Let \(M\) be the spacetime manifold, and for the purposes of this post, let the group of diffeomorphisms, \(\mathrm{Diff}(M)\), be the group of spacetime symmetries2. As warm up let us think about scalar fields that are just functions on \(M\), i.e., sections of the trivial line bundle. The only natural way a diffeomorphism \(\Lambda : M \to M\) can act on this scalar field, \(\phi : M \to \mathbb R\), is by pulling it back by \(\Lambda^{-1}\),
\[(\Lambda \cdot \phi) (p) = \phi(\Lambda^{-1} p) \text{ for every } p \in M,\]which is precisely how physics textbooks3 write the transformation rule for scalar fields. Let us see what happens in a nontrivial example.
Vector fields are sections of the tangent bundle \(TM \to M\). Diffeomorphisms act on vector fields in a natural way by pushforwards. In particular, for a vector field \(v\), and a diffeomorphism \(\Lambda\), the action is given by
\[(\Lambda \cdot v)(p) = (\Lambda_\ast v)(\Lambda^{-1} p) \text{ for every } p \in M,\]where \(\Lambda_\ast\) is the pushforward of vector fields induced by \(\Lambda\). Once again, this can be identified as the familiar transformation law for vector fields from physics textbooks.
However, note that the pushforward — seen as a bundle map — does two distinct things: (1) moves fibers around, and (2) acts on fibers by a linear transformation. How precisely these two decouple, is the content of the universal property of pullback bundles.
Per this universal property, the bundle map \(\Lambda_\ast = \kappa \circ \lambda\) factors through the pullback bundle \(\Lambda^\ast TM \cong TM\), by a unique map \(\lambda\). If \(v\) is a vector field, then the action of \(\kappa\) is to move the fiber around, i.e., \((\kappa\cdot v)(p) = v(\Lambda^{-1} p)\). And the action of \(\lambda\) is a linear transformation of fibers \((\lambda \cdot v)(p) = \lambda(p) v(p)\), where \(\lambda(p) \in \mathrm{GL}(T_p M)\).
Before moving to more generic situations, it is nice to pause and note that the transformation rule for scalar fields I wrote in the introduction, is a special case of what happens to vector fields. Fibers are moved around but the action on fibers is trivial.
If you are eagle-eyed (or if you already know where this story is going), you may ask if the bundle automorphism \(\lambda\) has an interpretation as a section of some bundle over \(M\) whose fibers are the group \(\mathrm{GL}(\mathbb{R}^m)\). This is correct and the interpretation is fleshed out in the following paragraphs.
To understand more general kinds of fields, we should look for a geometric structure that unifies and generalizes the examples above. This geometric structure is that of a principal \(G\)-bundle and associated vector bundles.
A principal \(\mathrm{SO}(m)\)-bundle that comes for free with every orientable manifold is the bundle of oriented orthonormal frames of the tangent bundle4, which I will denote \(\mathrm{SO}(M) \to M\). An almost tautological remark is that the tangent bundle is isomorphic to the associated bundle \(\mathrm{SO}(M) \times_\mathbf{m} \mathbb{R}^m\), where \(\mathbf{m}\) is the defining representation of \(\mathrm{SO}(m)\). Real scalar fields can be seen as sections of the associated line bundle \(\mathrm{SO}(M) \times_\mathbf{1} \mathbb{R}\), where \(\mathbf{1}\) is a one-dimensional real representation of \(\mathrm{SO}(m)\).
An abstract but more geometric way of thinking about vector bundles is to regard the associated frame bundle — a principal \(G\)-bundle for an appropriate structure group \(G\) — as the basic object, which gives rise to a whole family of vector bundles with the same topology5, indexed by linear representations of \(G\). From this point of view, we can construct a large class of fields on \(M\), as sections of \(\mathrm{SO}(M) \times_\rho V\) for every representation \(\rho : \mathrm{SO}(m) \to \mathrm{GL}(V)\).
To see how diffeomorphisms act on these fields we can run parallel to the discussion above on vector fields, once we note that vector bundle maps induce maps of associated frame bundles. So, a diffeomorphism induces the principal bundle map
which factors uniquely through the pullback bundle, \(\Lambda^\ast \mathrm{SO}(M) \cong \mathrm{SO}(M)\) as
As before \(\kappa\) moves fibers around, and \(\lambda\) acts on the fibers as a bundle automorphism6. Principal bundle automorphisms are sections of the adjoint bundle, so \(\lambda\) is a section of the associated bundle \(\mathrm{SO}(M) \times_\textrm{Ad} \mathrm{SO}(m)\).
On sections, \(\varphi\), of the associated vector bundle \(\mathrm{SO}(M) \times_\rho V\), a diffeomorphism acts as a composition of a coordinate change \(\kappa\), and a bundle automorphism \(\lambda\) as follows
\[(\Lambda \cdot \varphi)(p) = (\lambda \cdot \varphi)(\Lambda^{-1} p) \text{ for every } p \in M,\]where \((\lambda \cdot \varphi)(p) = \rho(\lambda(p))\varphi(p)\).
The upshot of all this abstract7 language is that the action of spacetime symmetries on any field that you can possibly think of is either — in case of tensors — a special case of the formula above, or — for spinors or superfields — a straightforward generalization of it. To wrap up, I will sketch how \(\mathrm{Diff}(M)\) acts on tensor fields and spinors.
