Half the particles in the universe bear his name

On January 1, 1894, Satyendra Nath Bose was born. Together with Albert Einstein, he formulated the Bose-Einstein statistics, one of the most significant discoveries in physics. He also made crucial contributions to mathematical physics.

Satyendra Nath Bose was a Renaissance man. While his primary work was in science, he harbored deep interests in literature and music. He was an aficionado of Hindustani classical music's acoustics and French literature and actively promoted the study of science in the Bengali language. His scientific papers spanned diverse fields, including quantum quantization, spectroscopy, chemistry, statistics, unified field theory, theoretical mechanics, thermoluminescence, and even the ionosphere. At the University of Dhaka, he established a laboratory for biochemistry and an X-ray research laboratory at the University of Calcutta. These initiatives facilitated the work of other researchers. However, not all of his experimental studies culminated in detailed research papers. His work on quantum theory was consolidated into four papers, yet, unfortunately, he did not leave behind any universally comprehensible writings detailing this revolutionary work.

The historical background and subsequent developments of how Satyendra Nath Bose arrived at his theory of quantization have been meticulously analyzed by Professor Partha Bose in the book S.N. Bose Collected Scientific Papers (1994). Two aspects of this evolution are notable. First, Bose may have speculated on the angular momentum of photon particles. Second, he considered that electrons in high-energy orbits within an atom do not spontaneously transition to lower-energy orbits without the intervention of incident radiation. This view might have clashed with Einstein's, who posited two methods of light emission from atoms: the spontaneous transition of electrons and the stimulated transition to lower-energy orbits by incident radiation. Modern science somewhat prioritizes Bose's stance. However, today's discussion focuses on Bose's seminal 1924 paper, which immortalized his name in science.

In 1921, Satyendra Nath Bose joined the University of Dhaka as a Reader and, partly encouraged by Meghnad Saha, began studying blackbody radiation. Many of you may know that quantum mechanics originated from solving a problem related to blackbody radiation. What is a blackbody? It is an idealized black object that absorbs all incident electromagnetic radiation but emits light according to its temperature—not at a single wavelength but across a spectrum of various wavelengths. In 1859, German scientist Kirchhoff introduced the concept of an ideal blackbody. He proposed that by painting a furnace black, insulating it with non-conductive materials, and leaving a single hole (with its walls also painted black), all light waves entering the hole or the black walls inside would be absorbed, while radiation emerging from the furnace could be observed.

Although an ideal blackbody has a specific geometric shape, broadly speaking, every object is to some degree a blackbody—the Sun, all the objects around us, and even the radiation from our bodies can be explained through the spectrum of a blackbody. By the late 19th century, classical physics posited that any object with a temperature would emit radiation. The higher the temperature, the more intense and higher energy the emitted radiation would be.

The spectrum of a blackbody at a given temperature can be plotted with frequency on the x-axis and the amount, brightness, or intensity of the radiation on the y-axis. Higher-frequency light waves have greater energy. The peak intensity of the spectrum shifts to the right, indicating higher frequencies.

At the end of the 19th century, scientists assumed that electromagnetic waves were generated inside the cavity of a blackbody (Figure 1). At that time, they were unaware of the structure of atoms or the positioning of electrons within them. They proposed that the waves originated from a type of oscillator situated on the walls. The frequency at which these oscillators vibrated would determine the electromagnetic waves they produced—one wave per frequency. These waves within the cavity were required to be standing waves, with the electric field (or vector) values at the cavity walls being zero (Figure 1).

How many waves of various frequencies can exist within a cavity of a particular shape? If that number is known, multiplying it by the average energy of a wave gives the radiated intensity. Since this is a three-dimensional cavity, the calculation is somewhat complex; this value is referred to as the "degrees of freedom" or "phase space," proportional to the square of the frequency (~f²).

Skipping the detailed derivation, the final result is as follows: for a specific frequency range (e.g., f to f + df), the phase space value is $8\pi f^2/c^3$, where $c$ is the speed of light. Can this expression be explained in simple terms? Suppose there are five possible waves within the frequency range of 100 to 1200. If we multiply 5 by the energy of each wave, the total energy density (or intensity) of those waves is obtained, represented on the y-axis of Figure 2. The Rayleigh-Jeans formula assumed the average energy of a wave to be $kT$ (where $k$ is the Boltzmann constant and $T$ is the temperature). Thus, the total intensity becomes $8\pi f^2 kT/c^3$. Based on the assumption that the frequency is continuous, integration over frequency yielded the average energy as $kT$.

The Rayleigh-Jeans formula was used to explain experimentally observed spectra. It matched experimental values for low frequencies but failed for high frequencies. This discrepancy was termed the "ultraviolet catastrophe" at the time. Why was it a catastrophe? Because at ultraviolet frequencies, the Rayleigh-Jeans formula predicted almost infinite intensity (Figure 2).

