Ladder operators in angular momentum

Summary

In this video, Professor MDA Science delves into the concept of ladder operators in quantum mechanics, focusing on their role in manipulating angular momentum. Ladder operators, denoted as 'j+' and 'j-', allow for discrete changes in angular momentum values, akin to stepping up or down a ladder. The video explores their mathematical properties, including non-hermiticity and their relationship with angular momentum operators 'j^2' and 'j3'. The practical application of ladder operators in quantum mechanics is highlighted, particularly in the context of angular momentum eigenstates, where they function as raising and lowering operators, respectively. The video concludes with a discussion on the significance of ladder operators in broader quantum mechanics topics like the quantum harmonic oscillator and second quantization.

Takeaways

• 🧑‍🏫 This video discusses ladder operators in the context of angular momentum in quantum mechanics.
• 🔢 Ladder operators allow discrete changes in angular momentum, which is quantized in quantum mechanics.
• 🪜 The ladder operators, labeled as raising (J⁺) and lowering (J⁻), increase or decrease the angular momentum by a discrete amount.
• 📏 Angular momentum in quantum mechanics consists of both orbital (L) and spin (S) components, with no classical analog for spin.
• 🔄 J₁, J₂, and J₃ do not commute, making them incompatible observables. Instead, J² and J₃ are used as a set of commuting observables.
• ➕ The raising operator J⁺ is defined as J₁ + iJ₂, and the lowering operator J⁻ is J₁ - iJ₂.
• 🧮 Ladder operators are not Hermitian and therefore are not observables, but they play a critical role in calculations.
• 📈 Acting on eigenstates of angular momentum, J⁺ raises the J₃ eigenvalue by h-bar, and J⁻ lowers it by h-bar.
• ⚙️ Angular momentum eigenstates have specific values, with J squared taking the form J(J+1) h-bar², and J₃ taking integer or half-integer values.
• 📚 Ladder operators are used in many areas of quantum mechanics, including the quantum harmonic oscillator and second quantization.

Q & A

What are ladder operators in the context of angular momentum in quantum mechanics?

Ladder operators are linear combinations of the angular momentum components J1 and J2 that allow changing the value of angular momentum by discrete amounts. They are used to move up and down a 'ladder' of discrete angular momentum values.

Why are ladder operators useful in quantum mechanics?

Ladder operators are useful because angular momentum in quantum mechanics is quantized and can only change in discrete steps. These operators help transition between different angular momentum states efficiently.

How are the raising and lowering ladder operators defined?

The raising operator J+ is defined as J1 + iJ2, and the lowering operator J- is defined as J1 - iJ2. They are linear combinations of the angular momentum components J1 and J2.

Do ladder operators commute with J3 and J2?

No, the ladder operators do not commute with J1 and J2 individually. However, J+ and J- commute with the total angular momentum operator J².

How do ladder operators affect the eigenvalues of angular momentum?

When acting on the eigenstates of the angular momentum operator J3, the raising operator J+ increases the eigenvalue by h-bar, and the lowering operator J- decreases it by h-bar.

What is the relationship between ladder operators and observables in quantum mechanics?

Ladder operators themselves are not observables because they are not Hermitian. However, they are useful in transforming states, while the total angular momentum (J²) and one component (usually J3) remain the observables.

What are the commutation relations involving ladder operators?

The key commutation relations are: [J², J+] = [J², J-] = 0, [J3, J+] = h-bar J+, [J3, J-] = -h-bar J-, and [J+, J-] = 2h-bar J3.

How does the norm of states change when acted upon by ladder operators?

The norm of a state when acted on by a ladder operator is proportional to a square root involving the angular momentum eigenvalue (J) and its projection (M). For example, J+ increases M by one unit of h-bar, while J- decreases M by one unit.

What are the allowed values for the eigenvalues of J² and J3 in quantum mechanics?

The eigenvalue of J² is J(J+1)h-bar², where J can be 0, 1/2, 1, 3/2, etc. For J3, the eigenvalue is M h-bar, where M takes values from -J to J in steps of 1.

How are ladder operators used in other areas of quantum mechanics?

Ladder operators are not only used in angular momentum problems but also play a critical role in the quantum harmonic oscillator and more advanced topics like second quantization.