An \((a, b)\)-tensor field on \(M\) is a section of the vector bundle \(TM^{\otimes a} \otimes T^\ast M^{\otimes b}\). Or in the language of principal bundles, it is a section of the vector bundle associated to \(\mathrm{SO}(M)\) with \(\rho = \mathbf{m}^{\otimes a} \otimes \mathbf{\bar m}^{\otimes b}\), where \(\mathbf m\) is the fundamental representation of \(SO(m)\) and \(\mathbf{\bar m}\) is its dual. In local coordinates, \(\rho(\lambda)\) will become the famous transformation rule for tensors.
Finally, if the spacetime manifold supports a spin structure then spinor fields can be understood as sections of an associated spinor bundle. In particular, given a choice of spin structure on \(M\), there is a principal \(\mathrm{Spin}(m)\)-bundle over it, \(\mathrm{Spin}(M) \to M\). Dirac spinor fields on \(M\) are sections of the associated vector bundle \(\mathrm{Spin}(M) \times_\sigma \Delta_m\), where \(\sigma : \mathrm{Spin}(m) \to \mathrm{GL}(\Delta_m)\) is a particular complex representation of the spin group called the spinor representation.
To see how a diffeomorphisms acts on a spinor fields, note that — modulo some assumptions on the topology of \(M\) — the \(\mathrm{SO}(M)\) bundle map induced by a diffeomorphism lifts to a \(\mathrm{Spin}(M)\) bundle map. As before, this bundle map can be decomposed, via the pullback bundle, into a piece that moves the fiber around and a piece that acts on fibers. So that the transformation rule for spinors can be written as
\[(\Lambda \cdot \psi)(p) = (\lambda \cdot \psi)(\Lambda^{-1} p) \text{ for every } p \in M,\]where the bundle automorphism, \(\lambda : \mathrm{Spin}(M) \to \mathrm{Spin}(M)\), acts on each fiber by the spinor representation \((\lambda \cdot \psi)(p) = \sigma(\lambda(p)) \psi(p)\).
And some waving of hands. ↩
Even when the setup is different, the story of spacetime symmetries will either — in case of flat space with Poincaré symmetry — be a special case of the one presented here or — for a supermanifold with supersymmetry — a straightforward generalization of it. ↩
Like Mark Srednicki’s Quantum Field Theory, for example. ↩
The tangent frame bundle is a priori a principal \(\mathrm{GL}(\mathbb{R}^m)\)-bundle, but the structure group can be reduced to \(\mathrm{SO}(m)\) by (1) picking a Riemannian metric, which is always possible for a smooth manifold, and (2) picking an orientation, which is possible because the manifold is assumed to be orientable. ↩
By the same topology, I mean that the local trivialization and transition functions are the same. ↩
Existence and uniqueness of the bundle automorphism \(\lambda\) is due to the universal property of pullback bundles. ↩
Honestly, the question of precisely how diffeomorphisms act on the tangent frame bundle turned out to be subtler than I had anticipated. For more details of the story I have sketched here, my favorite references are Loring Tu’s Differential Geometry, and Thomas Friedrich’s Dirac Operators and Riemannian Geometry. ↩
Consider the problem of classifying linear maps between two finite dimensional complex vector spaces, \(A : V \to W\), up to change of basis. If you are very smart, you will eventually conclude that this is done completely by the set of numbers \((\dim V,\) \(\dim W,\) \(\dim \ker A,\) \(\dim \mathrm{im}\, A)\), subject to the rank–nullity theorem.
But if you are geometrically minded, you might think that linear maps are just matrices, so the space of all linear maps must be \(R_{m, n} = \mathrm{Mat}_{n \times m}(\mathbb{C}) \cong \mathbb{C}^{mn}\), where \(\dim V = m\) and \(\dim W = n\). And identifying linear maps that are equivalent under change of bases is done by quotienting this space by the group \(G_{m, n}\) \(= \mathrm{GL}(V) \times \mathrm{GL}(W) \cong\) \(\mathrm{GL}_m(\mathbb{C}) \times \mathrm{GL}_n(\mathbb{C})\). And then, you will conclude that the classification problem is solved completely by the space \(M_{m, n} = R_{m, n} / G_{m, n}\).
Fantastic! Let us try to compute this space in some simple cases.
First, note that for \(A \in R_{m, n}\) and \((S, T) \in G_{m, n}\), the action is given by
\[(S, T)\cdot A = T A S^{-1},\]which is exactly a change of basis.
Now, let us do the simplest case: \(m = n = 1\). We have \(R_{1, 1} \cong \mathbb{C}\) and \(G_{1, 1} \cong \mathbb{C}^* \times \mathbb{C}^*\), and the action \(G_{1, 1} \curvearrowright R_{1, 1}\) is given by \((s, t)\cdot a = (t / s) a\) for every \(a \in \mathbb{C}\) and \(s, t \in \mathbb{C}^*\). It is an easy exercise to show that \(M_{1, 1} = \{[0], [1]\}\).
The good news is that \(M_{1, 1}\) has two points—corresponding to the two orbits of \(G_{1, 1}\) in \(R_{1, 1}\)—which matches with the linear algebra answer. The bad news is that the quotient topology makes \(M_{1, 1}\) the Sierpiński 2-point space.
Indeed, the point \(\{[0]\} \subset M_{1, 1}\) is closed because the orbit is a singleton, whose complement is open in \(\mathbb C.\) It is not open because the orbit is not open in \(\mathbb C\). \(\{[1]\} \subset M_{1, 1}\) is open because its orbit \(\mathbb C \setminus \{0\} \subset \mathbb C\) is open. It is not closed because the complement of its orbit is a singleton and therefore not open in \(\mathbb C\). Furthermore, as a result, the closure of \({\{[1]\}}\) is equal to the whole space, \(M_{1, 1}\).
