.. _chapter9:
Chapter 9: Translational and Rotational Symmetry in the Two-Body Problem
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
.. note::
*The atoms or elementary particles themselves are not real; they form a world of potentialities or possibilities rather than one of things or facts.* ― Werner Heisenberg
Summary
-------
.. attention::
Up to now, we have tiptoed our way in quantum mechanics, first
introducing single particles that can only assume two states, then
introducing the combination of two such particles. Next we
introduced the position-space (and momentum-space) representations
of the quantum state vector, using the eigenstates of the
:math:`\hat{x}` operator as a complete basis. What we did may seem
like a far cry from an understanding of quantum mechanics. Do not
despair! In spite of the simplicity of the approach, we have
learned a great deal about the formalism and this chapter is mostly
an extension of what we learned but now considering the
three-dimensional nature of space.
In particular, we find that the *probability of finding a particle
in the volume* :math:`d r^{3}=dx dy dz` *at position*
:math:`{\mathbf{r}}` is now given by:
.. math::
d^{3} r|\langle\mathbf{r} \mid \psi\rangle|^{2}
Similar to :ref:`chapter6`, we introduce translation operators,
this time along three different directions:
.. math::
\begin{array}{l}
\hat{T}\left(a_{x} \mathbf{i}\right)|x, y, z\rangle=\left|x+a_{x}, y, z\right\rangle \\
\hat{T}\left(a_{y} \mathbf{j}\right)|x, y, z\rangle=\left|x, y+a_{y}, z\right\rangle \\
\hat{T}\left(a_{z} \mathbf{k}\right)|x, y, z\rangle=\left|x, y, z+a_{z}\right\rangle
\end{array}
Again, using our usual approach, this allows us to introduce the
components of the linear momentum (vector) operator, as generators
of each of those translations (see box below). The momentum
operators are Hermitian and we can use their eigenstates to expand
any states (we say that the eigenstates constitute a *complete
basis*).
We also introduce a two-body Hamiltonian with potential energy that
only depends on distance between the two bodies:
.. math::
\hat{H}=\frac{\hat{\mathbf{p}}_{1}^{2}}{2 m_{1}}+\frac{\hat{\mathbf{p}}_{2}^{2}}{2 m_{2}}+V\left(\left|\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{2}\right|\right)
We then realize that we are now working in a quantum space of two
particles, as we have already seen in :ref:`chapter5` for the
combination of two spin-1/2 particles. Here, we combine two
particles that are represented by a position space
vector. Naturally, we *combine* the space spanning the possible
state of each particle using the direct (or tensor) product:
.. math::
\left|\mathbf{r}_{1}, \mathbf{r}_{2}\right\rangle=\left|\mathbf{r}_{1}\right\rangle_{1} \otimes\left|\mathbf{r}_{2}\right\rangle_{2}
We also remember that the operators that operate in different
individual space commute with one another (since they operate on
different *things*). This leads us to introduce a new operator that
operates on the *combined* space where the two particles live. This
operator is the total linear momentum and it corresponds to the
generation of translations on the two-body space:
.. math::
\hat{\mathbf{P}}=\hat{\mathbf{p}}_{1}+\hat{\mathbf{p}}_{2}
We demonstrate mathematically an important result: the Hamiltonian
mentioned above commutes with the operator
:math:`\hat{\mathbf{P}}`. This is not surprising since translating
rigidly the two-body system does not change its energy (this is
because the potential only depends on the distance between the
particles!):
.. math::
[\hat{H}, \hat{\mathbf{P}}]=0
Students by now should be conditioned! We have two operators that
commute and it follows that **they share the same eigenstates**
(who would have believed this?). This also means that because the
Hamiltonian is invariant under translation, we find to conservation
of total momentum of the system (using results derived in
:ref:`chapter4`):
.. math::
\frac{d\langle\mathbf{P}\rangle}{d t}=\frac{i}{\hbar}\langle\psi|[\hat{H}, \hat{\mathbf{P}}]| \psi\rangle=0
Next, we develop the classical argument of changing variables from
the position each particles to variables corresponding to the
position of the center of mass and of the position relative to the
center of mass:
.. math::
\begin{array}{l}
\hat{\mathbf{r}}=\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{2} \\
\hat{\mathbf{R}}=\frac{m_{1} \hat{\mathbf{r}}_{1}+m_{2} \hat{\mathbf{r}}_{2}}{m_{1}+m_{2}}
\end{array}
Finally, this yields, using the total momentum operator and the momentum operator of the relative motion:
.. math::
\hat{H}=\frac{\hat{\mathbf{P}}^{2}}{2 M}+\frac{\hat{\mathbf{p}}^{2}}{2 \mu}+V(|\hat{\mathbf{r}}|)
The two variables can then be separated and we solve two separate
problems (the first one being, trivially, the motion of a free
particle that students typically solve during kindergarten):
.. math::
\hat{H}_{\mathrm{cm}}=\frac{\hat{\mathbf{P}}^{2}}{2 M} \quad\textrm{and}\quad \hat{H}_{\mathrm{rel}}=\frac{\hat{\mathbf{p}}^{2}}{2 \mu}+V(|\hat{\mathbf{r}}|)
We just end up with solving a one-body (like) problem corresponding
to the motion relative to the CM:
.. math::
\hat{H}=\frac{\hat{\mathbf{p}}^{2}}{2 \mu}+V(|\hat{\mathbf{r}}|)
This problem is called the *single-body system in a central
potential*.
