Angular momentum eigenvalues

Summary

In this educational video, Professor MDA Science delves into the derivation of angular momentum eigenvalues in quantum mechanics. The video explains that angular momentum is quantized and cannot take any arbitrary value. It covers the mathematical derivation of these eigenvalues, introducing the concepts of ladder operators and the commutation relations of angular momentum components. The script guides viewers through the process of finding quantized eigenvalues for both the square of the angular momentum operator and its z-component, ultimately revealing the quantization of angular momentum and its significance in quantum mechanics.

Takeaways

• 🔬 The video discusses the derivation of angular momentum eigenvalues in quantum mechanics, emphasizing the quantization of angular momentum.
• 📐 Quantum mechanical quantities are represented by operators, with the eigenvalue equation being central to understanding measurement outcomes.
• 🧮 The angular momentum operator \( \vec{J} \) is composed of three components \( J_x, J_y, J_z \) that obey specific commutation relations.
• 🌀 The video introduces \( J^2 \) and \( J_z \) as compatible observables, which means they can have a common set of eigenstates.
• 🚫 The eigenvalues \( \lambda \) and \( \mu \) of \( J^2 \) and \( J_z \), respectively, are quantized and cannot take arbitrary values.
• 📈 The mathematical derivation includes the use of ladder operators \( J_+ \) and \( J_- \), which raise and lower the eigenvalues of \( J_z \).
• 💡 The expectation value of \( J^2 \) is always positive or zero, providing a constraint on the eigenvalues of angular momentum.
• 🔄 The application of ladder operators leads to the conclusion that there must be a maximum and minimum value for \( \mu \), which are quantized.
• 🔄 The allowed values of \( \mu \) are integer multiples of \( \hbar \), and the number of these values for a given \( \lambda \) is \( n + 1 \), where \( n \) is a non-negative integer.
• 🌐 The final result is that the eigenvalues of \( J^2 \) are \( j(j+1)\hbar^2 \) and the eigenvalues of \( J_z \) are \( m\hbar \), with \( m \) ranging from \( -j \) to \( +j \) in integer steps.

Q & A

What is the main focus of the video by Professor MDA Science?

The main focus of the video is to derive the angular momentum eigenvalues in quantum mechanics, explaining that angular momentum is quantized and not just any value is possible.

What is the significance of the eigenvalue equation in quantum mechanics?

The eigenvalue equation in quantum mechanics is significant as it helps determine the possible outcomes of a measurement, representing physical quantities through operators.

Why are the mathematical derivations in the video considered 'heavy'?

The mathematical derivations are considered 'heavy' because they involve complex mathematical steps and concepts, which require a deep understanding of quantum mechanics and linear algebra.

What is the role of the raising and lowering operators in the derivation of angular momentum eigenvalues?

The raising (j+) and lowering (j-) operators play a crucial role in determining the quantization of angular momentum by creating new eigenstates with incremented or decremented eigenvalues, respectively.

What does the commutation relation of angular momentum components imply?

The commutation relation of angular momentum components implies that they do not commute, meaning they cannot be simultaneously measured to arbitrary precision, which is a fundamental aspect of quantum mechanics.

Why is it necessary to find a common set of eigenstates for j squared and j3?

It is necessary to find a common set of eigenstates for j squared and j3 because they are compatible observables, allowing for a consistent description of the quantum state in terms of angular momentum.

What does the result that lambda must be larger than or equal to zero signify?

The result that lambda (the eigenvalue of j squared) must be larger than or equal to zero signifies that the square of the angular momentum, a physical observable, can only take non-negative values, reflecting the properties of real numbers.

How does the square of the eigenvalue mu relate to lambda?

The square of the eigenvalue mu is smaller than or equal to lambda, indicating that the eigenvalue associated with one component of the angular momentum vector (j3) is less than or equal to the eigenvalue associated with the square of the total angular momentum.

What is the significance of the quantization of angular momentum eigenvalues?

The quantization of angular momentum eigenvalues signifies that angular momentum can only take discrete values, which is a fundamental principle in quantum mechanics and has implications for the understanding of atomic and subatomic particles.

What are the implications of the video's findings for orbital and spin angular momentum?

The findings of the video imply that both orbital and spin angular momentum are quantized, which is essential for understanding the behavior of electrons in atoms and the properties of subatomic particles.

