Orbital angular momentum eigenvalues

Summary

This video by Professor M. Dust discusses the eigenvalues of orbital angular momentum in quantum mechanics, focusing on their quantized nature and the constraints that apply specifically to orbital angular momentum. The lecture explains the relationship between quantum numbers j and m, and how for orbital angular momentum, only integer values are allowed, unlike in general angular momentum where half-integer values are possible. The presentation includes a mathematical breakdown using spherical coordinates, wave functions, and commutation relations, and contrasts orbital angular momentum with spin angular momentum. It also hints at further discussions on eigenfunctions in a follow-up video.

Takeaways

• 📚 The eigenvalues of orbital angular momentum are quantized, labeled by quantum numbers j and m, which can be integers or half-integers.
• 🔬 Orbital angular momentum is a specific case of angular momentum that is consistent with the general properties of angular momentum, but with additional constraints.
• ⚙️ General angular momentum is described by an operator J with components that obey specific commutation relations, and the J^2 and J3 operators are compatible observables.
• 📐 For J^2, the eigenvalue is j(j + 1)h^2, where j can be 0, 1/2, 1, 3/2, and so on. For J3, the eigenvalue is mh, with m values from -j to j.
• 🌍 Orbital angular momentum relates to the motion of particles in 3D space and uses a similar approach to general angular momentum but imposes extra constraints on the quantum numbers.
• 🔄 Orbital angular momentum is described by L^2 and Lz, with similar eigenvalue equations to general angular momentum but using spherical coordinates for particles in 3D space.
• 🌐 Orbital angular momentum eigenfunctions depend on the angular coordinates, while radial functions remain undetermined by the angular momentum operators alone.
• 🌀 The allowed eigenvalues for orbital angular momentum are only integers, meaning both l and m must be integers, unlike general angular momentum where half-integer values are possible.
• 🧭 The azimuthal angle φ imposes a boundary condition that leads to the quantization of m as an integer.
• 🔄 Orbital angular momentum does not allow for half-integer values of l or m, but these values do appear in the case of spin angular momentum, which is addressed in related videos.

Q & A

What are the quantum numbers used to describe angular momentum in quantum mechanics?

The quantum numbers used to describe angular momentum in quantum mechanics are 'j' and 'm', which can be either integers or half-integers.

What distinguishes orbital angular momentum from general angular momentum?

Orbital angular momentum is associated with the motion of particles in three-dimensional space, and unlike general angular momentum, it only allows integer values for the quantum numbers 'l' and 'm'.

What is the significance of the operator J squared (J²) in quantum mechanics?

J² is an operator that represents the total angular momentum. It commutes with all components of angular momentum (J1, J2, J3), and its eigenvalue equation gives the quantized values of angular momentum.

How are the eigenvalues of J³ (the third component of angular momentum) defined?

The eigenvalues of J³ are quantized and are given by the expression m * ħ, where 'm' can take values between -j and j in steps of 1.

What is the role of spherical coordinates in solving problems of orbital angular momentum?

Spherical coordinates simplify the representation of orbital angular momentum, as the angular part of the wave function depends only on the angles (theta and phi) rather than the radial distance (r).

What are the eigenvalue equations for orbital angular momentum in spherical coordinates?

The eigenvalue equation for L² (orbital angular momentum squared) involves partial derivatives with respect to theta and phi, while the eigenvalue equation for Lz (the z-component) only involves a partial derivative with respect to phi.

Why are half-integer values not allowed for orbital angular momentum?

Half-integer values are not allowed for orbital angular momentum because the wave function must be continuous over the azimuthal angle, phi. This condition forces 'm' to be an integer.

What happens when we combine angular momentum operators with other operators in quantum mechanics?

Combining angular momentum operators with additional operators (like the Hamiltonian for energy) creates a complete set of commuting observables, which is necessary to fully specify the quantum state of a system.

What is the significance of the equation e^(i 2π m) = 1 for orbital angular momentum?

This equation ensures that the wave function is continuous over a full 360-degree rotation in the azimuthal angle. It leads to the result that 'm' must be an integer for orbital angular momentum.

Can half-integer values for angular momentum exist in any physical systems?

Yes, half-integer values for angular momentum do exist, but they occur in spin angular momentum rather than orbital angular momentum. Spin allows both integer and half-integer values.

