Useful notation for angular momentum eigenvalues

Summary

In 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

00:00

🔬 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.

05:01

📏 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.

10:03

🌐 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.

15:06

🌀 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.

Keywords

💡Angular Momentum

Angular momentum is a fundamental concept in physics, referring to the rotational motion of an object around an axis. In the video, it is discussed as a key property in both classical and quantum mechanics. The video emphasizes that while angular momentum is important in classical mechanics for describing the orbital motion of objects, it becomes even more crucial in quantum mechanics where it includes not only orbital but also spin angular momentum, which has no classical equivalent. The script mentions that angular momentum is quantized in quantum mechanics, which is a significant departure from classical mechanics.

💡Eigenvalues

Eigenvalues are the scalar values that result from the operation of an operator on an eigenstate in quantum mechanics. The video script discusses angular momentum eigenvalues, which are the possible values that the quantum mechanical angular momentum operators can have. These eigenvalues are quantized, meaning they can only take on certain discrete values, not any arbitrary real number. The script explains that these eigenvalues are crucial for understanding the behavior of systems in quantum mechanics.

💡Quantization

Quantization refers to the process by which certain physical quantities can only take on discrete values rather than a continuous range. In the context of the video, angular momentum is quantized, which means that it can only have specific values. The script mentions that this quantization is a key feature of quantum mechanics and is mathematically derived in a companion video, highlighting the importance of quantization in understanding angular momentum in quantum systems.

💡Orbital Angular Momentum

Orbital angular momentum is the angular momentum associated with the motion of an object in a circular or helical path around an axis. The video script distinguishes between orbital and spin angular momentum, noting that orbital angular momentum is the classical concept that is also present in quantum mechanics. It is typically labeled with the letter 'l' and is associated with the shape and orientation of the orbital paths of particles in quantum systems.

💡Spin Angular Momentum

Spin angular momentum is a quantum mechanical property that arises from the intrinsic rotation of particles, such as electrons. Unlike orbital angular momentum, spin does not have a classical analog and is a purely quantum mechanical concept. The video script explains that spin angular momentum is labeled with the letter 's' and is a key aspect of quantum systems, especially in understanding the behavior of electrons in atoms and molecules.

💡Commutation Relations

Commutation relations describe the algebraic properties of operators in quantum mechanics, specifically how the order in which they are applied affects the result. The video script mentions that the components of the angular momentum operator obey certain commutation relations, which are essential for defining the properties of angular momentum in quantum mechanics. These relations are fundamental to understanding why angular momentum components do not commute and have implications for the compatibility of observables.

💡Operator

In quantum mechanics, an operator is a mathematical function that acts on the state of a quantum system to produce another state or a measurable property of the system. The video script discusses the angular momentum operator 'j', which is a vector operator composed of three components (j1, j2, j3). These operators are crucial for describing the angular momentum properties of quantum systems, and their eigenvalues correspond to the quantized values of angular momentum.

💡Eigenstates

Eigenstates are the states of a quantum system that correspond to the eigenvalues of an operator. The video script explains that the eigenstates of the angular momentum operator are labeled by the eigenvalues of 'j squared' and 'j3', which are quantized. These eigenstates are essential for understanding the behavior of quantum systems under the influence of angular momentum, as they represent the possible states that the system can be in when measured.

💡Planck's Constant

Planck's constant, denoted by 'h', is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. The video script discusses how Planck's constant, and the reduced Planck's constant 'h bar', have units of angular momentum. This constant is crucial for understanding the quantization of angular momentum, as the eigenvalues of the angular momentum operators are proportional to 'h bar', indicating their discrete nature.

💡Half-Integer

In the context of the video, half-integer refers to the possible values that the quantum number 'j' can take, which are non-integer values obtained by dividing an odd integer by two. The script explains that 'j' can take on both integer and half-integer values, which is a key distinction between orbital and spin angular momentum. For example, spin angular momentum can have 'j' values of one-half, indicating a fundamental difference from classical mechanics where angular momentum is always an integer.

💡Observables

Observables in quantum mechanics are physical quantities that can be measured, such as position, momentum, or energy. The video script discusses how certain operators, like 'j squared' and 'j3', can be used to define a set of compatible observables for angular momentum. These observables are essential for building the theory of angular momentum in quantum mechanics, as they can be simultaneously measured without affecting each other's outcomes.

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

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