The only things I knew about this space before seeing this example were that
And that’s why I was horrified when I saw it come up in, what is essentially, an elementary linear algebra problem.
I learned about this in Marcus Reineke’s great series of lectures on Quivers and Cohomological Hall Algebras at the Galileo Galilei Institute in May 2024.
]]>Here is the outline, PDF with outline attached, and a link to the original (arXiv:1001.2933).
]]>For those who don’t know, NiSERCast is an outreach project started by a few NISER students, including myself, consisting of a podcast in which students have conversations with professors about their research and their life in academia. We were only able to release one episode, and when the second wave of the coronavirus hit India, we lost momentum and have unfortunately been on a hiatus since May 2021.
It was the February of 2020 when Spandan, who was also my roommate at the time, brought up the idea of a student led podcast based around conversations with NISER professors. As we imagined it at the time, it was going to be like MIT OpenCourseWare’s Chalk Radio, but more informal, less professional, and with lower production values. Over a weekend, Spandan, and I whipped up a very quick and dirty website and configured an Atom feed for the podcast, and then we sent out a wider call for volunteers via the Science Activities Club (SAC).
I must say, I was quite overwhelmed by the initial response. Many people, both my seniors and juniors, were extremely enthusiastic about the podcast. There were meetings and discussions in which the basics were fixed—the division of responsibilities: hosting, production, editing, social media; the basic format; the schedule; and perhaps most importantly, the name. We also talked to professors and made a list of potential guests. Finally, we decided to start recordings in March, after our midsemester exams which were scheduled for the last week of February.
And then, of course, the coronavirus hit.
We picked up the project again, almost exactly a year later, in March 2021, after social distancing norms were relaxed a little by the institute. Student volunteers fell into two major categories: (1) hosts, who were responsible for inviting professors as guests and, well, hosting the podcast, and (2) people responsible for recording, editing and releasing episodes. Before approaching the Dean of Student Affairs (DoSA) with a proposal, we did the following:
After making an intro jingle, a small redesign of the NiSERCast website, and getting required permissions from the DoSA, we were ready to start recording.
A rough format for episodes was decided after a discussion among all volunteers. To keep the podcast accessible to laymen—especially high school students who might be considering a career in the sciences—we decided that each episode should have two hosts, at least one of whom should be from outside the professor’s department. Initially, we decided to let the professor and the hosts talk for about two hours, which could then be cut down for a one hour episode. In retrospect, I feel like this was a mistake. With our courseload and other academic engagements, and having only one group of four people to look after recordings, editing, and releases, cutting a two hour conversation down to an hour every two weeks, in addition to supervising the recordings every week, was just not feasible. To be fair, however, these glitches would have probably evened themselves out if we were able to do a few more episodes.
If you have listened to NiSERCast, you may have noticed that it does not sound like a professionally produced podcast. However, I am going to describe the recording setup we used anyway, for reference.
The technical side of things was handled by Anirudh, Jyotirmaya, Spandan and I. Each of the three participants talked into an Ahuja AWM-490V1 wireless microphone, whose outputs were fed into Audacity for recording via a Yamaha MG10XU mixer and a Focusrite 2i2 Scarlett audio interface. Hosts, professors and the producers also wore headphones to monitor audio levels. We got the mics, the mixer and a few cables from the Drama and Music Club, the Focusrite audio interface belonged to me, we asked hosts to bring headphones, and we had to get things like ¼-inch to XLR cable, aux cables, and a five-way audio splitter on our own to make everything work.
I rewrote the NiSERCast website, more or less from scratch, in March 2021. It is built on top of Jekyll—which provides an easy templating system and manages the Atom feed—and is deployed with GitLab Pages while being hosted in a GitLab repository.
For the Atom feed, I started with the basic template that a new Jekyll project comes with and added extra tags according to Apple’s, Google’s and Spotify’s feed requirements. To make sure that everything is working as expected before submitting the feed to Apple, Google and Spotify, I used feed validators at Podbase and W3C.
After releasing the first episode, we hit a small bureaucratic bump in the road. As this podcast is an outreach activity bearing NISER’s name that is going to be released to the general public, we were also supposed to talk to the NISER Outreach Committee and get their permission separately. This could have been pointed out when we sent our initial proposal to the DoSA, but we caught the Outreach Committee’s attention only after the first real episode was released. The fact that our first guest, Prof. Varadharajan Muruganandam, is very opinionated did not help either.
Anyway, after a period of uncertainty, in which we were not even sure that we would be allowed to continue the podcast, and a few weeks’ delay, we got the clearance to continue as long as we made it clear that the opinions expressed were personal opinions of the guests and did not represent NISER’s views.
When the second wave of the coronavirus hit, almost all of the volunteers chose to return to their homes; because of a variety of personal reasons, and poor management of the pandemic by NISER’s administration, we could not keep producing the podcast. After the campus reopened last December, Jyotirmaya, Spandan and I were halfway into our final year and were extremely busy with our master’s theses and PhD applications, and it became nearly impossible for us to organize recording sessions from scratch again.
If you have made it this far into this post: thank you for putting up with my ramblings. I hope that this has been useful in some way for people who want to continue the podcast, or who decide to start a similar project at NISER. All in all, I am extremely proud of what we were able to accomplish while working with various constraints and limited resources. And I must express my deepest gratitude and thanks to everyone involved in the project.