One of the main concepts we introduce in this chapter is related to
the *orbital* angular momentum operators. See box below about the
angular momentum operators we have studied so far. Similar to
classical mechanics, the *orbital* angular momentum operators are
noted :math:`(\hat{L}_{x},\hat{L}_{y},\hat{L}_{z})` and are defined
as :math:`\hat{\mathbf{L}}=\hat{\mathbf{r}} \times
\hat{\mathbf{p}}` (note that this equation looks like what you have
studied in classical mechanics but, here, this is an **operator**
equation!). The problem we are interested is **rotationally
invariant** (we see this because the Hamiltonian
:math:`\hat{H}_{\mathrm{rel}}` only includes *distances*, in
position and momentum spaces -- as we formally proved in the
course), this can be translated as:
.. math::
\left[\hat{H}, \hat{L}_{z}\right]=0
and
.. math::
\left[\hat{H}, \hat{L}^2\right]=0
in addition, we established that
.. math::
\left[\hat{L}^2, \hat{L}_{z}\right]=0
These last three commutation relations consitute a **central result
of this chapter**. We have *three operators that commute with each
other* and it follows that one can find eigenstates that are common
to all the three operators :math:`\hat{H}`, :math:`\hat{L}^2`, and
:math:`\hat{L}_{z}`. In other words, we can write the eigenvalue
problem as (we also remember that all three operators are Hermitian
and thus have real eigenvalues):
.. math::
\begin{aligned}
\hat{H}|E, l, m\rangle &=E|E, l, m\rangle \\
\hat{\mathbf{L}}^{2}|E, l, m\rangle &=l(l+1) \hbar^{2}|E, l, m\rangle \\
\hat{L}_{z}|E, l, m\rangle &=m \hbar|E, l, m\rangle
\end{aligned}
It is a good time to take a minute and understand the significance
of the equations above. What they really mean is that we now know
it is possible to know the energy, the total momentum, and the
projection of the angular momentum on the *z*-axis at the same
time.
In this chapter, we also work out the Schrodinger equation in
position space (see box below on how this is done):
.. math::
\left\langle\mathbf{r}\left|\frac{\hat{\mathbf{p}}^{2}}{2 \mu}\right| \psi\right\rangle+\langle\mathbf{r}|V(|\hat{\mathbf{r}}|)| \psi\rangle=-\frac{\hbar^{2}}{2 \mu}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r} \frac{\partial}{\partial r}\right)\langle\mathbf{r} \mid \psi\rangle+\frac{\left\langle\mathbf{r}\left|\hat{\mathbf{L}}^{2}\right| \psi\right\rangle}{2 \mu r^{2}}+V(r)\langle\mathbf{r} \mid \psi\rangle=E\langle\mathbf{r} \mid \psi\rangle
.. |br| raw:: html
Solving this equation is a bit intimidating but it is mostly a
matter of remembering how to represent operators into their
position-space representations. In addition, using spherical
coordinates, we can separate the radial, azimuthal, and polar parts
as:
.. math::
\langle\mathbf{r} \mid E, l, m\rangle=R(r) \Theta(\theta) \Phi(\phi)
The radial part of the Schrodinger equation becomes (we will come
back to the azimuthal and polar parts later):
.. math::
\left[-\frac{\hbar^{2}}{2 \mu}\left(\frac{d^{2}}{d r^{2}}+\frac{2}{r} \frac{d}{d r}\right)+\frac{l(l+1) \hbar^{2}}{2 \mu r^{2}}+V(r)\right] R(r)=E R(r)
We tranform this equation by writing :math:`R(r)=\frac{u(r)}{r}` to find:
.. math::
\left[-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+V(x)\right]\langle x \mid E\rangle=E\langle x \mid E\rangle
Where the effective potential is:
.. math::
V_{\mathrm{eff}}(r)=\frac{l(l+1) \hbar^{2}}{2 \mu r^{2}}+V(r)
It is informative to understand the origin of this effective
potential. The second part is just the original central potential
(for example, the electrostatic potential). The first part
corresponds to a *centrifugal barrier* ("barrier" because it is a
repulsive part, away from the center). However, it is important to
understand that this "potential" comes originally from a *kinetic
energy* term (it originates from the fact the system rotates). This
has far reaching consequences since that term depends on *l* (and
is, in fact, zero for :math:`l=0`). We will see the consequence on
the ground-state of the hydrogen atom in the next chapter.