Outlines

00:00

🔬 Introduction to Angular Momentum Eigenvalues

Professor MDA Science introduces the topic of angular momentum eigenvalues in quantum mechanics. The video aims to derive the quantized values of angular momentum, explaining that physical quantities in quantum mechanics are represented by operators and that the eigenvalue equation is crucial for understanding measurement outcomes. The lecture will cover the mathematical derivation of angular momentum eigenvalues, emphasizing that the process is similar to that used for the quantum harmonic oscillator. The professor also mentions a companion video for further discussion on angular momentum's features and conventions.

05:05

📐 Deriving the Eigenvalue Equations

The lecture continues with a refresher on angular momentum in quantum mechanics, focusing on the vector operator J composed of J1, J2, and J3. The angular momentum operator obeys specific commutation relations, which include the Levi-Civita symbol and Einstein notation. The video explains that J squared, the sum of the squares of the components, commutes with each individual component, allowing for a set of compatible observables. The professor then derives the eigenvalue equations for J squared and J3, labeling the eigenstates by the eigenvalues lambda and mu. The goal is to find the quantized values of lambda and mu, which are constrained to be positive or zero for lambda.

10:07

🔍 Constraints on Angular Momentum Eigenvalues

The video delves into proving constraints on the eigenvalues of angular momentum. It begins by demonstrating that lambda, the eigenvalue of J squared, must be non-negative. This is shown by considering the expectation value of J squared and using the properties of operators. The professor then proves that the square of the eigenvalue mu, associated with J3, must be less than or equal to lambda. This is done by using a result from a previous video on ladder operators and calculating the expectation value of a specific operator expression. The lecture concludes that the eigenvalues of angular momentum are not arbitrary but must adhere to these constraints.

15:08

🌐 The Ladder to Quantization

The lecture explores the implications of the ladder operators, J+ and J-, on angular momentum eigenstates. It explains that applying J+ to an eigenstate increases the eigenvalue of J3 by a constant, while applying J- decreases it. The video argues that there must be a maximum and minimum value for mu, the eigenvalue of J3, to prevent reaching forbidden regions of the real axis. The professor uses diagrams to illustrate how applying the raising operator repeatedly could lead to a contradiction unless there is a maximum value for mu. This maximum value is reached when applying J+ results in the state being 'killed,' thus terminating the ladder and maintaining the constraints on mu.

20:10

🎓 Final Conditions and Conclusions

The final part of the lecture presents the final conditions for the eigenvalues lambda and mu. It is shown that mu must be an integer multiple of a constant, h bar, and that lambda must be of the form n(n+1) times h bar squared, where n is a non-negative integer. The lecture concludes that angular momentum is quantized, with lambda and mu taking on specific quantized values. The professor summarizes the derivation, explaining that the allowed values of j (from lambda) and m (from mu) are integral to understanding angular momentum in quantum mechanics. The video ends with a reminder to watch a companion video for further exploration of angular momentum properties and encourages viewers to subscribe for more content.

Keywords

💡Angular Momentum

Angular momentum is a physical quantity that describes the rotational motion of an object around an axis. In quantum mechanics, it is represented by operators and is quantized, meaning it can only take on discrete values. The video discusses the derivation of angular momentum eigenvalues, which are the possible outcomes of a measurement of angular momentum. The script mentions that angular momentum includes both orbital and spin angular momentum, with the latter not having a classical analog.

💡Operators

In quantum mechanics, operators are mathematical functions that act on the state of a system to produce a new state or a measurable outcome. The script explains that physical quantities in quantum mechanics are represented by operators, and the angular momentum operators are denoted by j1, j2, and j3. These operators are crucial for deriving the eigenvalues associated with angular momentum.

💡Eigenvalue Equation

The eigenvalue equation is a fundamental concept in quantum mechanics that relates an operator to its corresponding eigenvalues and eigenstates. The script uses the eigenvalue equation to explore the quantization of angular momentum, showing that the eigenvalues are not arbitrary but follow specific patterns. The equation is used to find the eigenvalues lambda and mu for the operators j squared and j3.

💡Quantization

Quantization refers to the process by which certain physical quantities can only take on discrete values rather than a continuous range. The video script discusses how angular momentum is quantized, meaning it cannot take just any value but is restricted to specific, quantized values. This is a key result of the mathematical derivation presented in the video.

💡Ladder Operators

Ladder operators, also known as raising and lowering operators, are used in quantum mechanics to change the eigenvalues of an operator by a fixed amount. The script mentions j+ and j- as ladder operators that can increase or decrease the eigenvalue of j3 by a constant amount, which is crucial for understanding the quantization of angular momentum.

