Useful notation for angular momentum eigenvalues
Summary
TLDRIn this educational video, Professor M.D.A. Science delves into the intricacies of angular momentum in quantum mechanics, emphasizing its significance beyond its classical counterpart. The video clarifies the quantization of angular momentum eigenvalues, introduces the vector operator \( \vec{J} \), and explains the commutation relations crucial for understanding its behavior. It simplifies complex mathematical concepts, focusing on the language and notation used in angular momentum discussions. The video also distinguishes between orbital and spin angular momentum, highlighting their unique properties and the importance of spin one-half in quantum mechanics, particularly in the study of electrons.
Takeaways
- 🌟 Angular momentum is a fundamental property in both classical and quantum mechanics, with quantum mechanics introducing additional concepts like spin angular momentum.
- 🔍 In quantum mechanics, angular momentum is represented by a vector operator \( \vec{J} \) composed of \( J_x, J_y, \) and \( J_z \), which obey specific commutation relations.
- 📐 The eigenvalues of angular momentum operators are quantized, meaning they can only take on specific, discrete values, unlike in classical mechanics.
- 🧮 The eigenvalues for \( J^2 \) (the square of the angular momentum operator) are given by \( j(j+1)\hbar^2 \), where \( j \) can be an integer or half-integer, and \( \hbar \) is the reduced Planck constant.
- 📉 The possible values for the eigenvalue of \( J_z \), denoted as \( m \), range from \( -j \) to \( j \) in integer steps, resulting in \( 2j+1 \) possible values.
- 🔄 The notation for angular momentum eigenstates is simplified to \( |jm\rangle \), where \( j \) and \( m \) represent the quantum numbers corresponding to the eigenvalues of \( J^2 \) and \( J_z \), respectively.
- 🌐 The language and notation used for angular momentum are crucial for understanding quantum mechanics, and it's important to be familiar with the conventions used.
- 🌐 The units of Planck's constant match those of angular momentum, confirming the consistency of quantum mechanical angular momentum with physical dimensions.
- 🔄 Orbital angular momentum is labeled with \( l \) and spin angular momentum with \( s \), each having its own set of eigenvalues and eigenstates.
- 🔄 The allowed values of \( j \) and \( m \) for orbital and spin angular momentum differ, with orbital angular momentum having integer values for \( l \) and spin angular momentum allowing both integer and half-integer values for \( s \).
Q & A
What is angular momentum and why is it important in quantum mechanics?
-Angular momentum is a key property in both classical and quantum mechanics. In quantum mechanics, it is even more important because it includes not only the orbital angular momentum from classical mechanics but also spin angular momentum, which has no classical analog. Angular momentum eigenvalues are frequently encountered in quantum mechanics.
What are the two types of angular momentum discussed in the script?
-The script discusses orbital angular momentum, which is analogous to classical mechanics, and spin angular momentum, which is unique to quantum mechanics.
What does the angular momentum operator consist of?
-The angular momentum operator consists of three components: j1, j2, and j3. These components obey specific commutation relations.
What is the significance of the commutation relations for the angular momentum components?
-The commutation relations indicate that the different angular momentum components do not commute, implying that they do not form a set of compatible observables.
What operator can be defined that commutes with each individual component of angular momentum?
-The operator j squared, which is defined as the sum of the squares of the individual components (j1^2 + j2^2 + j3^2), commutes with each individual component.
What are the eigenvalue equations for j squared and j3?
-The eigenvalue equation for j squared is j^2ψ = λψ, where λ is the eigenvalue. For j3, the eigenvalue equation is j3ψ = μψ, where μ is the eigenvalue.
What are the allowed values for the eigenvalues λ and μ?
-λ is given by j(j + 1)ħ², where j can be an integer or half-integer. μ is given by mħ, where m can take values from -j to j in steps of one.
How are the units of Planck's constant related to angular momentum?
-Planck's constant has units of angular momentum, which is consistent with the units of the eigenvalues of j squared and j3.
What is the notation used for angular momentum eigenstates?
-Eigenstates are typically denoted as |jm⟩, where j is the eigenvalue of j squared and m is the eigenvalue of j3.