Outlines

00:00

📘 Introduction to Orbital Angular Momentum and Eigenvalues

In this section, Professor Dust introduces the concept of orbital angular momentum in quantum mechanics, emphasizing the quantized nature of eigenvalues. The eigenvalues are associated with quantum numbers j and m, which can be either integers or half-integers. The focus is on orbital angular momentum, where not all values of j and m are allowed. A brief refresher on general angular momentum is provided, including the key commutation relations between angular momentum components and how the operator j² is used in quantum mechanics.

05:01

🌐 Orbital Angular Momentum in Spherical Coordinates

This paragraph delves into the projection of the eigenstate onto spherical coordinates, leading to the wave function ψlm. The orbital angular momentum operator l² is introduced, and the corresponding differential operators in spherical coordinates are discussed. The eigenvalue equations for l² and lz are outlined, with the solution presented in terms of the angular variables (θ and φ). These equations highlight the separable nature of the wave function into radial and angular components, introducing the idea that additional operators are needed to fully describe a quantum state.

10:01

🌀 Separable Solutions and Eigenfunctions of Orbital Angular Momentum

Here, the focus shifts to the separable nature of the wave function in terms of radial and angular components. The eigenfunctions of the operators l² and lz are written in terms of spherical harmonics, denoted by Ylm. The relationship between the eigenvalues and the angular components is discussed, specifically noting that the angular momentum operators alone are insufficient to completely define a quantum state. The need for additional observables, such as the Hamiltonian in the case of the hydrogen atom, is highlighted.

15:02

🔢 Allowed Values of Orbital Angular Momentum

In this section, the allowed eigenvalues for orbital angular momentum are derived, starting with the differential equation for lz. By enforcing the continuity of the wave function along the azimuthal angle φ, the conclusion is reached that both l and m must be integers. This constraint differentiates orbital angular momentum from general angular momentum, where half-integer values are also allowed.

💡 Conclusion: Distinction Between Orbital and Spin Angular Momentum

The final paragraph discusses the differences between orbital angular momentum and spin angular momentum. While half-integer eigenvalues are allowed for spin angular momentum, they are not permitted for orbital angular momentum, which can only take integer values for both l and m. The section concludes by encouraging viewers to explore the topic of spin in more depth and check out the related videos on eigenfunctions.

Keywords

💡Eigenvalues

In quantum mechanics, eigenvalues refer to the measurable outcomes of an operator, such as angular momentum. In the video, eigenvalues are critical because they represent quantized values that can be observed for physical properties like angular momentum. For instance, the eigenvalues of the angular momentum operator are discussed as being quantized and dependent on quantum numbers.

💡Quantum Numbers

Quantum numbers (denoted as 'j' and 'm' in the video) are values that describe the properties of quantum systems, such as angular momentum. These numbers are crucial for understanding the allowed eigenvalues of the system. The video explains how the quantum numbers for orbital angular momentum are constrained to integer values, in contrast to general angular momentum, which can take half-integer values.

💡Angular Momentum

Angular momentum is a fundamental quantity in both classical and quantum mechanics that describes the rotational motion of a particle. In quantum mechanics, angular momentum is quantized and has discrete eigenvalues. The video explores both general angular momentum and the specific case of orbital angular momentum, showing how the quantum numbers dictate its quantized nature.

💡Orbital Angular Momentum

Orbital angular momentum refers to the angular momentum of a particle due to its motion through space, such as an electron orbiting a nucleus. The video focuses on the eigenvalues of orbital angular momentum, explaining that they are subject to stricter constraints than general angular momentum. Only integer values of the quantum numbers are allowed for orbital angular momentum.

💡Commutation Relations

Commutation relations are mathematical expressions that describe how operators (such as angular momentum components) interact in quantum mechanics. The video mentions that the components of angular momentum do not commute, meaning that they cannot all be measured precisely at the same time. This leads to the definition of a compatible set of observables, like J^2 and J_3.

💡Operators

Operators in quantum mechanics are mathematical objects that correspond to physical observables, such as momentum or energy. In the video, the angular momentum operator is discussed in detail, including its components J_1, J_2, and J_3, and how they obey specific commutation relations that lead to quantized eigenvalues for angular momentum.