]]>Oh, and did I mention? I am going to try writing everything. From scratch. In C. And I’m writing this post to convince myself that it is not a bad idea.
Writing code for a computational physics class is weird. You are supposed to
import numpy as np right at the beginning to “make your life easier”,
but then you are supposed to forget that extremely optimized versions of
algorithms that you will naively implement over the next few weeks were
imported into your program by the very first line of code you wrote. If you don’t
want to import numpy, you could, of course, do something like class
Array(list), but then, in addition to the housekeeping, you have to bear with
extremely slow linear algebra operations in everything you write. Course
instructors don’t want you to use “any” libraries, but in practice this rule
gets extremely fuzzy: What *is* a library? Does the standard library count?
Is it okay to use objects from a library but not its algorithms? Where do you
draw the line?
The other option, if you don’t want to use “any” libraries and still have your code be relatively quick, is to use a compiled language like C++ or Rust (or Julia, if you don’t mind being a dirty cheater1).
But writing good object oriented code with C++ is hard. Getting used to Rust’s memory safety system is hard. Writing C code, on the other hand, is… also hard.
But it is also simple. C++ is very complicated.
Just look at Kernighan and Ritchie’s The C Programming Language side-by-side with its C++ counterpart (Stroustrup’s The C++ Programming Language):
K&R fits a surprisingly accessible and complete introduction to C, several working examples, common idioms, and a reference for the core language and standard library, in a measly 250 pages. But this “simplicity” also speaks to a lack of features, most notably garbage collection, operator overloading, and OOP patterns like inheritance, that most people take for granted when writing code in a higher level language.
There is also the whole thing about the inescapable legacy of C in all of modern programming, which I will not recount here. I am also going to skip the metaphysics and philosophy that a more experienced C programmer would have spent several paragraphs on, but here is a quote from Why learn C?:
To want to master C is to care about what is powerful, clever, and free. To become a programmer with all the vast powers of technology at his or her fingertips and the responsibility to do something to benefit the world.
Having written a few numerical linear algebra algorithms, I have to admit, I have started to understand why people say things like this. C forces you to care deeply about your code, especially when it comes to memory. I have found myself trying to minimize heap allocations as much as possible, something I would not have given a second thought to if I were using Julia (even though the documentation suggests that I should).
In the end, there is no perfect language2. And for an application as broad as numerical analysis, there can be no perfect tool for the job3.
There is no single reason why I have decided to stick4 with C.
As far as I can tell, there is no “killer feature”. But putting together the
learning experience C provides, the challenge of working with a very simple
language, its privileged status among programmers, and (perhaps most
importantly) the bragging rights that come with writing C, spending
some extra time and effort fighting code, and beating it into shape with gdb
and valgrind seems worthwhile.
In this post, I start by building some elementary quantum thermodynamics: we shall look at the quantum version of the first law of thermodynamics, some general facts about quantum thermodynamic processes, and then define an effective temperature for quantum systems and a quantum analogue of the isothermal process. Once we have the definition of an isothermal process, we shall see with an easy computation that when the working substance is a quantum two-level system, the internal energy is not invariant during the process.
If the energy eigenstates of our quantum system are labelled by \(|n\rangle\) with corresponding eigenenergies \(E_n\), the Hamiltonian (in the energy basis) can be written as
\[\hat{H} = \sum_n E_n |n \rangle \langle n|. \tag{1}\]It is reasonable to identify the internal energy \(U\), of the quantum system with the expectation value of the Hamiltonian, hence
\[U = \langle \hat{H} \rangle = \sum_n E_n P_n, \tag{2} \label{internal-energy}\]where \(P_n = \langle n | \hat{\rho} |n \rangle\) is the occupation probability for the \(n\)th eigenstate, and \(\hat{\rho}\) is the density matrix.
We recall the usual expression of the first law: \(dU = dQ + dW\), and take the exterior derivative of \(\eqref{internal-energy}\) to get a quantum analogue2
\[dU = \sum_n \left(E_n dP_n + P_n dE_n\right). \tag{3} \label{quantum-first-law}\]In order to identify the analogues of heat exchanged and work done, we recall the Gibbs formula for entropy: \(S = - \sum_n P_n \ln P_n\), and because the heat exchanged is \(dQ = TdS\), we identify
\[dQ = \sum_n E_n dP_n. \tag{4} \label{quantum-heat}\]Consequently, the work done is identified as
\[dW = \sum_n P_n dE_n. \tag{5} \label{quantum-work}\]In heat engines based on quantum systems, a change in energy levels is associated with work done by the engine, and a change in occupation probabilities is associated with the heat exchanged between the engine and the heat bath.
Statistical mechanics tells us that when a quantum system is in equilibrium with a heat bath at temperature \(T = 1/k_B \beta\), the density matrix is given by
\[\rho(\beta) = \frac{e^{-\beta \hat{H}}}{Z(\beta)}, \tag{6}\]where \(Z(\beta) = \text{Tr}(e^{-\beta \hat{H}})\) is the partition function. Occupation probabilities \(P_n\) can be obtained from the diagonal elements of the density matrix,
\[P_n = \langle n |\hat{\rho}(\beta)| n\rangle = \frac{e^{-\beta E_n}}{\sum_m e^{-\beta E_m}}, \tag{7} \label{canonical-distribution}\]and we note that when the system is at a fixed temperature \(T\), the occupation probabilities must satisfy the above distribution.