💡Commutation Relations

Commutation relations describe the algebraic relationship between operators, specifically how the order in which they are applied affects the outcome. The script states that the angular momentum components j1, j2, and j3 obey certain commutation relations, which are essential for defining the angular momentum operator in quantum mechanics.

💡Eigenstates

Eigenstates are the states of a quantum system that correspond to the eigenvalues of an operator. The video script discusses finding a common set of eigenstates for the operators j squared and j3, which are labeled by their respective eigenvalues lambda and mu. These eigenstates are fundamental to understanding the quantization of angular momentum.

💡Expectation Value

The expectation value is a statistical measure used in quantum mechanics to predict the average result of a measurement. In the script, the expectation value of j squared is used to prove that the eigenvalue lambda must be positive or zero, which is a key step in the derivation of angular momentum eigenvalues.

💡Raising and Lowering

Raising and lowering refer to the actions of the ladder operators j+ and j- on the eigenstates of angular momentum. The script explains that applying these operators can increase or decrease the eigenvalue of j3, respectively. This process is essential for understanding the quantization of angular momentum and the limits on the possible values of the eigenvalues.

💡Quantum Harmonic Oscillator

The quantum harmonic oscillator is a model in quantum mechanics used to describe systems with oscillatory behavior. The script mentions that the strategy used to derive the eigenvalues of angular momentum is similar to that used for the quantum harmonic oscillator, indicating a broader application of the concepts discussed in the video.

Highlights

Introduction to deriving angular momentum eigenvalues in quantum mechanics

Physical quantities in quantum mechanics are represented by operators

Exploration of the eigenvalue equation of angular momentum operators

Angular momentum is quantized and cannot take just any value

Derivation of angular momentum eigenvalues is similar to that of the quantum harmonic oscillator

Definition of a general angular momentum in quantum mechanics

The importance of the commutation relations in angular momentum

Introduction of the operator j squared and its commutation with j3

Eigenvalue equations for j squared and j3 and their common eigenstates

Proof that the eigenvalue lambda is always positive or zero

Derivation that the square of the eigenvalue mu is smaller than or equal to lambda