Solving the radial equation requires more information about the
system (that is: the function :math:`V(r)`) as we shall see in the
next chapter. For now, let's conclude this chapter with a study of
the solution of the angular part of the equation. In spherical
coordinates, we write:
.. math::
\langle r, \theta, \phi \mid E, l, m\rangle=R(r) Y_{l, m}(\theta, \phi)
Where :math:`Y_{l, m}(\theta, \phi)` are known as the **spherical harmonics**:
.. math::
Y_{l, m}(\theta, \phi)=\frac{(-1)^{l}}{2^{l} l !} \sqrt{\frac{(2 l+1)(l+m) !}{4 \pi(l-m) !}} e^{i m \phi} \frac{1}{\sin ^{m} \theta} \frac{d^{l-m}}{d(\cos \theta)^{l-m}} \sin ^{2 l} \theta
This last equation is one of the reasons professors are happy to
let students use cribsheet. However, students should be aware of
the general shape of those functions (as we describe in
:ref:`mathfunc`.)
.. note::
**From operators to position-space representations**
Starting from the operator version of the Schroginder equation
.. math::
\left\langle\mathbf{r}\left|\frac{\hat{\mathbf{p}}^{2}}{2 \mu}\right| \psi\right\rangle+\langle\mathbf{r}|V(|\hat{\mathbf{r}}|)| \psi\rangle = E\langle\mathbf{r} \mid \psi\rangle
* Linear momentum (radial coordinate):
We use:
.. math::
\hat{p}_{r} \rightarrow \frac{\hbar}{i}\left(\frac{\partial}{\partial r}+\frac{1}{r}\right)
to find
.. math::
-\frac{\hbar^{2}}{2 \mu}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r} \frac{\partial}{\partial r}\right)\langle\mathbf{r} \mid \psi\rangle=\frac{\left\langle\mathbf{r}\left|\hat{p}_{r}^{2}\right| \psi\right\rangle}{2 \mu}
This allows us to write:
.. math::
-\frac{\hbar^{2}}{2 \mu}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r} \frac{\partial}{\partial r}\right)\langle\mathbf{r} \mid \psi\rangle+\frac{\left\langle\mathbf{r}\left|\hat{\mathbf{L}}^{2}\right| \psi\right\rangle}{2 \mu r^{2}}+V(r)\langle\mathbf{r} \mid \psi\rangle=E\langle\mathbf{r} \mid \psi\rangle
* Since we can choose eigenstates that are eigenstates of :math:`\hat{\mathbf{L}}^{2}`, we have, for the eigenstates:
.. math::
\frac{\left\langle\mathbf{r}\left|\hat{\mathbf{L}}^{2}\right| \psi\right\rangle}{2 \mu r^{2}} =\frac{l(l+1) \hbar^{2}}{2 \mu r^{2}} \langle\mathbf{r} \mid \psi\rangle
Finally, we get, for the radial part:
.. math::
\left[-\frac{\hbar^{2}}{2 \mu}\left(\frac{d^{2}}{d r^{2}}+\frac{2}{r} \frac{d}{d r}\right)+\frac{l(l+1) \hbar^{2}}{2 \mu r^{2}}+V(r)\right] R(r)=E R(r)
.. note::
**Angular solution**
Because we could separate angular and radial coordinates, we find the angular dependence as solutions to
.. math::
\hat{\mathbf{L}}^{2} \rightarrow-\hbar^{2}\left[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{\sin ^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}\right]
and
.. math::
\left\langle r, \theta, \phi\left|\hat{L}_{z}\right| \psi\right\rangle=\frac{\hbar}{i} \frac{\partial}{\partial \phi}\langle r, \theta, \phi \mid \psi\rangle
The solutions of those equations are the Spherical Harmonics, as described in :ref:`mathfunc`.