The effective temperature of a quantum system is defined by ‘inverting’ \(\eqref{canonical-distribution}\). In order to understand what this means, we look at a system with only two states \(|g\rangle\) and \(|e\rangle\) with energies \(E_g\) and \(E_e\) respectively. If the system is in equilibrium with a heat bath, the occupation probabilities satisfy
\[\frac{P_e}{P_g} = e^{-\beta (E_e - E_g)}, \tag{8}\]and the relation can easily be inverted to get the temperature in terms of the occupation probabilities
\[T = \frac{E_e - E_g}{k_B} \left(\ln \frac{P_g}{P_e} \right)^{-1}. \tag{9}\]Based on the above expression, we can obtain an effective temperature even when the system is not in equilibrium with a heat bath: \(k_B T_{eff} = (E_e - E_g) / \ln (P_g/P_e)\).
We see that for a two-level system, the effective temperature is uniquely defined for all occupation probabilities \(P_g\) and \(P_e\). However, this might not be so when the system has more than two levels. In general, a unique effective temperature is defined only when the occupation probabilities satisfy the canonical distribution in \(\eqref{canonical-distribution}\).
Once we know what a temperature means for an arbitrary quantum system, we can define a quantum isothermal process in the obvious way. In a quantum isothermal process, energy levels and occupation probabilities must change simultaneously to always satisfy the canonical distribution for a fixed temperature.
Before looking at what happens to a two-level system in an isothermal process, we shall note a fact that will make certain computations easier: the work done \(dW\), heat exchanged \(dQ\), and therefore the change in internal energy \(dU\), are invariant under a uniform shift of all energy levels.
If we assume that all energy levels shift uniformly: \(E_n' = E_n + \delta\), the first thing we note is that that \(dE_n' = dE_n\) because \(\delta\) is a constant. Next, we consider the occupation probabilities,
\[P_n' = e^{-\beta (E_n + \delta)} \left(\sum_m e^{-\beta (E_m + \delta)}\right)^{-1} = e^{-\beta E_n} \left(\sum_m e^{-\beta E_m}\right)^{-1} = P_n, \tag{10}\]and observe that \(dP_n' = dP_n\). Finally, from the quantum analogues of heat \(\eqref{quantum-heat}\), work \(\eqref{quantum-work}\); and the first law \(\eqref{quantum-first-law}\), the result follows.
In particular we note that, for the two-level system, assuming \(E_g = 0\) has no effect on \(dU.\)
In an isothermal process \(i \to f\), we can assume that \(E_g^f = E_g^i = 0\) and \(E_e^f = \zeta E_e^i\), where \(\zeta > 0\) is some constant that can be used to parametrize the evolution of the system under the process. For our final trick, we consider the internal energy of the two-level system
\[U(\zeta) = \sum_n P_n E_n = \frac{e^{-\beta \zeta E_e}}{1 + e^{-\beta \zeta E_e}} \zeta E_e, \tag{11}\]and its derivative with respect to \(\zeta\)
\[\begin{align} \frac{dU(\zeta)}{d\zeta} & = \frac{e^{-\beta \zeta E_e}}{1 + e^{-\beta \zeta E_e}} \zeta E_e \left(\frac{1}{\zeta} - \frac{\beta}{1 + e^{-\beta \zeta E_e}}\right) \\ & = U(\zeta) \left(\frac{1}{\zeta} - \frac{\beta}{1 + e^{-\beta \zeta E_e}}\right), \tag{12} \end{align}\]which, in general, is non-zero. Thus, we have shown that the internal energy is not invariant3 in a quantum isothermal process, and I have fulfilled my promise.
This blog post was inspired by an appendix in Quantum Thermodynamic Cycles and Quantum Heat Engines (PhysRevE 76.031105, arXiv:quant-ph/0611275) by Quan et al. In the article, the authors have done a more general computation which can also be applied to a quantum harmonic oscillator and a particle in an infinite square well, but they assume (without any justification) that all the energy levels change in the same ratio in an isothermal process. This is not a problem for the two-level system because there are only two energy levels.
I know that isothermal processes with a (classical) non-ideal gas as working substance are not, in general, isoenergetic. The title only is a thinly veiled excuse to write about quantum thermodynamic processes. ↩
For more details on the quantum version of the first law, refer to Quantum Heat Engine With Multi-Level Quantum Systems (PhysRevE 72.056110, arXiv:quant-ph/0504118) by Quan et al. ↩
I am not completely unapologetic for the double negatives in this post. ↩
Elementary texts explain the quantization rule by appealing to the constructive interference of electron waves (the only orbits permitted are the ones whose circumference is an integral multiple of the de Broglie wavelength of the electron). However, we note that de Broglie formulated his hypothesis more than a decade after the Bohr model was proposed.
In this post, I want to start with the information available to Bohr at the time (evidence for quantized energy levels from atomic spectra and the empirical Rydberg formula) and get to the magical quantization rule with the correspondence principle (as Bohr did).
The following section will start with Bohr’s original postulates and establish some notation, I will discuss the origin of the quantization in the next section.
We start with Bohr’s assumptions in his own words:
That an atomic system can, and can only, exist permanently in a certain series of states corresponding to a discontinuous series of values for its energy, and that consequently any change of the energy of the system, including emission and absorption of electromagnetic radiation, must take place by a complete transition between two such states. These states will be denoted as the stationary states of the system.