💡Spherical Coordinates

Spherical coordinates are a system for defining the position of a point in three-dimensional space using a radius and two angles (theta and phi). The video explains how working with orbital angular momentum is easier in spherical coordinates and discusses how the eigenvalue equations of the angular momentum operator simplify when projected into this coordinate system.

💡Wavefunction

A wavefunction describes the quantum state of a system and provides information about the probability distribution of a particle's position and momentum. In the video, the eigenstates of the angular momentum operator are expressed as wavefunctions, with particular emphasis on the separation of radial and angular components when working in spherical coordinates.

💡Spin Angular Momentum

Spin angular momentum is a type of intrinsic angular momentum that particles possess, separate from orbital angular momentum. The video hints at the existence of half-integer quantum numbers for spin angular momentum, which is not allowed for orbital angular momentum. This distinction is key to understanding how different types of angular momentum behave in quantum systems.

💡Eigenfunctions

Eigenfunctions are solutions to an operator's eigenvalue equation, representing the quantum states associated with specific eigenvalues. In the video, the eigenfunctions of orbital angular momentum are discussed, particularly how they depend on angular variables in spherical coordinates. The separation of the angular and radial parts of the eigenfunctions is a key concept.

Highlights

Introduction to eigenvalues of orbital angular momentum in quantum mechanics.

Angular momentum eigenvalues are quantized, and labeled by quantum numbers j and m, which can be integers or half-integers.

Orbital angular momentum is linked to the motion of particles in three-dimensional space.

Not all values of j and m are allowed for orbital angular momentum; there are additional constraints.

General angular momentum is defined by the operator j, composed of components j1, j2, and j3, which obey specific commutation relations.

Introduction of the operators j² and j3, where j² commutes with every angular momentum component.

Eigenvalue equation for j²: Eigenvalue is j(j+1)h², with j taking values such as 0, 1/2, 1, 3/2, etc.

For j3, the eigenvalue is mh, with m ranging from -j to +j in integer steps.

Orbital angular momentum is represented by the operator l with components lx, ly, and lz, obeying general angular momentum commutation relations.

The operator l² has an eigenvalue equation similar to j², and lz has an eigenvalue equation similar to j3.

Orbital angular momentum describes particle motion in 3D space, best solved in spherical coordinates.

Eigenvalue equations for orbital angular momentum focus on angular variables theta and phi, rather than r.

The phi-dependent part of the eigenfunction must be continuous, enforcing that m must be an integer for orbital angular momentum.

The allowed values of l and m for orbital angular momentum must be integers, unlike general angular momentum where half-integer values are allowed.

Final discussion highlights that half-integer values of angular momentum exist, but only for spin angular momentum, not orbital angular momentum.

Transcripts

hi everyone

this is professor m dust science and

today i want to talk about the

eigenvalues of orbital angular momentum

in another one of our videos on rigorous

quantum mechanics

we know that for any type of quantum

angular momentum

the eigenvalues are quantized we label

the eigenvalues with the quantum numbers

j

and m which can be integers

or half integers today we're going to

look at what happens to these quantum

numbers in a very particular but very

important case

the orbital angular momentum associated

with the motion of particles

in three-dimensional space

unsurprisingly we will find that the

results are consistent with those for

the general angular momentum

but we will find out that there are

additional constraints

such that not all values of j and m are

allowed for orbital angular momentum

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

of general angular momentum

in quantum mechanics consider an

operator j

made of three components j1 j2 and j3

if these three components obey these

commutation relations

then we call j an angular momentum

and as a quick reminder this is the

levitivita symbol and i'm using einstein

notation

so this expression implies a sum over

the repeated indices

k as the angular momentum components

don't commute

then they don't form a set of compatible

observables

instead we define a new operator j

squared which is equal to j1 squared

plus j2 squared plus j3 squared

that commutes with every angular

momentum component

given this result in the theory of

angular momentum we define

as our set of compatible observables the

operators j

squared and one of the other components

which is conventionally chosen to be

j3 from the video on the eigenvalues of

a general angular momentum

which you can find linked in the

description we know what the eigenvalue

equations of j squared and j3 are

for j squared we have this eigenvalue

equation

where the eigenvalue is j times j plus 1

h bar squared

and j can take any of the values zero

one half

one three halves two and so on in steps

of one half

and for j3 we have this eigenvalue

equation

where the eigenvalue is m h bar and

m can take any of the values minus j

minus j plus 1

all the way to j minus 1 and j in steps

of 1.