In 1900, Max Planck made a groundbreaking decision. He demonstrated that if electromagnetic radiation is treated not as continuous but as a collection of discrete packets (quanta or quantum), a formula could be derived that accurately explained the blackbody spectrum. Each quantum would have energy $hf$, where $h$ is Planck's constant.

Planck hypothesized that electromagnetic radiation comprises quanta with energies $hf, 2hf, 3hf, \dots$. By summing these as a series, he showed that the average energy of a quantum is not $kT$ but rather $hf/(e^{hf/kT}-1)$. The details of how Planck arrived at this result are beyond today's discussion. However, it is worth mentioning that he used the concept of entropy in this problem. Entropy measures the number of ways energy (or particles) can be arranged or distributed. A closed system naturally progresses toward higher entropy and remains at maximum entropy in thermal equilibrium. Using this principle, Planck showed that the energy of discrete quantum particles could be described by $hf$.

Planck's approach introduced the quantum (discrete or separate) concept, but many scientists were not entirely comfortable with it. This discomfort arose because Planck derived his formula by blending classical physics with the new quantum theory. Classical physics calculated the number of waves in the cavity based on continuous wave frequencies, which is a traditional approach. However, Planck applied the quantum idea of discrete particles in calculating energy. Radiation could not simultaneously behave as both a continuous wave and discrete particles—it had to be treated as one or the other. This inconsistency persisted for the next 20 years, with scientists like Debye, Einstein, Natanson, Kamerlingh Onnes, and Pauli attempting, though not fully succeeding, to resolve it.

In March 1924, Meghnad Saha visited Dhaka and discussed with Satyendra Nath Bose the research by these scientists aimed at resolving the logical inconsistencies in Planck's method. By then, Saha had incorporated the quantum nature of light into his research, particularly the concept that the momentum of a photon is $p = hf/c$.

Within two months, Satyendra Nath Bose formulated Planck's equation without any logical inconsistencies. On June 4, 1924, he sent a research paper to Einstein, accompanied by a letter:

I am anxious to know your opinion about it. You will notice that I have derived the coefficient $8\pi f^2/c^3$ in Planck's formula without relying on classical electrodynamics; I have merely assumed that the volume of the elementary phase space is ( h^3 ...

This represents a revolutionary step. Planck assumed that the walls of a blackbody generate electromagnetic waves, with the frequency of the waves being equal to the frequency of the oscillators. By 1924, however, it was understood that these oscillators were atoms. Satyendra Nath Bose disregarded the concept of oscillators altogether, instead proposing that the cavity of a blackbody is filled with quantum particles of light (which we now call photons). He then determined how many positions (or states) these particles could occupy within a specific frequency range (from $f$ to $f + df$). To do this, he used the three-dimensional momentum components $p_x, p_y, p_z$ and the spatial coordinates $x, y, z$. The magnitude of the three-dimensional momentum is $p = hf/c$. Using this framework, Bose found that the number of states for a frequency range $df$ within a spatial volume $V$ is $4\pi f^2 V df / c^3$.

Next, he accounted for the two polarizations of a photon (due to angular momentum), multiplying the above expression by 2. From a paper by Chandrasekhara Raman, we learn that Bose’s original research paper contained an explanation for this multiplication, but it was omitted in Einstein’s translation. Later, it was confirmed that photons possess two angular momentum states: one aligned with their direction of motion and the other opposite to it.

In the following step, Bose employed a new method of enumeration. According to this approach, one quantum particle of light cannot be distinguished from another (they are identical), and multiple quantum particles can occupy the same quantum state. If there are $N_S$ quanta to be distributed across $A_S$ states (as determined by Equation 1), Bose illustrated this by considering cases such as $p_0$ states containing zero quanta, $p_1$ states containing one quantum, $p_2$ states containing two quanta, and so forth. Adding these yields the total number of quanta $N_S$. For example, consider two states with zero quanta, three states with one quantum each, and four states with two quanta each. The total number of quanta is $0 \times 2 + 1 \times 3 + 2 \times 4 = 11$.

If a given energy level (or frequency range) contains 1,000 quanta distributed across 10 states, then each state can contain anywhere from 0 to 1,000 quanta, without restriction on the number of quanta in a single state.

For a specific frequency range $df$ (or energy), the same distribution applies to the subsequent frequency range $df$. Furthermore, for each specific distribution of $df$, all distributions of other $df$ values are possible. For instance, if the number of ways $W$ is 10 for the first $df$ and 5 for the second, the combined number of ways is $10 \times 5 = 50$. Thus, if $s$ represents the number of $df$ series or ranges, then $s = 1$ for the first and $s = 2$ for the second. The product of $s$-values for many ranges provides the total distribution.