That the radiation absorbed or emitted during a transition between two stationary states is ‘‘unifrequentic’’ and possesses a frequency \(f\), given by the relation \(E' - E'' = hf\), where \(h\) is Planck’s constant and where \(E'\) and \(E''\) are the values of the energy in the two states under consideration.
We note that both of the above assumptions are at odds with classical electrodynamics because (1) accelerating charges radiate away their energy, and electrons in an orbit around the nucleus must be accelerating; and (2) the frequency of radiation given off by a periodically accelerating charge should be the same as the frequency of acceleration.
A third assumption is the corresponence principle which asserts that, as microscopic systems become macroscopic, the results must go over to the classical.
In order to estimate the energy of a stationary state, we equate the Coulomb force on the electron and the centripetal force
\[\frac{e^2}{4 \pi \epsilon_0} \frac{1}{r^2} = \frac{mv^2}{r} \tag{1}\]which leads to the kinetic energy given by
\[T = \frac{1}{2} m v^2 = \frac{1}{2} \frac{e^2}{4 \pi \epsilon_0} \frac{1}{r}. \tag{2}\]The total energy is then given by
\[\begin{align} E & = T + V \\ & = \frac{1}{2} \frac{e^2}{4 \pi \epsilon_0} \frac{1}{r} - \frac{e^2}{4 \pi \epsilon_0} \frac{1}{r} \\ & = - \frac{1}{2} \frac{e^2}{4 \pi \epsilon_0} \frac{1}{r} \tag{3} \end{align}\]We note that the only variable in the above equation is \(r\). Hence if the atomic energy levels are quantized, the atomic radius must also be quantized. Due to the second postulate, frequency of radiation \(f\), emitted due to a transition from energy \(E'\) to \(E''\) (\(E' > E''\)) is given by
\[\begin{align} hf & = E' - E'' \\ & = \frac{1}{2} \frac{e^2}{4 \pi \epsilon_0} \left( \frac{1}{r'} - \frac{1}{r''} \right) \label{e:frequency1} \tag{4} \end{align}\]where \(r'\) and \(r''\) are the atomic radii of the electron in states of energy \(E'\) and \(E''\) respectively. Equation \eqref{e:frequency1}, along with the Rydberg formula
\[\frac{1}{\lambda_{nm}} = R_H \left(\frac{1}{m^2} - \frac{1}{n^2}\right)\\ \label{e:rydberg} \tag{5}\]where \(n\) and \(m\) are integers, and \(R_H\) is a constant referred to as the Rydberg constant which has the units of inverse length, can be used to deduce the quantization of the atomic radii. Since, the frequency and wavelength are related by \(f \lambda = c\), where \(c\) is the speed of light, Equation \eqref{e:rydberg} can be written in terms of the frequency as
\[h f_{nm} = c R_H \left(\frac{1}{m^2} - \frac{1}{n^2}\right) \label{e:rydberg2} \tag{6}\]Expressions in Equations \eqref{e:frequency1} and \eqref{e:rydberg2} lead us to guess that the orbital radii are quantized as
\[r_n = a_0 n^2 \tag{7}\]where \(n\) is an integer and \(a_0\) is a constant with units of length, which for \(n = 1\) gives the radius of the electron orbit in the lowest energy stationary state and is called the Bohr radius.
Now, we need to calculate the Bohr radius. We stop here to note that this is the point when elementary texts ‘assume’ the angular momentum quantization rule, \(L_z = m_e v_n (a_0 n^2) = n \hbar\) as a third postulate to derive an expression for \(a_0\). In this post however, we will use the correspondence principle to derive and expression for \(a_0\) and consequently the quantization rule.
First, we note that the kinetic energy of the electron is quantized
\[T = \frac{1}{2} m_e v_n^2 = \frac{1}{2} \frac{e^2}{4 \pi \epsilon_0} \frac{1}{a_0 n^2}. \tag{8} \label{e:kinetic}\]Next we recall that, according to classical electrodynamics the frequency of radiation emitted from an oscillating charge is equal to the frequency of oscillation of the charge. According to correspondence principle, the frequency of radiation for transition between adjacent states given by the second postulate, must approach the orbital frequency of the electron in its stationary state for highly energetic states, that is, for large \(r_n\).
The frequency of oscillation of the electron in an orbit of radius \(r_n\) is
\[\begin{equation} f_{orbit} = \frac{v_n}{2 \pi (a_0 n^2)} \label{e:forb} \tag{9} \end{equation}\]Next, we substitute for the velocity from Equation \eqref{e:kinetic}
\[f_{orbit}^2 = \frac{1}{4 \pi^2 n^4 a_0^2} \left[\frac{1}{m_e} \left(\frac{e^2}{4 \pi \epsilon_0}\right) \frac{1}{n^2 a_0}\right] \label{e:forb2} \tag{10}\]The frequency of radiation emitted for transition between adjacent states (between stationary states characterized by \(n\) and \(n + 1\)) can be calculated from the second postulate, and \(f_{radiation}^2\) is given by
\[\begin{align} f_{radiation}^2 & = \left(\frac{1}{2} \left(\frac{e^2}{4 \pi \epsilon_0} \right) \frac{1}{h a_0} \left[\frac{1}{n^2} - \frac{1}{(n + 1)^2}\right] \right)^2 \\ & = \left(\frac{1}{2} \left(\frac{e^2}{4 \pi \epsilon_0} \right) \frac{1}{h a_0} \left[\frac{2n + 1}{n^2 (n + 1)^2} \right] \right)^2. \tag{11} \end{align}\]As \(n\) becomes large, the expression becomes
\[\lim_{n \rightarrow \infty} f_{radiation}^2 = \left[ \left(\frac{e^2}{4 \pi \epsilon_0} \right) \frac{1}{h a_0} \frac{1}{n^3}\right]^2 \tag{12}\]which can be equated to \(f_{orbit}\) and the resulting equation solved for \(a_0\) to obtain
\[a_0 = (4 \pi \epsilon_0) \frac{\hbar^2}{m_e e^2} \tag{13}\]Finally, from Equations \eqref{e:forb} and \eqref{e:forb2}, the velocity of the electron in the \(n\)th Bohr orbit is
\[v_n = \frac{\hbar}{m_e a_0} \frac{1}{n} \tag{14}\]and so the angular momentum is
\[\begin{align} L_z & = m_e v_n r_n \\ & = m_e \left(\frac{\hbar}{m_e a_0} \frac{1}{n}\right) (n^2 a_0) \\ & = n \hbar. \tag{15} \end{align}\]This blog post was inspired by Burkhardt and Leventhal’s treatment of the problem in Topics in Atomic Physics. The primary source, of course, is Bohr’s seminal article titled On the Quantum Theory of Line Spectra.