What is the difference between orbital and spin angular momentum in terms of their possible values of j?
-Orbital angular momentum can only have integer values of j, while spin angular momentum can have both integer and half-integer values.
Why is the concept of spin one-half angular momentum central in quantum mechanics?
-Spin one-half angular momentum is central in quantum mechanics because the electron, a fundamental particle in many systems, is a spin one-half particle.
Outlines
🔬 Introduction to Angular Momentum in Quantum Mechanics
Professor M.D.A. begins by introducing the concept of angular momentum, emphasizing its importance in both classical and quantum mechanics. In quantum mechanics, angular momentum is not only limited to the orbital component but also includes spin angular momentum, which lacks a classical equivalent. The video aims to discuss the language and conventions used in angular momentum, eigenvalues, and to differentiate between orbital and spin angular momentum. The summary of angular momentum in quantum mechanics involves a vector operator J composed of three components, which obey specific commutation relations. The operator J includes both orbital and spin angular momentum, labeled by l and s respectively. The video explains that while the individual components of angular momentum do not commute, the operator J squared does commute with each component, allowing for a set of compatible observables to be constructed using J squared and one of the components, conventionally J3. The eigenvalue equations for these observables are discussed, with the eigenvalues being quantized and taking specific forms related to the quantum number j and its projection m.
📏 Units and Conventions in Angular Momentum
The paragraph delves into the units of angular momentum and Planck's constant, highlighting that both have units of angular momentum. Planck's constant, defined with an exact value since 2019, is fundamental in quantum mechanics, historically linking the energy of a photon to its frequency. The units of energy (joules) and frequency (inverse seconds) combine to give Planck's constant units of joules times seconds, which aligns with the dimensions of angular momentum (mass times length squared divided by time). The paragraph then discusses the notation and conventions used for angular momentum eigenvalues and eigenstates. It explains that the general eigenvalue equations for J squared and J3 can be replaced with their allowed values, simplifying the notation. The eigenstates are commonly denoted as |jm⟩, where j and m are the eigenvalues for J squared and J3, respectively. The paragraph also emphasizes the importance of understanding the meaning behind these notations and the physical implications they represent.
🌐 Examples of Angular Momentum Eigenvalues and States
This section provides examples to illustrate the calculation of angular momentum eigenvalues and the labeling of eigenstates for different values of j. It starts with j=0, where the eigenvalues are zero, leading to a single eigenstate. For j=1/2, there are two possible m values (-1/2 and +1/2), each corresponding to different eigenvalues and eigenstates. The process continues with j=1, where three m values (-1, 0, +1) result in three eigenvalues and eigenstates. The paragraph demonstrates how this pattern continues for other j values, allowing for the construction of any required combination of eigenvalues and states. It also differentiates between the notation for general angular momentum (j) and specific cases like orbital (l) and spin (s) angular momentum, explaining the corresponding changes in eigenvalue notation and eigenstate labeling.
🌀 Distinction Between Orbital and Spin Angular Momentum
The final paragraph contrasts orbital and spin angular momentum, emphasizing their differences in quantum mechanics. While orbital angular momentum is analogous to its classical counterpart and is described by integer values of l, spin angular momentum is a purely quantum mechanical property that can take both integer and half-integer values. The paragraph focuses on the example of spin one-half, which is crucial for understanding the behavior of electrons and complex materials. It explains that while ml (for orbital angular momentum) must be an integer, ms (for spin angular momentum) can be a half-integer, such as ±1/2. The video concludes by reinforcing the importance of understanding these concepts and encourages viewers to explore further topics involving angular momentum, such as the hydrogen atom, and to watch the companion video for a detailed mathematical derivation of angular momentum eigenvalues.
Mindmap
Keywords
💡Angular Momentum
💡Eigenvalues
💡Quantization
💡Orbital Angular Momentum
💡Spin Angular Momentum
💡Commutation Relations
💡Operator
💡Eigenstates
💡Planck's Constant
💡Half-Integer
💡Observables
Highlights
Angular momentum is a key property in both classical and quantum mechanics, with quantum mechanics introducing spin angular momentum.