Outlines

00:00

📚 Introduction to Ladder Operators in Quantum Mechanics

Professor MDA Science introduces the concept of ladder operators in the context of angular momentum in quantum mechanics. Ladder operators are tools that allow for the manipulation of angular momentum values in discrete steps, reflecting the quantized nature of angular momentum. The video promises to delve into the mathematical details of these operators, which are not only crucial for understanding angular momentum but also applicable in other areas such as the quantum harmonic oscillator and second quantization. A brief reminder of angular momentum in quantum mechanics is provided, discussing the operator J composed of three components (J1, J2, J3) that obey certain commutation relations. The video also touches on the distinction between orbital and spin angular momentum, emphasizing the generality of the discussion by using the operator J.

05:03

🔍 Exploring Ladder Operators and Their Properties

The script explains that ladder operators, specifically J+ and J-, are linear combinations of J1 and J2 and are not Hermitian, meaning they are not observables themselves. However, they are adjoint to each other, with J+ being the raising operator and J- the lowering operator. The video then explores the flexibility in incorporating J1 and J2 into quantum mechanics theory, often preferring to work with J+ and J- due to their convenience. The script derives expressions for J squared in terms of J+ and J-, which are essential for understanding how these operators act on the angular momentum states. The commutator relations involving J squared, J3, J+, and J- are also discussed, laying the groundwork for understanding how these operators interact within the quantum mechanical framework.

10:04

🌐 Action of Ladder Operators on Eigenstates

The video script discusses how ladder operators act on the eigenstates of angular momentum. It starts by establishing the eigenvalue equations for J squared and J3, which are essential for describing angular momentum in quantum mechanics. The eigenvalues lambda and mu are shown to have specific forms related to the quantum number j and its magnetic quantum number m. The script then demonstrates that applying J+ to an eigenstate results in a new state that is also an eigenstate of J squared but with an increased eigenvalue for J3 by h-bar, thus confirming J+ as a raising operator. Conversely, J- is shown to decrease the eigenvalue of J3 by h-bar, making it a lowering operator. This section provides a clear understanding of how ladder operators can 'raise' or 'lower' the quantum states within the quantized energy levels.

15:05

📉 Detailed Analysis of Ladder Operator Actions

This part of the script provides a detailed analysis of the action of ladder operators on eigenstates. It discusses the norm of the state generated by applying the raising operator and shows how it relates to the original state with an increased eigenvalue for J3. The script also covers the action of the lowering operator, demonstrating a similar process but with a decrease in the eigenvalue of J3. The section concludes with a discussion on rewriting the results in terms of the allowed values of lambda and mu, providing a reference for future use. The importance of understanding these actions is emphasized, as they are fundamental to the study of angular momentum in quantum mechanics.

20:06

🎓 Conclusion and Further Applications

The script concludes by summarizing the main results of the video, emphasizing the utility of ladder operators in the study of angular momentum in quantum mechanics. It encourages viewers to refer back to these results as they continue their studies, suggesting that the concepts covered are not only fundamental but also frequently used. The video also hints at the broader applications of ladder operators in other areas of quantum mechanics, such as the quantum harmonic oscillator and second quantization. The presenter invites viewers to explore these topics further and ends with a prompt for viewers to subscribe for more content.

Keywords

💡Ladder Operators

Ladder operators, also known as raising and lowering operators, are fundamental in quantum mechanics for their ability to change the quantum state of a system by discrete amounts. In the context of angular momentum, these operators allow for transitions between different quantized states. The script discusses how ladder operators are used to 'go up and down a ladder of discrete angular momentum values,' illustrating their utility in manipulating quantum states.

💡Angular Momentum

Angular momentum is a key concept in both classical and quantum physics, describing the rotational motion of objects. In quantum mechanics, as highlighted in the script, angular momentum is quantized, meaning it can only take on discrete values. The video emphasizes the importance of understanding angular momentum in quantum systems, as it is central to the operation of ladder operators.

💡Quantization

Quantization refers to the process by which certain physical quantities can only change in discrete, rather than continuous, amounts. The script mentions that 'angular momentum is quantized' and can only change 'in discrete steps,' which is why ladder operators are so useful as they allow for these discrete changes in angular momentum states.