]]>In this post, I am using the ‘particle physics’ metric signature \((+, -, -, -)\) and taking the speed of light, \(c = 1\).
Our starting point is going to be the fact that the electromagnetic field tensor, \(F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\) (where \(\partial_\mu\) is the four-gradient and \(A_\mu\) is the four-potential) transforms like a Lorentz tensor, i.e. \(F'_{\mu \nu} = \Lambda_\mu^\alpha \Lambda_\nu^\beta F_{\alpha \beta}\) for an arbitrary Lorentz transformation \(\Lambda \in SO(1, 3)\).
The components of a (proper) Lorentz transformation are given by
\[\begin{align} \Lambda_0^0 & = \gamma, \\ \Lambda_0^i = \Lambda_i^0 & = - \gamma \beta_i, \\ \Lambda_i^j & = \delta_i^j + \frac{\beta_i \beta_j}{\beta^2} \left(\gamma - 1\right). \end{align}\]We stop here to note the relation between the electromagnetic field tensor \(F_{\mu \nu}\) and the classical fields \(\vec{E}\) and \(\vec{B}\). We choose the Cartesian coordinates for simplicity, but note that this choice doesn’t lead to any loss of generality because the final expressions derived are coordinate independent.
\[E_i = F_{0 i} = - F_{i 0}, \\\]and
\[B_i = - \frac{1}{2} \epsilon^{i j k} F_{jk} \implies F_{jk} = -\epsilon_{i j k} B_i,\]where \(\epsilon_{ijk}\) is the Levi-Civita tensor. The property \(\epsilon_{ijk} \epsilon^{ljk} = 2 \delta_i^l\) is used invert the relation in the second line. We also note that \(\epsilon_{ijk} = \epsilon^{ijk}\) because for space components the (Cartesian) metric is simply the identity.
For electric field, the transformation law for \(F_{0 i}\) components is used,
\[\begin{align} E'_i & = F'_{0 i} \\ & = \Lambda_0^\mu \Lambda_i^\nu F_{\mu \nu} \\ & = \Lambda_0^0 \Lambda_i^l F_{0 l} + \Lambda_0^k \Lambda_i^0 F_{k 0} + \Lambda_0^k \Lambda_i^l F_{k l} \end{align}\]where the summation is broken up into space and time parts and the antisymmetry of the electromagnetic tensor is used to make the diagonal terms go away. Next we simplify the first two terms in the expression above:
\[\begin{align} \Lambda_0^0 \Lambda^l_i F_{0 l} + \Lambda_0^k \Lambda_i^l F_{k l} & = \gamma \left(\delta_i^l + \frac{\beta_i \beta_l}{\beta^2} (\gamma - 1)\right) E_l + (-\gamma \beta_k)(-\gamma \beta_i) (-E_k) \\ & = \gamma E_i + \gamma (\gamma - 1) \frac{\beta_i \beta_l}{\beta^2} E_l - \gamma^2 \beta_i \beta_k E_k \\ & = \gamma E_i + \gamma \beta_i \beta_k E_k \left(\frac{\gamma - 1}{\beta^2} - \gamma\right) \\ & = \gamma E_i - \frac{\gamma^2}{\gamma + 1} \beta_i (\vec{\beta} \cdot \vec{E}). \end{align}\]Now, we consider the third term:
\[\begin{align} \Lambda_0^k \Lambda_i^l F_{k l} & = (-\gamma \beta_k) \left(\delta_i^l + \frac{\beta_i \beta_l}{\beta^2} (\gamma - 1)\right) (- \epsilon^{mkl} B_m) \\ & = \gamma \epsilon^{imk}\beta_k B_m - \gamma(\gamma - 1) \frac{\beta_i}{\beta^2} \left(\epsilon^{mkl} B_m \beta_k \beta_l\right) \\ & = -\gamma (\vec{\beta} \times \vec{B})_i \end{align}\]In \(\left(\epsilon^{mkl} B_m \beta_k \beta_l\right)\), the sum runs over all the three indices, which makes the term vanish due to the complete antisymmetry of the Levi-Civita tensor.