In quantum mechanics, angular momentum eigenvalues are quantized and play a crucial role in understanding the system's behavior.
The angular momentum operator J is composed of three components (J1, J2, J3) that obey specific commutation relations.
J squared and one of its components (J3) are compatible observables used to build the theory of angular momentum in quantum mechanics.
Eigenvalues of J squared and J3 are quantized and can only take specific values related to the quantum number j.
The eigenvalues lambda and mu are related to the quantum numbers j and m, with j taking half-integer or integer values and m ranging from -j to j.
Planck's constant, and the reduced Planck's constant, both have units of angular momentum, which is essential for understanding the dimensions of angular momentum.
Eigenstates are labeled by the quantum numbers j and m, and they represent states with specific angular momentum values.
Notation for angular momentum eigenstates is simplified to jn, where j and m are replaced by their allowed values.
The language used to describe angular momentum states is crucial for understanding quantum mechanics, especially the distinction between j, l, and s for general, orbital, and spin angular momentum.
Examples are provided to illustrate how to calculate and label angular momentum eigenvalues and eigenstates for different quantum numbers.
Orbital angular momentum is analogous to classical mechanics, while spin angular momentum is a purely quantum mechanical concept.
Different notations are used for orbital (l) and spin (s) angular momentum to avoid confusion when multiple types are considered.
The allowed values of j and m for orbital and spin angular momentum are derived from the commutation relations of the angular momentum operator.
Orbital angular momentum is limited to integer values of j, while spin angular momentum can have both integer and half-integer values.
Spin one-half angular momentum is particularly important in quantum mechanics due to the electron's spin being a spin one-half particle.
Understanding angular momentum is essential for exploring topics such as the hydrogen atom and the differences between orbital and spin angular momenta.
Transcripts
hi everyone
this is professor m da science and today
i'm going to talk about angular momentum
eigenvalues in another one of our videos
on rigorous quantum mechanics
angular momentum is a key property in
classical mechanics where the orbital
angular momentum
is central in many problems if anything
angular momentum is actually even more
important
in quantum mechanics where we not only
have the orbital variant but we also
have
spin angular momentum which does not
have a classical analog
this means that in quantum mechanics you
will come across angular momentum
eigenvalues
all the time there is a companion video
linked in the description
where we derive angular momentum
eigenvalues mathematically
showing that they are quantized what i
want to do in this video is to discuss
the language that is typically used
when discussing angular momentum
eigenvalues among other things we will
discuss some widely used but actually
somewhat confusing conventions
we will look at a few examples of
allowed angular momentum
and we will also anticipate some of the
key differences between orbital and spin
angular momentum
so let's go as always let's start with
the summary of angular momentum in
quantum mechanics
we consider a vector operator j made of
three operators j1 j2 and j3
this operator j will be an angular
momentum operator if the three
components obey
these commutation relations and here i'm
using the
levy chevita symbol and the convention
that we sum
over repeated indices k if these ideas
don't sound familiar to you
then you should first check out the
video that introduces angular momentum
j is a general angular momentum and it
includes the orbital angular momentum
that we're familiar with from classical
mechanics
and that we typically label with the
letter l
and it also includes a spin angular
momentum that doesn't have a classical
analog and for which we use the letter
s as we can see
up here the different angular momentum
components don't commute
which implies that they don't form a set
of compatible observables
but we know from the video on angular
momentum that we can define
another operator j squared equal to j1
squared plus j2 squared plus j3 squared
that does commute with each individual
component
this means that in the quantum theory of
angular momentum
we can build a set of compatible
observables by considering j
squared and one of the components ji and
we conventionally use j3
we now build the theory of angular
momentum in quantum mechanics with these
two compatible observables j
squared and j3 the key equations
are the eigenvalue equations which for j
squared takes this form where the
eigenvalue is lambda
and for j3 we have this eigenvalue
equation
where the eigenvalue is mu
these eigenstates here and here are a
common set of eigenstates for the two
commuting observables
and i label them with the eigenvalues
lambda and mu
so that this is clear in the confining