💡Eigenstates

Eigenstates are the states of a quantum system that correspond to definite values of an observable quantity. The script discusses how ladder operators act on eigenstates of angular momentum, changing their eigenvalues in a predictable manner. This is crucial for understanding how quantum systems evolve and interact.

💡Eigenvalues

Eigenvalues are the specific values that an observable quantity can take on when measured in an eigenstate. The script explains how the eigenvalues of angular momentum, denoted by lambda and mu, change when ladder operators are applied, either increasing or decreasing by a fixed amount (h-bar), which is a fundamental constant in quantum mechanics.

💡Hermitian Operators

Hermitian operators are a class of operators in quantum mechanics that are equal to their own adjoint. The script notes that ladder operators are not Hermitian, which means they are not observables themselves. However, they are essential for manipulating the eigenstates of observables like angular momentum.

💡Commutators

Commutators are used in quantum mechanics to determine whether two operators can be simultaneously measured with precision. The script discusses how the commutators of ladder operators with angular momentum components are crucial for understanding their properties and how they transform quantum states.

💡Raising Operator (j+)

The raising operator, denoted as j+, is one of the ladder operators used to increase the eigenvalue of a quantum state by a discrete amount. The script explains that 'acting with the raising operator multiple times allows us to go up a ladder of eigenvalues,' which is essential for understanding quantum transitions in systems with quantized angular momentum.

💡Lowering Operator (j-)

The lowering operator, denoted as j-, is the counterpart to the raising operator, used to decrease the eigenvalue of a quantum state. The script mentions that 'acting with the lowering operator... allows us to go down a ladder,' which is crucial for understanding how quantum systems can transition to lower energy states.

💡Normalization

Normalization is a process in quantum mechanics where wave functions are scaled so that their integral over all space equals one, ensuring probabilistic interpretations are valid. The script touches on normalization when discussing the action of ladder operators on eigenstates, indicating that the new states generated must also be normalized to be valid quantum states.

💡Orthonormal Set

An orthonormal set is a collection of vectors (or quantum states) that are orthogonal (perpendicular) to each other and normalized. The script refers to eigenstates forming an orthonormal set, which is important for the mathematical manipulation and interpretation of quantum states, particularly in the context of ladder operators.

Highlights

Ladder operators are essential in quantum mechanics for changing angular momentum values discretely.

Angular momentum in quantum mechanics is quantized and can only change in discrete steps.

Ladder operators are used extensively in quantum mechanics, including in the quantum harmonic oscillator and second quantization.

Angular momentum operator J is composed of three components J1, J2, and J3, which must obey certain commutation relations.

J squared commutes with each individual component, suggesting a set of commuting observables can be built.

Ladder operators are defined as linear combinations of J1 and J2, with J+ being the raising operator and J- the lowering operator.

Ladder operators are not Hermitian and thus are not observables; they are each other's adjoint operators.

The squared angular momentum J^2 can be expressed in terms of J+ and J-, which is useful for studying angular momentum.

Computation relations between J^2, J3, J+, and J- are crucial for understanding the action of ladder operators.

Ladder operators act on eigenstates of angular momentum to produce new eigenstates with altered eigenvalues.

J+ raises the eigenvalue of J3 by h-bar, while J- lowers it, hence their names raising and lowering operators.

The norm of a state changed by a ladder operator can be calculated, revealing the change in the eigenvalue of J3.

Eigenstates of angular momentum can be labeled by j and m, with j determining the possible values of m.

The action of ladder operators on eigenstates can be rewritten using the allowed values of lambda and mu.

Ladder operators are not only useful for angular momentum but also have applications in other areas of quantum mechanics.

The video concludes with a summary of the main results, emphasizing the utility of ladder operators in quantum mechanics.