the common eigenstates are labeled by j

and m as shown here

and here these results only depend on

the defining commutation relations

for the general angular momentum up here

in this video we want to explore what

happens when we consider the special

case of orbital angular momentum

orbital angular momentum is the angular

momentum associated with the motion

particles that we are familiar with from

classical mechanics

as a particular instance of an angular

momentum the properties of orbital

angular momentum

must be consistent with these general

properties for a general angular

momentum

however what we will discover is that

there are extra constraints

associated with orbital angular momentum

that mean for example that only a subset

of the allowed values of

j are actually possible for orbital

angular momentum

so let's now turn to orbital angular

momentum

using the usual notation we call the

orbital angular momentum operator l

which is a vector operator made of three

components

lx ly and lz

these three components obey the general

angular momentum computation relations

which look like this for lx and y

like this for l y l set and

like this for l z lx we can straight

away rewrite the eigenvalue equations

for general angular momentum

for l squared we get this

and for lz this

orbital angular momentum describes the

motion of particles in the

three-dimensional euclidean space

so working with it is easiest if we use

the position representation

as we know from the videos on the

position representation we need to

project these equations onto the

position basis

and this amounts to projecting the two

sides of these two equations

onto the basis states

on top of that it is usually more

convenient to work in terms of spherical

coordinates

where positions are described by a

distance r from the origin

by a polar angle theta and by an

azimuthal angle

phi that first requires projecting onto

the horizontal

plane and then measuring the angle from

this axis

so let's start with the eigenstates lm

the projection of the eigenstate gives

the so-called

wave function of the system which i

write

as psi lm of r

i am labeling the wave function with the

same quantum numbers

l and m to identify the corresponding

eigenstate

and as we'll work in spherical

coordinates then we can write the vector

r

in terms of the corresponding

coordinates

for the operators we actually already

derived all the relevant quantities in

the position representation using

spherical coordinates

in the corresponding video that is

linked in the description as always

the first operator is l squared we found

in that video that it is equal to minus

h bar squared

then the partial derivative with respect

to theta twice

then 1 over the tangent of theta

multiplying the partial derivative

with respect to theta and then 1

over sine squared of theta multiplying

the partial derivative of phi twice

the second operator is lz

and it has a particularly simple form

minus either

times the partial derivative with

respect to phi

okay so with the result from the

previous slide we obtained these two

eigenvalue equations in the position

representation

in the first one this is the l squared

operator in the position representation

and in spherical coordinates this here

is

the corresponding eigenstate lm

when written in the position basis we

can still call this wavefunction an

eigenstate

but most of the time we call it an

eigenfunction to reflect the fact that

in this position representation

quantum states are given by functions

this here is the eigenvalue which states

unchanged

and this again the eigenfunction

corresponding to the eigenstate

lm in the second equation

we have the lz operator the two

eigenfunctions here

and here both corresponding to the

eigenstate lm

and the eigenvalue m h bar

so the theory of orbital angular

momentum amounts to solving

these two differential equations to

determine the eigenvalues and

eigenfunctions in this video we will

look at the properties of the

eigenvalues

and you can find the discussion on the

eigenfunctions in the corresponding

video that is also linked

in the description

so what we have here are the eigenvalue

equations for l squared and l z

written in the position representation

the first important thing to note is

that the differential operators in both

of these equations

only depend on the angles these are and

phi

but don't depend on r this means that we

can use a separable trial solution for

psi

which is the product of a function that

only depends on

r and then a function that only depends

on the angular variables

the part that only depends on the

angular variables is conventionally

written with a capital y

and with the label l as a sub index and

the label m

as a super index plugging in this trial

equation

into the l squared equation here and

here

gives this very long expression

as the differential operators don't

affect the r-dependent part

we can move it before the operator and

then it cancels with the corresponding

part on the right-hand side

this means that we can rewrite the

eigenvalue equation for l squared like

this

and following the same procedure with

the eigenvalue equation for

lz we find this second equation for

ylm these equations imply that

the ylm are the common eigenfunctions of

l squared

and lz if we look again

at the full wave function psi up here

then we see that the eigenvalue

equations for

l squared and lz only tell us about the

angular part but don't tell us about the

radial part

that means that any radial function f

would be consistent with the angular

momentum eigenvalue equations