Learning Material
-----------------
Copy of Slides
~~~~~~~~~~~~~~
.. raw:: html
The slides for Chapter 9 are available in pdf format here: 📂.
.. raw:: latex
The slides for Chapter 9 are available in pdf format \href{https://www.dropbox.com/s/7h3ao18xvo2yg80/chapter9.pdf}{here}.
Screencast
~~~~~~~~~~
.. raw:: html
.. raw:: latex
This lecture is available as a YouTube recording at this \href{"https://www.youtube.com/embed/9G5uBR5S2QM"}{link}.
.. admonition:: Key Learning Points
* Position-space representation in 3D:
.. math::
\hat{x}|\mathbf{r}\rangle=x|\mathbf{r}\rangle \quad \hat{y}|\mathbf{r}\rangle=y|\mathbf{r}\rangle \quad \hat{z}|\mathbf{r}\rangle=z|\mathbf{r}\rangle
where
.. math::
|\mathbf{r}\rangle=|x, y, z\rangle
* Completeness relation in 3D (position-space representation):
.. math::
|\psi\rangle=\iiint d x d y d z|x, y, z\rangle\langle x, y, z \mid \psi\rangle=\int d^{3} r|\mathbf{r}\rangle\langle\mathbf{r} \mid \psi\rangle
* Orthogonality (with: :math:`|\mathbf{r}\rangle=|x, y, z\rangle`
.. math::
\left\langle x, y, z \mid x^{\prime}, y^{\prime}, z^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right) \delta\left(y-y^{\prime}\right) \delta\left(z-z^{\prime}\right)
or, more succinctly:
.. math::
\left\langle\mathbf{r} \mid \mathbf{r}^{\prime}\right\rangle=\delta^{3}\left(\mathbf{r}-\mathbf{r}^{\prime}\right)
* Generators of translation in 3D:
.. math::
\begin{array}{l}
\hat{T}\left(a_{x} \mathbf{i}\right)=e^{-i \hat{p}_{x} a_{x} / \hbar} \\
\hat{T}\left(a_{y} \mathbf{j}\right)=e^{-i \hat{p}_{y} a_{y} / \hbar} \\
\hat{T}\left(a_{z} \mathbf{k}\right)=e^{-i \hat{p}_{z} a_{z} / \hbar}
\end{array}
and because the different generators of translation commute, we have:
.. math::
\hat{T}(\mathbf{a})=e^{-i \hat{p}_{x} a_{x} / \hbar} e^{-i \hat{p}_{y} a_{y} / \hbar} e^{-i \hat{p}_{z} a_{z} / \hbar}=e^{-i \hat{\mathbf{p}} \cdot \mathbf{a} / \hbar}
* Commutation relationship (using Kronecker delta):
.. math::
\left[\hat{x}_{i}, \hat{p}_{j}\right]=i \hbar \delta_{i j}
* Position-space representation of the momentum operator vector:
.. math::
\langle\mathbf{r}|\hat{\mathbf{p}}| \psi\rangle=\frac{\hbar}{i} \nabla\langle\mathbf{r} \mid \psi\rangle
* position-space representation of eigenstates of the momentum operator vector:
.. math::
\langle\mathbf{r} \mid \mathbf{p}\rangle =\frac{1}{(2 \pi \hbar)^{3 / 2}} e^{i \mathbf{p} \cdot \mathbf{r} / \hbar}
* Angular momentum operators, commutation relationships (using the Levi-Civita symbol :
.. math::
\left[\hat{L}_{i}, \hat{L}_{j}\right]=i \hbar \sum_{k=1}^{3} \varepsilon_{i j k} \hat{L}_{k}
* Position-space representation of angular momentum operator (in spherical coordinates):
.. math::
\hat{L}_{z} \rightarrow \frac{\hbar}{i} \frac{\partial}{\partial \phi}
and
.. math::
\hat{\mathbf{L}}^{2} \rightarrow-\hbar^{2}\left[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{\sin ^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}\right]