Transcripts

hi everyone

this is professor mda science and today

we're going to be talking about lateral

apparatus in angular momentum

in another one of our videos on rigorous

quantum mechanics in the context of

angular momentum and as their name

suggests ladder operators allow us to

change the value of the angular momentum

by some discrete amount

therefore going up and down a ladder of

discrete angular momentum values the

reason why lateral apparatus are so

useful

is that in quantum mechanics angular

momentum is quantized

and so it can only change in discrete

steps now

this video is going to be a little

mathematical because we're going to look

into the details of ladder operators but

it will be really useful because we're

going to use them all the time

in other videos on angular momentum and

on top of that

similar concepts are used in other areas

such as the quantum harmonic oscillator

or even in more advanced topics such as

second quantization

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

reminder of angular momentum in quantum

mechanics

and for the full details you should

check the corresponding video linked in

the description

in quantum mechanics an angular momentum

operator j

is an operator made of three separate

components j1

j2 and j3 the key requirement for these

components

is that they must obey these computation

relations

and you will remember from the video on

angular momentum that epsilon ijk

is the levitivitas 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 so this is a definition

of a general angular momentum in quantum

mechanics this includes

orbital angular momentum which we

typically denote by the letter

l and which plays the same role

as the equivalent quantity in classical

mechanics

however in quantum mechanics we also

have this spin angular momentum

without a classical analog if we want to

specify that we're working with spin

then we typically use the letter s

today we will keep the discussion

general so we'll work in terms of the

general angular momentum j

up here and all results we derive will

be valid for

any type of angular momentum

as the different angular momentum

components don't commute they

don't form a set of compatible

observables so we cannot use j1 j2 and

j3

as the basic building blocks of the

theory

instead we found in the video on angular

momentum

that the operator j squared does commute

with each individual component this

suggests that we can build a set

of commuting observables by combining j

squared and one of the components ji

the conventional choice is to use j3

although in principle we could have

chosen any other component

so moving forward we will describe the

physical observables of angular momentum

through these two operators j1 and j2

are still needed for the development of

the theory

but since they are not part of the set

of compatible observables

then we have some flexibility in how

exactly we work

with j1 and j2 it turns out that in the

theory of angular momentum

it is often more convenient to work with

a specific set

of linear combinations of j1 and j2

rather than using j1 and j2 directly

i therefore introduce you to the lateral

writers

as you probably guessed the lateral

braces are linear combinations of j1 and

j2

and there are two of them the first one

is called the racing operator

we label it with j plus and it is

defined as equal to j1

plus ij2 the second one is called

the lowering operator we label it with j

minus and it is defined as equal to j1

minus i j2

at this stage we're just going to accept

that these operators are called

racing and lowering operators and

collectively called lateral apparatuses

but the rationale for these names will

become clear later in the video

ladder operators play a key role in the

study of angular momentum in quantum

mechanics

so the objective of this video is to

work through some of the properties that

will prove

extremely useful in many other videos

the very first property of lateral

parasites is that they are

not hermitian so they are not

observables

in fact j plus and j minus are each

other's adjoint operators

to see this consider j plus dagger

it equals j 1 plus i j j2 dagger

and we can expand this bracket and then

using the fact that j1 and j2 are

emission

we get this this is the definition of

j minus of course the opposite follows

and j

minus dagger is equal to j plus

the angular momentum of races that we

had been working with

until now where j squared and then

j3 j1 and j2 but as we discovered today

we can actually be flexible in how we

incorporate j1 and j2 into the theory

and very often it will prove more

convenient to work in terms of j

plus and j minus which are linear

combinations of j 1 and j

2. the fact that j plus and j minus

aren't observables is not a problem

because we're anyway restricted to j

squared and j 3 as the set of commuting

observables

therefore from now on we are going to

consider either this set here

or this new set of four operators

which contains the same information

given that this second set with j plus

and j minus

is very useful in many problems in the

rest of this video we're going to

explore the properties of the ladder

operators

the squared angular momentum is

generally given by this expression

in terms of the angular momentum

components

however given that we'll mostly use

these four operators to study angular

momentum in quantum mechanics we should

write j

squared in terms of j plus and j minus

rather than in terms of j1 and j2

to accomplish this consider j plus times

j minus

explicitly writing the definitions of j

plus and j minus gives

this and carrying out the multiplication

we get

these four terms

the first two carry through unchanged

and these last two terms can be grouped

into this

commutator the commutator is equal to ih

bar j3

so that we get j1 squared plus j2

squared

plus h bar j3

we can use the expression of j squared

in terms of the angular momentum

components up here

to rewrite the sum of these two terms as

j squared minus j 3 squared

and we get to the final expression j

squared

minus j 3 squared plus h bar

j 3. repeating the same exercise with

the product

j minus times j plus

we get j squared minus j 3 squared minus

h bar j three

we can now add these two expressions

and this equals two j squared minus

two j three squared and we can finally

rearrange to isolate j squared and we

get that j

squared is equal to one half j plus j

minus plus j minus j plus

plus j three squared

of course you don't need to remember any

of these

but we're going to use them constantly

in our study of angular momentum so you

should either be ready to derive them as

needed

or simply keep a list for reference

let's look at the computation relations

j

squared trivially commutes with both j

plus and j minus

to see this let's first write out j plus

and j minus

so we get this and then as j squared

commutes with both j1

and j2 we get 0.