so the angular momentum operators are

not enough

to fully specify the quantum state psi

of the system

in turn this means that although l

squared and lc form a set of compatible

observables

they don't form a complete set of

commuting observables

and i emphasize the word complete here

to fully specify a quantum state we need

additional operators beyond the angular

momentum

operators we're not going to worry about

this for now

because the aim of the videos on angular

momentum is to learn

about these specific operators however

when we use them to solve specific

problems

then we will need to combine them with

additional operators

you can find a very good example of this

in our videos on the hydrogen atom

where we need to add the operator

associated with the total energy of the

system

called the hamiltonian to the two

angular momentum apparatus to define a

complete set of commuting observables to

be able to fully specify

the quantum states of the hydrogen atom

right so we are now ready to discuss the

allowed eigenvalues of orbital angular

momentum

as we'll see it turns out that for this

we only need

the equation for lz the key point is

that

in this equation the differential

operator only depends on phi

this means that we can use a separable

tri solution for y

which is the product of a function that

only depends on theta

and then a function that only depends on

phi

plugging in this trial solution into the

lz equation here

and here gives this

as the differential operator doesn't

affect the theta dependent part

we can move it to the other side and

then it cancels with the corresponding

part on the right-hand side we can also

cancel the h-bars and overall

this means that the eigenvalue equation

for a z

only affects the phi part of the wave

function

and we can rewrite it like this

this is now a differential equation for

the function g of a single variable

phi it has the standard form df by dx

equal to alpha f of x for constant alpha

which can be solved by separation of

variables

we can then integrate both sides and

we get the logarithm of f as equal to

alpha x

plus an integration constant a

exponentiating gives

f of x equal to a times e to the power

alpha x where a is a constant

using this solution for our equation we

find that

glm of phi is equal to some constant a

times e to the power i n phi

we can therefore write the angular

momentum eigenfunction

y as equal to f times

e to the i m phi where i have absorbed

the normalization constant

into f

we're now ready for the final step in

figuring out

the allowed values of l and m

let's start writing again the latest

expression we got

for the wave function the wave function

must be continuous for these equations

to be obeyed

for example if the phi dependent part

was not continuous

then acting with the derivative here

would produce a delta function which

would then be incompatible

with the right hand side so let's

enforce continuity along the phi

coordinate

we know that the azimuthal angle phi is

defined between 0 and 2 pi

and this means that the eigenfunction at

0 must be equal to the eigenfunction

at 2 pi plugging in the expression that

we just got for the eigenfunction

we get this

the f's cancel so we end up with

e to the i 2 pi m equal to 1.

the only solution to this equation is

for integer m

which means that for orbital angular

momentum m

must be an integer as m

is given by this list of values then if

m is an integer

this means that l must also be an

integer

so what does this mean in the videos on

general angular momentum

we figured out that for j squared the

eigenvalues take

this form and they're labeled by the

number j

which can be any of zero one half one

three halves

2 and so on and in a similar way we

figured out that for

j 3 the eigenvalues take this form

where m can be any of the values in this

list

so for a general angular momentum j and

m can be

either integer or half integer

now for orbital angular momentum we have

l squared with the corresponding

eigenvalues

and lz with the corresponding

eigenvalues

but now in the special case of orbital

angular momentum

the only allowed values of l are integer

and the only allowed values of m are

also integer

this result for orbital angular momentum

is consistent with the result for

general angular momentum

but it has a stricter constraint on the

allowed eigenvalues

l and m because half integer values are

not possible

so right so this is it for the

eigenvalues of orbital angular momentum

however before we conclude let's have a

quick final discussion

we know from the general theory of

angular momentum that half integer

values for j

are possible but we just figured out

that they don't exist for orbital

angular momentum

so a question that you may have now is

where the half intercept angular

momentum eigenvalues

really exist at all and the answer is

it turns out that for spin angular

momentum half integer values

do exist if you want to learn more about

this

a good starting point would be to check

out our videos on spin

orbital angular momentum exhibits

quantized eigenvalues

as all angular momentum must however the

eigenvalues that are allowed for orbital

angular momentum are only a fraction of

those that are possible for general

angular momentum

you can learn more about other

possibilities in our videos about spin

and now that we know all there is to

know about the eigenvalues

i encourage you to check out the video

on the eigenfunctions

of orbital angular momentum and as

always if you liked the video

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