video on angular momentum eigenvalues
we found out that the eigenvalues lambda
and mu
cannot take just any arbitrary value but
instead can only take a set of special
values
lambda is always given by j j plus one h
bar squared
where j can only take the values zero
one half one three halves two
five halves three and so on in steps of
one half
the values zero one two three and so on
are obviously
integers and we call the values one half
three halves five halves and so on
half integers half integer simply means
that we pick
an odd integer and divided by two
once we have determined the value of j
from this list
then mu is always given by this form
m h bar and m can only take
the values minus j minus j plus one
minus j plus two and so on in steps of
one
all the way to j minus one and finally
j and this list here contains
two j plus one possible values for him
so this is it for the angular momentum
eigenvalues
in the companion video i go into a lot
of detail over the mathematical
derivation as to why
these are the only allowed eigenvalues
of j squared and j3
today's video you'll be happy to hear is
much lighter on the maths front
and i instead focus on some important
concepts that we need to be familiar
with
relating to these eigenvalues
so let's start with a sanity check and
just confirm that these eigenvalues have
the correct units for an angular
momentum
the key insight is to realize that
planck's constant
and similarly the reduced planck
constant
both have units of angular momentum
planck's constant is a fundamental
physical constant
which since 2019 is actually defined
to have the exact value h equals
6.62607015
times 10 to the minus 34 joules
times second in si units
historically planck's constant was the
proportionality constant between
the energy and the frequency of a
quantum of light
called a photon you can straight away
check that the units of energy
being joules and those of frequency
being inverse seconds
leads to the units of h being joules
times second
so what is the dimension of the planck's
constant then it is basically equal to
energy times time
there are various ways in which we can
spell out energy so let's just
pick the infinitesimal work done by
displacing a particle
using a force f by some infinitesimal
distance
dr this means that the dimension of
energy
is force times length
and in turn using newton's second law
we get that force is equal to mass times
acceleration so that the dimension of
force is equal to
mass times length divided by time
squared
putting everything together we get that
the dimension of planck's constant is
equal to
mass times length squared divided by
time
if we now write the familiar formula for
orbital angular momentum
l as r cross p then
we have that the dimension of r is
length
and the dimension of momentum is mass
times length divided by time
so overall we get that the dimension of
angular momentum
is mass times length squared divided by
time
comparing the dimension of planck's
constant with the dimension of angular
momentum we see that they
are indeed the same therefore the fact
that the eigenvalue of j
squared is proportional to h bar squared
is consistent with j
squared being the square of an angular
momentum
and similarly the fact that the
eigenvalue of j 3
is proportional to h bar is also
consistent
with j 3 being an angular momentum
component
the next topic i want to discuss relates
to conventions
about notation we started with these
general eigenvalue equations for j
squared and for j3 with the
corresponding eigenvalues lambda
and mu but now that we know that lambda
can only take these values
with j in turn given by this
and that mu can only take these values
where
m is given by this
it then makes sense to replace all
lambdas and mu's
by their allowed values in particular we
can do this for the common set of
eigenstates
and we could write them like this
however in practice we simplify the
notation and write these eigenstates as
jn in this language the eigenvalue
equation for j squared becomes
this and the eigenvalue equation for
j3 becomes this
another important simplification that is
used when working with angular momentum
is that for a system in eigenstate jn
we typically say that this is a state
with angular momentum
j now this language is used constantly
so it is very important to become
familiar with it
but it is also essential that you don't
forget what an eigenstate jm
really means it means that the j
squared eigenvalue is j times j plus 1
times h bar squared
and that the j 3 eigenvalue is m h bar
we're now ready to look at a few
examples of angular momentum eigenvalues
and eigenstates
let's start by placing the relevant
quantities at the top
starting with j then the j squared
eigenvalue
then m then the j3 eigenvalue and then
the corresponding eigenstate the first
possible j value in the list is j equals
zero
the eigenvalue is trivially zero m is
also zero
and so is the j3 eigenvalue so this is
an easy case
there is a single eigenstate which we
label by zero zero
let's now look at the next possible j
value in the list which is j
equals one half for the j
squared eigenvalue we plug in one half