Explanation of ladder operators and their properties

Demonstration that there must be a maximum value for mu, called mu max

Argument for the quantization of angular momentum eigenvalues

Final conditions for lambda and mu and their quantized values

Introduction of quantum numbers j and m for angular momentum

Conclusion on the quantization of angular momentum and its implications

Transcripts

hi everyone

this is professor mda science and today

we're going to derive

the angular momentum eigenvalues in

another one of our videos on rigorous

quantum mechanics

we know that physical quantities in

quantum mechanics are represented by

operators

and that the most important equation

associated with theseophoresis

is the eigenvalue equation for example

it can tell us about the different

outcomes of a measurement

today we will explore the eigenvalue

equation of angular momentum operators

we will learn that angular momentum

cannot take just any value it is

actually quantized

this video will cover the mathematical

derivation

of angular momentum eigenvalues and

although it is a little bit heavy

on the mathematical side it has some

really neat very visual steps that will

give you a very clear

understanding of what is going on it is

also very similar to the strategy used

to figure out the eigenvalues of the

quantum harmonic oscillator

so what you will learn today will be

useful beyond angular momentum

you can also find a companion video

linked in the description where you will

find

a discussion of the more interesting

features and conventions

associated with the eigenvalues of

angular momentum

so let's go let's start with a quick

refresher of angular momentum in quantum

mechanics

we consider a vector operator j that is

made of three operators

j1 j2 and j3 the operator j

is an angular momentum operator if the

three components

obey these commutation relations

you will remember from the video on

angular momentum that epsilon ijk is the

levitivita symbol

and that we're using einstein notation

so that an expression like this

implies a sum over repeated indices in

this case

over the k indices this is the

definition of a general angular momentum

in quantum mechanics which includes the

orbital angular momentum that we're

familiar with from classical mechanics

as well as the spin angular momentum

that doesn't have a classical analog

today we will work with the general

angular momentum so the results will in

principle be

relevant for both orbital and spin

variants

now another important thing to remember

is that as the different angular

momentum components don't commute

they don't form a set of compatible

observables

what we found in the video that

introduces angular momentum

was that we can define a new operator j

squared

equal to j 1 squared plus j 2 squared

plus j

3 squared and that this operator j

squared 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

which is typically chosen to be j3

as for any observable the very first

thing we need to do is to find

their eigenvalues for j squared we have

this eigenvalue

equation where the eigenvalue is lambda

and for j3 we have this eigenvalue

equation

where the eigenvalue is mu

as j squared and j3 are compatible

observables

we can always find a common set of

eigenstates for these two operators

and what i've done here is to use such a

common set of eigenstates

here and here and label them by the

eigenvalues

lambda and mu

the objective of this video is pretty

straightforward

we want to find out what the eigenvalues

lambda and mu

are what we will discover as we do this

is that lambda and mu cannot take just

any value

but can only take some very special ones

this means that angular momentum in

quantum mechanics

is quantized and on top of that the

derivation is actually quite fun

and really eye-opening so i'm sure

you're all priming with excitement so

let's just

get started first we will cover a few

prerequisites

the raising operator j plus and the

lowering of rater j

minus are collectively called the ladder

operators

and what i have here is a long list of

the properties

all these results should be familiar to

you because we derived them in detail in

the video on ladder operators

i'm writing this list here because we'll

use many of these results to figure out

what are the possible eigenvalues of the

angular momentum operators

what i will do is to quote the results

from this list as needed

during the video we don't have to

remember these results they're simply

tools that we want to use in our

derivations

so if you're happy with this then just

continue with me

however if you first want a reminder on

where these results come from or if

you've never seen them before

then i really encourage you to first

check out the video on ladder operators

that is linked in the description

and continue with this video after that

the very first result i want to prove is

that the eigenvalue lambda

is always positive or zero

to do that consider the expectation

value of j squared

with respect to an arbitrary state psi

using the definition of j squared we can

expand this into three terms

the expectation value of j1 squared the

expectation value of j2 squared

and the expectation value of j3 squared

the angular momentum components are

emission which means that ji squared

equals ji

dagger ji and in turn this means that we

can rewrite

these three terms like this

this is now the definition of the norm

squared of j1 psi

this is a norm squared of j 2 psi

and this is the norm squared of j 3 psi

as a norm is larger than or equal to

0 then this whole expression is larger

than

or equal to zero overall this means that

psi j squared psi

is larger than or equal to zero for any

state psi

in particular this must be true when the

state psi

is an eigen eigenstate of the operator j

squared

but in this case we can use the

eigenvalue equation

to write this as lambda lambda mu

lambda is a scalar so we can take it out

into the left

so that we get lambda times the bracket

lambda mu

lambda mu being larger than or equal to

zero this is simply one

so in conclusion lambda must be larger

than

or equal to zero

okay so this is our first constraint on

the angular momentum eigenvalues

all eigenvalues of j squared are

positive numbers or zero

if we think about this result for a

moment it actually makes sense

j squared is a physical observable that

is given by the square of the angular

momentum operators

squares of real numbers are always

positive or 0 so it makes sense that the