next let's consider the commutator of j3

with jplus

in this case things are not so easy so

let's again start by expanding j

plus and then

we can separate this expression into two

commutators

this is i h bar j two and this is

minus i h bar j 1

so we get h bar multiplying i j

2 plus j 1 and this term is simply j

plus so we get h bar j plus

we can use an analogous derivation to

show that the commutator of j3

with j minus is equal to minus h-bar

j minus the final commutator we can look

at

is that between j plus and j minus

writing out both ladder operators we get

this we can now expand this

into these four terms

and feel free to pause here for a second

to double check this

but it should be clear after a moment

these two terms are 0 because they

involve the same angular momentum

component

this is ihbar j3 and this is

minus i h bar j 3

so that overall we get 2 h bar j 3.

so as a summary of the computation

relations we have that the j

squared operator commutes with all of j3

j plus and j minus

then the j3 operator obeys these

computation relations with the j

plus and j minus operators

and finally the j plus and j minus

operators

obey this computation relation

again you don't need to remember any of

these

but do keep them close because we'll

also use them

all the time

we're now ready to investigate how the

ladder operators act on eigenstates of

the angular momentum observable so this

is the exciting part

as we've been discussing we're

considering j squared and j3

as the two compatible observables to

describe angular momentum

so we'll start by writing down their

eigenvalue equations

the eigenvalue equation for j squared is

this

where the eigenvalue is lambda

and for j3 is this where the eigenvalue

is mu

as j squared and j3 are compatible

observables we can always find a common

set of eigenstates

and we'll take these states labeled by

the eigenvalues

lambda and mu to be this common set

now before we proceed let's discuss

these equations in a little bit more

detail

as we discuss in the video on angular

momentum eigenvalues

it turns out that lambda and mu can only

take

very special values in particular

lambda is always given by j times j

plus 1 times h-bar squared where j is

equal

to one of zero one half one three halves

and so on

in steps of one half

mu on the other hand is always given by

the expression

m times h bar and for a given j

m can only take the values minus j

minus j plus one and so on in steps of

one

up to a maximum value of j

despite this the properties we want to

derive today

don't depend on these special values

that lambda and mu take

so we're going to keep the discussion

general and use lambda and mu

directly for completeness at the end of

the video we will quote

all the results we obtain using the

special values that lambda and mu can

actually take

let's start by constructing a new state

j plus minus

acting on the eigenstate lambda mu

the first result we consider is that if

lambda mu is an eigenstate of j

squared with eigenvalue lambda then this

new state is

also an eigenstate of j squared with the

same eigenvalue

to see this let's act with j squared on

this state

and as j squared commutes with both j

plus and j

minus we can exchange the order here and

we get this

now we can use the eigenvalue equation

for j squared

to get lambda lambda mu and this leads

to lambda times

j plus minus lambda mu

so we see that by acting with j squared

on this state

we get lambda times the same state

and this confirms that j plus minus

lambda mu is also an eigenstate of j

squared

with eigenvalue lambda

let's now make some room and consider j3

the result for j3 is that if lambda mu

is an eigenstate of j3 with eigenvalue

mu

then these new states are also

eigenstates of j3

but this time with different eigenvalues

that we are about to calculate

to do this let's start with j3 acting on

the state

generated from j plus and

looking at the computator up here we can

copy it

and we can rearrange it to j 3

j plus is equal to j plus j 3 plus

h bar j plus using this expression we

get this

and here we can use the eigenvalue

equation for j3 to get

mu lambda mu so that overall we get

mu plus h bar multiplying j plus

lambda mu so what does this mean

we see that by acting with j 3 on this

state

we get a scalar mu plus h bar

times the same state this confirms that

j

plus lambda mu is also an eigen state of

j 3 and the eigenvalue is now

mu plus h bar repeating the same

exercise with j minus we would find that

j

3 acting on j minus lambda mu equals mu

minus h bar multiplying j minus lambda

mu

so in conclusion j minus lambda mu is

also an eigen state of j

3 and this time the eigenvalue is mu

minus h bar

so these up here are the two equations

we have just derived