Finally, we put everything together to yield,
\[E'_i = \gamma \left(E_i - (\vec{\beta} \times \vec{B})_i \right) - \frac{\gamma^2}{\gamma + 1} \beta_i \left(\vec{\beta} \cdot \vec{E}\right),\]or in vector notation,
\[\vec{E'} = \gamma \left(\vec{E} - \vec{\beta} \times \vec{B}\right) - \frac{\gamma^2}{\gamma + 1} \vec{\beta} \left(\vec{\beta} \cdot \vec{E}\right).\]In order to derive the transformation law for magnetic fields, a similar procedure is employed. We begin by noting that,
\[\begin{align} B'_i & = -\frac{1}{2} \epsilon^{ijk} F'_{jk} \\ & = -\frac{1}{2} \epsilon^{ijk} \Lambda_j^\mu \Lambda_k^\nu F_{\mu \nu} \\ & = -\frac{1}{2} \epsilon^{ijk} \left( \Lambda_j^0 \Lambda_k^l F_{0 l} + \Lambda_j^l \Lambda_k^0 F_{l 0} + \Lambda_j^l \Lambda_k^m F_{l m} \right) \end{align}\]Next, the first two terms inside the brackets are simplified,
\[\begin{align} \Lambda_j^0 \Lambda_k^l F_{0 l} + \Lambda_j^l \Lambda_k^0 F_{l 0} & = (\Lambda_j^0 \Lambda_k^l - \Lambda_j^l \Lambda_k^0) F_{0 l} \\ & = \left[(-\gamma \beta_j) \left(\delta_k^l + (\gamma - 1) \frac{\beta_l \beta_k}{\beta^2} \right) - \left(\delta_j^l + (\gamma - 1) \frac{\beta_l \beta_j}{\beta^2} \right) (-\gamma \beta_k)\right] F_{0 l} \\ & = \gamma \left( -\delta_k^l \beta_j + \delta_j^l \beta_k \right) F_{0l}, \end{align}\]so that,
\[\begin{align} -\frac{1}{2} \epsilon^{ijk} (\Lambda_j^0 \Lambda_k^l F_{0 l} + \Lambda_j^l \Lambda_k^0 F_{l 0}) & = \frac{1}{2} \gamma \epsilon^{ijk} \left(\delta_k^l \beta_j - \delta_j^l \beta_k \right) F_{0 l} \\ & = \frac{1}{2} \gamma \left( \epsilon^{ijl}\beta_j - \epsilon^{ilk} \beta_k \right) F_{0 l} \\ & = \frac{1}{2} \gamma \left( \epsilon^{ijl}\beta_j + \epsilon^{ijl} \beta_j \right) F_{0 l} \\ & = \gamma \epsilon^{ijl} \beta_j E_l \\ & = \gamma \left(\vec{\beta} \times \vec{E}\right)_i. \end{align}\]For the final step, we expand
\[\begin{align} \Lambda_j^l \Lambda_k^m F_{lm} & = \left(\delta_j^l + (\gamma - 1) \frac{\beta_j \beta_l}{\beta^2} \right) \left(\delta_k^m + (\gamma - 1) \frac{\beta_k \beta_m}{\beta^2} \right) (- \epsilon_{nlm} B_n) \\ & = - \epsilon_{njk} B_n - \frac{\gamma - 1}{\beta^2} (\epsilon_{njm} \beta_k \beta_m + \epsilon_{nlk} \beta_j \beta_l) B_n. \end{align}\]Where the last term vanishes due to the antisymmetry of the Levi-Civita. Next, the \(-\epsilon^{ijk}/2\) factor is put in to get
\[\begin{align} -\frac{1}{2} \epsilon^{ijk} \Lambda_j^l \Lambda_k^m F_{lm} & = \frac{1}{2} \left[ \epsilon^{ijk} \epsilon_{njk} B_n + \frac{\gamma - 1}{\beta^2} (\epsilon^{ijk} \epsilon_{njm} \beta_k \beta_m + \epsilon^{ijk} \epsilon_{nlk} \beta_j \beta_l) B_n \right] \\ & = B_i + \frac{1}{2} \frac{\gamma - 1}{\beta^2} \left( \beta_m \beta_m B_i + \beta_l \beta_l B_i - \beta_i \beta_n B_n - \beta_i \beta_n B_n \right) \\ & = B_i + (\gamma - 1) B_i - \frac{\gamma - 1}{\beta^2} \beta_i \beta_n B_n \\ & = \gamma B_i - \frac{\gamma^2}{\gamma + 1} \beta_i \left(\vec{\beta} \cdot \vec{B}\right). \end{align}\]In the above simplification, we have used the following properties of the Levi-Civita tensor: \(\epsilon_{ijk} \epsilon^{imn} = \delta_j^m \delta_k^n - \delta_j^n \delta_k^m\) and \(\epsilon_{jmn} \epsilon^{imn} = 2 \delta^i_j\)
Putting everything together leads to
\[B'_i = \gamma \left(B_i + (\vec{\beta} \times \vec{E})_i \right) - \frac{\gamma^2}{\gamma + 1} \beta_i \left(\vec{\beta} \cdot \vec{B}\right),\]and
\[\vec{B'} = \gamma \left(\vec{B} + \vec{\beta} \times \vec{E}\right) - \frac{\gamma^2}{\gamma + 1} \vec{\beta} \left(\vec{\beta} \cdot \vec{B}\right),\]in vector notation.
I found this particular way of deriving the transformation laws pretty neat; and it was a nice exercise in tensor notation and index juggling. If you spot a mistake in the algebra, please let me know.
]]>