Test your knowledge
-------------------
1. What pairs of operators below commute?
A. :math:`\hat{x}` and :math:`\hat{p}_y`
B. :math:`\hat{x}` and :math:`\hat{p}_x`
2. What pairs of operators below commute?
A. :math:`\hat{S}_x` and :math:`\hat{\boldsymbol{S}}^2`
B. :math:`\hat{S}_x` and :math:`\hat{S}_y`
3. Among these vector states, which ones are not eigenvectors of operator :math:`\hat{r}=(\hat{x}, \hat{y}, \hat{z})`?
A. :math:`\bra{x,y,z}`
B. :math:`\ket{x-\delta,y,z}`
C. :math:`1/\sqrt{2} (\ket{x,y,z}-\ket{-x,y,z})`
4. Among the possibilities below, select all correct normalization relationships.
A. :math:`\langle p_x, p_y, p_z | p'_x, p'_y, p'_z \rangle = \delta(p_x-p'_x)\delta(p_y-p'_y)\delta(p_z-p'_z)`
B. :math:`\langle p_x, p_y, p_z | p'_x, p'_y, p'_z \rangle = \delta(x'-x)\delta(y'-y)\delta(z'-z)`
C. :math:`\langle p_x, p_y, p_z | p'_x, p'_y, p'_z \rangle = \delta_{p'_x,p_x}\delta_{p'_y,p_y}\delta_{p'_z,p_z}` (where :math:`\delta_{ij}` is Kronecker's delta)
5. Imagine a two-body system where the interaction potential between the two bodies only depends on their separation distance. What are the correct statements from the list below?
A. We can rewrite the Hamiltonian in the center-of-mass frame so that it becomes effectively a one-body problem. This is an approximation.
B. The system is invariant under translation. This is translated mathematically as :math:`\comm{\hat{H}}{\hat{\boldsymbol{P}}}=0` (where :math:`\hat{\boldsymbol{P}}` is the total linear momentum of the system).
C. The system is invariant under rotation. However, because IQM is so weird, one cannot claim that :math:`\comm{\hat{H}}{\hat{\boldsymbol{L}^2}}=0` (where :math:`\hat{\boldsymbol{L}}` is the total angular momentum of the system).
D. None of the other claims is correct.
6. We know that a system is invariant under rotation. How do you translate this mathematically?
A. :math:`\comm{\hat{H}}{\hat{\boldsymbol{L}}^2}=\comm{\hat{H}}{\hat{{L}}_x}=\comm{\hat{H}}{\hat{{L}}_y}=\comm{\hat{H}}{\hat{{L}}_z}=0`.
B. :math:`\comm{\hat{H}}{\hat{\boldsymbol{L}}^2}=\comm{\hat{H}}{\hat{L}_z}=0`; and :math:`\comm{\hat{L}_x}{\hat{{L}}_y}=0`.
C. :math:`\comm{\hat{\boldsymbol{L}}^2}{L_z}=0`.
D. None of the other mathematical descriptions corresponds to a system that is invariant under rotation.
7. (select the most accurate answer) The orbital momentum operator :math:`\hat{L}_z`
A. is the generator of rotations around the :math:`z` axis.
B. can be expressed as :math:`\hat{L}_z=\hat{x}\hat{p}_y-\hat{y}\hat{p}_x`.
C. Both answers are correct
8. Consider a system of two bodies whose interaction only depends on the separation between the two particles
A. The system is invariant under translation and but not under rotation of both particles
B. The system is invariant under translation but not under rotation because :math:`\comm{\hat{L}_z}{\hat{L}_x} \neq 0`
C. The Hamiltonian of the system can be written such that the rotational energy of the entire system appears as an effective potential, in a way similar to the existence of a fictitious centrifugal force in a classical system
D. None of the other claims is correct
9. What are the spherical harmonics?
A. Spherical harmonic oscillators
B. Solutions of the angular part of the general solution of the 2-body problem in the presence of a spherically symmetric potential
C. Spherical model explaining Bohr's model of the hydrogen atom.
D. Functions whose Fourier transforms are obtained by direct product of vector spaces of two coupled harmonic oscillators.
.. hint::
Find the answer keys on this page: :ref:`answerkeys`. Don't cheat! Try solving the problems on your own first!
Homework Assignment
-------------------
Solve the following problems from the textbook: 9.4, 9.7, 9.13, 9.17, 9.23
Recitation Assignment
---------------------
Solve the following problems from the textbook: 9.3, 9.8, 9.12, 9.16, 9.20