if we start with j plus the first

equation tells us that j

plus lambda mu is an eigenstate of j

squared with eigenvalue

lambda the second equation tells us that

it is also

an eigenstate of j3 with eigenvalue

mu plus h bar this means that we can

rewrite

this state as lambda mu plus h bar

and in general we will have some

normalization constant in front which

we're going to call

n plus so what does j

plus do when acting on an eigenstate of

j

squared and j3 it simply gives us

another eigenstate of these two

operators

but the j3 eigenvalue has been raised

by a constant h-bar and this

is the reason why we call j plus a

raising operator

we can do the same by looking at j minus

acting on lambda mu

which is an eigen state of j squared

with eigenvalue lambda

and an eigen state of j 3 with

eigenvalue

mu minus h bar this means that we can

write the action of j

minus on an eigen state of angular

momentum like this

and as the eigenvalue of j3 is lowered

by a constant h bar

we call j minus a lowering operator

acting with the racing operator multiple

times

allows us to go up a ladder of

eigenvalues every time adding an

extra h-bar while acting with the

lowering operator we go

down a ladder removing one h-bar each

step

hence the collective name ladder

operators

let's look in more detail at the new

state generated by applying

the lateral operators if we start with

the raising operator

then let's consider the norm of j plus

lambda mu

by definition this is equal to this

and the add joint of j plus is simply j

minus so we can rewrite it like this

and using this expression above we can

rewrite j minus j plus

in terms of j squared and j 3 and we get

these three terms

and feel free to pause here for a second

to double check this

but it should be clear after a moment

using the eigenvalue equations we get

lambda lambda mu

mu squared lambda mu and mu lambda mu

so that we get these new three terms

the eigenstates form an orthonormal set

so we end up with lambda minus mu

squared

minus mu h bar

at the same time we can write j plus

lambda mu

as equal to n plus lambda mu plus h bar

and its norm

is then simply equal to absolute value

of n plus squared

this expression and this expression are

equal

because they're both the norm of j plus

lambda mu

this means that j plus acting on this

state

gives this square root

multiplying the state in which the

eigenvalue mu

has increased by h bar

we can of course repeat the same

exercise with j minus and this is

the corresponding result

again no need to remember any of this

but it will come in handy in our study

of angular momentum

in quantum mechanics

the very last thing i want to do before

we conclude

is to rewrite this final result in terms

of the allowed values of

lambda and mu for angular momentum

eigenvalues

so that you can use this final answer

for reference

let's write down the eigenvalue

equations for j squared

and j3 and as we discussed earlier in

the

videos on eigenvalues of the angular

momentum operators

we learned that the eigenvalue lambda is

always given by

j times j plus 1 h bar squared where j

can only be zero one half one three

halves and so on in steps of one half

and so once we have determined j then

the values of mu

are also fixed and can be written as m

h bar wherein can only take one of the

values

minus j minus j plus one all the way to

j

in steps of one knowing that the

eigenvalues

always have this form then we could

update the eigenstates from

lambda mu to these new labels

however to simplify notation this is

typically simply written

as jm in terms of these

the action of the raising operator on an

eigenstate jm

is simply equal to this normalization

times a new eigenstate j n plus 1.

writing down the corresponding

expression for the lowering operator

we get this

hooray we're done here

i simply have a list of the main results

that we have obtained today

we'll use them in many of the other

videos on angular momentum in quantum

mechanics so an easy way to keep them

handy

might be for you to copy them down or

simply to take a screenshot

ladder operators are extremely useful

when working with angular momentum in

quantum mechanics

so although this video might have

appeared a little too dry the first time

you watch it

we're going to use these results all the

time so it was time well invested

check out the what next section in the

description to see lateral apparatus in

action more generally lateral operators

play a really important role in other

parts of quantum mechanics

most popularly in the quantum harmonic

oscillator and if you want to explore

more advanced topics that build on

similar ideas

i encourage you to watch the playlist on

second quantization

and as always if you liked the video

please subscribe