Boltzmann used a similar enumeration method, but it did not include the concept of indistinguishable quanta or particles. Polish scientist Ladislaus Natanson developed similar statistics in 1911, but his work was overlooked because he did not view blackbody radiation from a quantum perspective. There is no indication that Bose was aware of Natanson’s work.

Let us assume we have three energy levels $E_1, E_2, E_3$. For $E_1$, there are three quanta distributed across two states. This means the three quanta can be distributed in four ways: $(0, 3), (3, 0), (1, 2), (2, 1)$. According to Equation 5, the number of distributions is $(3 - 1 + 2)! / (3! \times (2 - 1)!) = 4$. Now suppose $E_2$ has four quanta distributed across three states. The total number of distributions is $(4 - 1 + 3)! / (4! \times (3 - 1)!) = 15$. Next, assume $E_3$ has five quanta distributed across three states. Here, the number of distributions is $(5 - 1 + 3)! / (5! \times (3 - 1)!) = 21$. Thus, for these three energy levels, the total number of distributions is $4\times 15\times 21=1,260.\mathrm{<br>}\mathrm{<br>}$

Any isolated or discrete system spontaneously reaches a state of thermal equilibrium, where the entropy attains its maximum value. Thus, the subsequent step was to determine the maximum entropy of the system under consideration. Boltzmann had provided a formula for calculating entropy: $S = k \ln(W)$, where $k$ is Boltzmann's constant, and $W$ represents the number of possible distributions, i.e., the ways in which quanta can occupy various states. To find this maximum value, Satyendra Nath Bose employed calculus and the Lagrange multiplier method—already an established technique—based on two conservation principles. The first principle stated that the total energy of the system remains conserved, while the second stated that the total number of quanta remains conserved. However, Bose demonstrated that only the first principle was required to derive Planck’s radiation formula. His calculations revealed that the conservation of quanta was unnecessary, proving that the number of quanta or photons is not conserved. This is because processes such as atomic emission and absorption can result in the creation or annihilation of photons.

The truly innovative aspect of Bose’s work was treating quanta as indistinguishable entities, making it impossible to differentiate one from another. This indistinguishability laid the foundation for focusing solely on how many quanta occupy each state rather than identifying which specific quanta occupy which states.

Bose’s work has two significant components: first, determining the density of states based on the momentum and spatial coordinates of quantum particles, entirely relying on quantum mechanics (as opposed to classical wave theory); and second, introducing a novel enumeration theory for calculating energy distributions of these particles. It is this second contribution for which he is most renowned, now known as Bose-Einstein statistics. However, when Einstein translated Bose’s paper, his recommendation letter to the journal editor only mentioned the first contribution.

In 1925, Einstein extended Bose’s statistics to gaseous molecules, hypothesizing that near absolute zero, atoms would behave like Bose particles, i.e., they would occupy a single quantum state. This state of matter is called the Bose-Einstein condensate (BEC). At temperatures below 2.17 Kelvin, liquid helium transitions into a superfluid phase, where all its atoms occupy the lowest energy state, their spin becomes zero, and the liquid flows without viscosity. In 2001, three scientists were awarded the Nobel Prize for successfully inducing this state in gases of rubidium and sodium atoms.

Today, particles with integer quantum spins are called bosons in honor of Satyendra Nath Bose, and they adhere to Bose-Einstein statistics. The term "boson" was first used by Paul Dirac in 1940. Examples include photons, W and Z bosons, gluons (all with spin 1), and particles like the Higgs boson and pions (spin 0). These are all bosons.

Satyendra Nath Bose formulated his statistics during the nascent stages of quantum mechanics, prior to Schrödinger’s wave function and Heisenberg’s uncertainty principle. He accomplished this groundbreaking work from the distant periphery of the world's major scientific research centers, driven by a resolve to contribute fundamentally to global scientific progress. Sitting at the University of Dhaka, he authored this timeless paper, a testament to the institution's illustrious legacy.

In 1964, Life magazine published a book titled The Scientist, which discussed how scientists worldwide were collectively shaping a scientific culture. This book, which was in our home in Dhaka, featured a rare and remarkable photograph of Satyendra Nath Bose. As a school student in 1968, I was astonished to see the image of a Bengali scientist in an international book. Writing about his work today, that astonishment remains undiminished.

Author: Professor, Moreno Valley College, California, USA
Bibliography:

• Partha Ghose, Bose Statistics: A Historical Perspective
• S.N. Bose Collected Scientific Papers (1994), S.N. Bose National Centre for Basic Sciences, Calcutta