eigenvalues of j squared

which are the possible outcomes of a

measurement of this quantity will also

be positive or zero

the next result i want to prove is that

the square of the eigenvalue mu

is smaller than or equal to lambda

the first ingredient we need is one of

the results from the video on ladder

operators

we can write j squared minus j3 squared

as equal

to one half times j plus

j minus plus j minus j plus

as the ladder operators are at joints of

each other

we can also write this expression using

the add joints

we now consider the right hand side of

this expression and calculate its

expectation value with respect to a

common eigenstate of j

squared and j3 we can expand this

expression into two terms

and this term is the norm squared of j

minus lambda mu

and this term is the norm squared of j

plus lambda mu

as norms are always larger than or equal

to zero then this whole expression

is larger than or equal to zero

if we now use the left-hand side of the

equality at the top

this result now implies that the

expectation value of j

squared minus j three squared with

respect to lambda mu

must also be larger than or equal to

zero

expanding this expression we get these

two terms

we can now use the eigenvalue equation

for j squared

here and the eigenvalue equation for j3

twice here and we end up with these two

terms

this bracket is equal to 1 and so is

this one

so that overall we get that mu squared

is smaller than

or equal to lambda

so this here is our second constraint on

the angular momentum eigenvalues

if we think about this result for a

moment we find that it also makes sense

j3 is one of the three components of the

vector operator j

so it makes sense that the square of the

eigenvalue associated with j3

is smaller than the eigenvalue

associated with the square of j

we will actually find later in this

video that this condition

is even more stringent than what we have

here

but to build some suspense i won't say

any more now so you'll have to keep

watching to find out

the next step starts with another result

from the video on ladder operators

the raising operator j plus acting on an

eigen state

lambda mu of j squared and j3 is

equal to this prefactor multiplying this

new eigenstate of j squared and j3

so what is the action of the raising

operator on an angular momentum

eigenstate

it generates another angular momentum

eigenstate

which has the same j squared eigenvalue

lambda

but whose j 3 eigenvalue mu

has increased by a constant h bar

now this is the result we got in the

video on ladder operators

and it is a valid general result but

crucially it also includes the

possibility of a particular result

which is that the action of j plus on

the state lambda mu

is equal to zero this will happen if

lambda

is equal to mu squared plus mu h bar

because in this case the prefactor here

vanishes

so why do we need these results what we

want to prove

next is that there must be some angular

momentum eigenvalues lambda and mu

for which this equality holds

to see this consider the real axis we

have

0 here and we want to draw allowed

values of the mu

eigenvalue on this real axis what we

know so far

is that mu squared is smaller than or

equal to lambda

this means that we can place the square

root of lambda here

and minus the square root of lambda here

and mu must always be

somewhere between these two values so we

can discard this region

and this region of the real axis as off

limits

for mu let's imagine that mu is here

within the allowed region this

corresponds to the eigenstate

lambda mu we can now apply the raising

operator to this state

and we get a new state at mu plus h bar

which is another angular momentum

eigenstate with the same lambda

but a mu that is larger by h bar

we can now apply j plus again to get a

new state at

mu plus 2 h bar which again is another

angular momentum eigenstate

and in principle we could continue up

the ladder by applying j

plus again and again and again however

you may already see that at some point

we will have a problem

let's zoom into this region near the

square root of

lambda we can draw the real axis again

in this zoomed in region and let's also

draw the points

square root of lambda minus h bar and

square root of lambda minus 2 h bar

let's imagine that we've been applying

the j plus operator sequentially

and after applying it p minus 1 times we

land here

and we can then apply it again but then

we end up with this new state at

mu plus ph bar so here is the problem

if we now try to apply it one more time

we would land somewhere in the forbidden

region

beyond square root of lambda

so something has gone horribly wrong in

our construction

because we've ended up with a state for

which mu

is larger than square root of lambda

which we know

is impossible so what does this mean

if we pick mu at random like we did here

and we apply j plus enough times then we

will always eventually get past

square root of lambda violating this

condition that we have up here

as this is not possible this means that

the mu that we started with is in fact

not an allowed value of mu because we've

just seen that it eventually leads to a

violation

of the upper bound looking at these

diagrams again

you may now think that in fact no value

of mu will actually be allowed

because in principle we can always apply

j plus to increase the value

past the upper bound square root of

lambda

so does this mean that there can be no

allowed value of mu

applying the raising operator as shown

in case 1

means that we can go up the mu ladder

indefinitely

which we've just found leads to a

violation of the upper limit of mu

and this is precisely where case 2 comes

in

it allows us to terminate the ladder

to see how this comes about let's

restate the problem

the fact that mu cannot be larger than

square root of

lambda implies that mu must have

some maximum value which i call mu max

but how can mu have a maximum value if

we can keep

increasing its value by h bar by simply

applying the raising operator like this

the answer of course is that at some

point the application of j

plus must kill the state

and how can it happen well this is

precisely what this case here

allows us to do if we ever reach a mu

such that this equation holds then the

next time that we apply

j plus we will kill the state and

terminate the ladder

this means that if we are to find

any allowed values of mu there must be a

mu for which this equation holds