in the usual expression
which multiplies to three over four h
bar squared
there are now two possible values of m
the first is minus j
so minus one half and the second is plus
j
so plus one half these two
m values give two possible eigenvalues
minus one-half h-bar and plus one-half
h-bar
there are two eigenstates for j equals
one-half which are
one-half minus one-half and one-half
plus one-half we now move on to the next
eigenvalue in the list
we get j equals 1. we can plug in the
value 1 in the j squared
eigenvalue expression and overall we get
2
h bar squared there are now three
possible values for
n the first is minus j so minus one
then we add one to get zero and
we add one again to get plus one and
this is it because plus one is now
already equal to j these three
n values give three possible j three
eigenvalues
minus h bar zero and plus h bar
and looking at the eigenstates we also
have three and they are labeled by
one minus one one zero and one plus one
you can now see how this would continue
for other allowed values of j
and the corresponding m so you should be
able to build basically any combination
that you need
for most of our discussion of angular
momentum we've used a general definition
of angular momentum
however we know that we can be more
specific and discuss the special
case of orbital angular momentum which
is analogous to the corresponding
quantity in classical mechanics
and we also have the spinal momentum
which is a purely quantum mechanical
quantity
we also know that we label a general
angular momentum using the letter j
but that when we want to specify orbital
angular momentum we instead use the
letter
l and for spin angular momentum we use
the letter
s as you can imagine we use similar
changes in annotation when we talk about
eigenvalues and eigenstates
the j squared eigenvalue label small j
is typically replaced by the label small
l
for orbital angular momentum and by the
label small layers
for spin angular momentum the
corresponding
eigenvalues read as usual for the
general case
and then for the orbital they are in
terms of l
and for spin in terms of this
for the j3 eigenvalue label m if there
is no possibility of confusion
then we also typically just use m for
both orbital
and spin angular momentum however if we
have a system in which multiple types of
angular momentum
must be included at the same time then
we can use
m subindex l for orbital
and m sub index s for spin to avoid any
kind of confusion
for the sake of clarity in the rest of
the video i will use
ml and ms but remember that we can have
either option
and it should be clear what is meant by
the context
in this more involved notation the
corresponding eigenvalues read
mh bar mlh bar and msh bar
and finally the eigenstates jm become
lml for orbital and sms for spin
feel free to explore our other videos
for examples of
orbital and spin angular momenta that
you can find
linked in the description
the very final thing i want to do is to
anticipate some of the discussion in the
videos on orbital and spin angular
momentum
let's consider again our eigenvalue
equation for j squared
and for j3 and remember again that small
j
can be zero one half one three halves
two and so on
and that for a given j then m can be
any of the values minus j minus j plus
one
all the way to j in steps of one
we derive these allowed values in the
companion video on angular momentum
eigenvalues
and the only assumption behind that
derivation is
the defining commutation relation of the
angular momentum
operator j that derivation tells us that
these here are the possible values of j
and m
but in fact the derivation does not tell
us that they must all occur
only that they can occur as we discuss
in the corresponding videos
it turns out that when we have orbital
angular momentum
then the only values of j that can occur
are the
integer values so if you use the
notation introduced in the previous
slide
this means that small l can only be 0
1 2 3 and so on in integer steps
and as a consequence ml must also be an
integer
the story is completely different with
spin angular momentum
in this case j can take both integer or
half integer values
perhaps the most important example of
spin angular momentum is that of spin
one half where again using the notation
from the previous slide we have
small s equal to one half and
then ms can be either minus one half or
plus one half
spin one half is central in quantum
mechanics because the electron happens
to be a spin one-half particle
this means that to study anything
ranging from the simplest atom
to the most complex material the theory
of spin one-half angular momentum
is a must in this video
we've done some necessary bookkeeping to
make sure that we have the right
language to work with angular momentum
you're now ready to explore
many topics in which angular momentum is
key from the differences between
orbital and spin angular momenta to the
hydrogen atom
and don't forget to check out the
mathematical derivation of angular
momentum eigenvalues
in the companion video and as always if
you liked the video
please subscribe
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