m
equals one
we have the spherical harmonic y one one
again leaving the duration for you we
get this pre-factor
this face factor and we'll multiply it
by sine theta
we can separate this expression into a
real part
plus an imaginary part
this here is a plot of the y 1 1
spherical harmonic the top diagram again
shows the real part while the bottom
diagram shows the imaginary part the
expression is really quite similar to
the one that we have for the spherical
harmonic y one minus one the imaginary
parts are in fact the same
and the only difference between the real
parts is this minus sign here
this means that the only difference in
the plots is that the real part is the
negative of the real part of the y one
minus one spherical harmonic and if you
remember in that one we had red in front
and blue at the back whereas here we
have blue in the front and red at the
back
the real and imaginary parts are again
offset by 90 degrees from each other
and we can again make them rotate to
better appreciate the full angular
dependence
as a summary we have here the three
spherical harmonics for l equals one
with the top diagram showing the real
parts and the bottom diagrams showing
their imaginary parts it's not really
trivial to get your head around these
plots so again do take as long as you
need to make sure that you are
absolutely happy with the plotting
so let's next consider l equals two
for this value of l there are five
possible values of n
minus two minus one zero one and two
the derivation of the mathematical form
of this fragile harmonics is analogous
to that of the previous two examples so
we will write the relevant expressions
directly
we can write y 2 minus 2 and y 2 2
together
and they are equal to these pre-factors
these phase factors and sine square
theta we next have y 2 minus 1 and y 2 1
these are the pre-factors
these are the phase factors and then we
have sine theta cosine theta
and finally we have y to 0 which is
purely real and is given by this
expression
these here are the five spherical
harmonics for l equals two with the top
diagram showing their real parts and the
bottom diagrams showing their imaginary
parts i'm not going to discuss them in
detail but i recommend that you spend
some time trying to relate their
mathematical form with the corresponding
figures
you may agree that they are actually
quite beautiful
but to fully appreciate their angular
dependence we can also make them rotate
about the vertical axis starting with y2
minus 2
then y2 minus 1
y20 doesn't have a 5 dependent so it
looks the same at any angle about the
vertical axis
and we can also make y 2 1 rotate and
finally y 2 2.
we could of course go on and plot
additional spherical harmonics and here
i have the l equals three spherical
harmonics we can again make them rotate
for better visualization starting with y
three minus three
then y three minus two
y three minus one
again the y three zero spherical
harmonic is purely real and doesn't
depend on fives the same in all
directions moving to y three one we can
again make it rotate
the same for y 3 2
and finally for y 3 3.
we've generated all of these figures
using python and you can have a go
yourself plotting spherical harmonics by
following the link to the jupiter
notebook in the description
and overall i hope you will agree that
these diagrams have a rather captivating
beauty about them
to finish i just want to point out that
you will encounter spherical harmonics
depicted in a variety of ways
i personally think that the approach
that we've taken today is particularly
clear because it really highlights the
angular dependence of the spherical
harmonics by plotting them on the unit
sphere
the top row here shows again the real
part of the l equals one spherical
harmonics that we've discussed earlier
in the video
and just as a reminder we've been
plotting the functions at every angle on
the unit sphere and capturing the value
with a color figure
red for positive and blue for negative
smoothly connected through white which
corresponds to zero however there is
another very common approach to plotting
these functions and you're bound to
encounter it elsewhere so i wanted to
very briefly describe it
we show this alternative approach in the
bottom row
in this case we use the magnitude of the
function along each direction to define
a radial distance and then plot the
function at that radial distance rather
than plotting it on the unit sphere
so we can again make all of these rotate
for ease of visualization
starting with y one minus one the y one
zero have no phi dependence so they look
the same along any phi angle
and we can also make the y one one
rotate
and as always i recommend that you spend
some time convincing yourself of this
alternative depiction as it is also used
very commonly
here i am now comparing the real parts
of the spherical harmonics for l equals
2 with our original plots in the top
column and the alternative plots in the
bottom column
we can again make them rotate starting
with the y 2 minus 2
then y y2 minus 1
as usual y to 0 is phi independent
but we can also make the y 2 1 rotate
and we can also do the same for y tick
too
so again do spend some time
familiarizing yourself with these
and finally we have the comparison to
the real part of the spherical harmonics
with l equals three and we make them
rotate because it's cool and we have the
y three minus three
the y three minus two
the y three minus one
the y three zero which you guessed right
is phi independent
and we can also rotate the y 3 1
the y 3 2
and the y 3 3.
final comment although we've been
plotting the real and imaginary parts of
the spherical harmonics you will often
encounter plots that show their absolute
value squared
we're not going to explicitly go into
those but you should be able to
construct and interpret these
alternative plots with similar
strategies to the ones that we've used
today
i hope you've enjoyed visualizing the
spherical harmonics and remember that
they are extremely useful in a range of
problems especially the hydrogen atom
and again remember that you can generate
these spherical harmonics yourself by
using the python code that we've linked
in the description and as always if you
liked the video please subscribe
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