and by definition that will be the

maximum mu

that we can reach which we have called

mu max

mathematically we can start with a state

with eigenvalue mu

and we now know that this mu cannot be

arbitrary

there must be an integer p for which

applying the raising operator p times we

end up

with a state with eigenvalue mu plus

ph bar and for which this value we end

up with

must be exactly equal to mu max

this is necessary because then when we

apply j

plus again we will kill the state

terminating the ladder and thus obeying

the upper bound

on the values of mu if we draw our real

actors again

then our discussion tells us not only

that mu must be smaller than the square

root of lambda

so that this region is off limits but

also that it can only take some very

special values

along the allowed region these special

values are mu max

which for a given lambda is then

uniquely given by this equation

and then successively mu max minus h bar

mu max minus 2 h bar and so on

so this here is our new constraint on

the angular momentum eigenvalues

mu is not only forced to be within minus

square root of lambda

and plus square root of lambda but it

can actually only take

some very specific values in this

allowed region

any other value of mu that is not an

integer multiple of h bar

below mu max would mean that applying j

plus enough times we would cross into

the forbidden region

thus disqualifying any such mu as a

possible eigenvalue

and this of course means you've probably

guessed it already that the new

eigenvalues are quantized

as you can imagine we can make an

analogous argument by considering the

action of the lowering operator

on an angular momentum eigenstate from

the video on ladder operators

the action of the lowering operator is

such that it decreases the mu

eigen value by h bar and again we can

have a second situation in which the

action of j

minus kills the state which happens if

lambda equals mu squared minus mu h bar

in this case starting with a state with

eigenvalue mu

we then apply the lowering operator q

times to end up with a state with

eigenvalue mu

minus qh bar and the j3 eigenvalue of

this final state

must equal some minimum mu which i call

mu min

such that mu min squared minus mu min h

bar

is equal to lambda

this is necessary to terminate the

ladder when we apply

j minus one final time

analogously to the case for the rating

operator we can draw the real axis

and we must terminate the ladder given

by the lowering operator

to ensure that we never cross into the

forbidden region

below minus square root of lambda

so these are our two new conditions one

of the values that mu takes must be mu

max

such that this equation is debate and

another one of the values that mu takes

must be mu min such that this

other equation is obeyed putting these

two equations together

we have that mu max squared plus mu max

h bar is equal to mu min squared

minus mu min h bar this in turn

means that mu max is equal to minus mu

min

let's now draw our real axis again

and place the barriers at square root of

lambda

and minus square root of lambda and we

now know that mu must actually be

somewhere between

mu min and mu max

we are now ready for the final step

let's imagine that this here is an

allowed meal

then we must reach this mu max by

applying the racing operator

some number of times say p times

this means that the distance between mu

and mu max is equal to an

integer multiple of h bar

similarly from the same mu we must reach

this mu min by applying the logging

operator some number of times

say q times this means that the distance

between

mu and mu min is also equal to an

integer multiple of

h bar so putting these two together

the distance between mu min and mu max

must be equal to an integer multiple of

h bar

so let's call that integer n

mathematically this means that mu max is

equal to mu min

plus n times h bar for some positive

integer

n

we're now almost there we just figured

that mu max is equal to mu min plus nh

bar where n is an integer

we also know that mu min is equal to

minus mu max

so we can rewrite this equation like

this

and this means that mu max is equal to n

over 2

times h bar for integer n

and it then follows that mu min is equal

to minus a half

in h bar going back to the relation

between lambda and

mu max we have that lambda must equal

n over 2 h bar all squared plus

n over 2 h bar squared which is equal to

n over 2 times n over 2 plus 1

times h bar squared

so these are the final conditions for

lambda and mu

lambda can only have this form

where n is 0 or a positive integer

and for a given value of lambda then mu

can be its minimum value or we can add

one h bar or another h bar and so on

until we get to one h bar short of the

maximum value

and then we can of course have the

maximum value

so for a given lambda the only allowed

values of mu are the ones on this list

and there will be a total of n plus one

allowed values of mu

for a given lambda

so we've done it we started with the

general eigenvalue equations for j

squared and for j3

where the eigenvalues lambda and mu are

general

and we have now determined that the only

allowed values of lambda

are of this form

and for a given lambda then that can be

n plus 1 different values of mu

which are given by this list in steps of

h bar

in the theory of angular momentum what

we do is we call the number

n over 2 j and as

n is a positive integer or zero then the

allowed values of j

are zero one half one three halves two

and so on in this language

lambda is equal to j times j plus one

times

h bar squared we also write mu as equal

to

m h bar and the allowed values of m

are minus j minus j plus one minus j

plus two

all the way to j minus 1 and j

we can finally take the original

eigenvalue equations with general lambda

and mu

and write them again using the allowed

values of

lambda and of mu

where i have re-labeled the common

eigenstates using

the quantum numbers j and m as is

usually done

the derivation was a little bit lengthy

but i think it was worth it because now

we've demonstrated that angular momentum

is indeed quantized

also don't forget to look at the

companion video

where we will take the eigenvalues of

the angular momentum operator that we've

derived today and look at some of the

more general properties

you can also learn more about the role

of angular momentum eigenvalues by

checking out our videos on orbital and

spin angular momentum and as always

if you liked the video please subscribe