Orbital angular momentum eigenvalues
Summary
TLDRThis 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
- ๐ The eigenvalues of orbital angular momentum are quantized, labeled by quantum numbers j and m, which can be integers or half-integers.
- ๐ฌ Orbital angular momentum is a specific case of angular momentum that is consistent with the general properties of angular momentum, but with additional constraints.
- โ๏ธ 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.
- ๐ 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.
- ๐ 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.
- ๐ 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.
- ๐ Orbital angular momentum eigenfunctions depend on the angular coordinates, while radial functions remain undetermined by the angular momentum operators alone.
- ๐ 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.
- ๐งญ The azimuthal angle ฯ imposes a boundary condition that leads to the quantization of m as an integer.
- ๐ 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 (Jยฒ) in quantum mechanics?
-Jยฒ 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 Jยณ (the third component of angular momentum) defined?
-The eigenvalues of Jยณ are quantized and are given by the expression m * ฤง, 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 Lยฒ (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 2ฯ 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
๐ 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 jยฒ is used in quantum mechanics.
๐ Orbital Angular Momentum in Spherical Coordinates
This paragraph delves into the projection of the eigenstate onto spherical coordinates, leading to the wave function ฯlm. The orbital angular momentum operator lยฒ is introduced, and the corresponding differential operators in spherical coordinates are discussed. The eigenvalue equations for lยฒ and lz are outlined, with the solution presented in terms of the angular variables (ฮธ and ฯ). 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.
๐ 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 lยฒ 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.
๐ข 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 ฯ, 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.
๐ก 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.
Mindmap
Keywords
๐กEigenvalues
๐กQuantum Numbers
๐กAngular Momentum
๐กOrbital Angular Momentum
๐กCommutation Relations
๐กOperators
๐กSpherical Coordinates
๐กWavefunction
๐กSpin Angular Momentum
๐กEigenfunctions
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 jยฒ and j3, where jยฒ commutes with every angular momentum component.
Eigenvalue equation for jยฒ: Eigenvalue is j(j+1)hยฒ, 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 lยฒ has an eigenvalue equation similar to jยฒ, 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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