Introduction to quantum mechanics for applications in quantum biology: the case of the avian compass

1 Introduction

Since the early 20th century, quantum mechanics has transformed our understanding of nature at its most fundamental level. Emerging from the pioneering works of Planck, Einstein, Bohr, Schrödinger, and Heisenberg, this theory was built to explain the behavior of matter and light in regimes where classical physics failed: from the discrete energy spectra of atoms to the wave-like behavior of particles. What began as an abstract mathematical formalism to describe microscopic phenomena soon became the cornerstone of modern physics, chemistry, and increasingly, biology. Quantum mechanics did not only reshape scientific thought; it redefined the very way we understand the physical world, revealing that probability, superposition, and uncertainty are intrinsic features of reality itself.

As quantum theory matured throughout the 20th century, it evolved from a purely theoretical framework into the foundation of technologies that shape our everyday lives. Transistors and semiconductors, the building blocks of computers, smartphones, and all modern electronics, rely directly on quantum mechanical principles governing the motion of electrons in solids. Lasers, used in medicine, biology, telecommunications, sensors, and data storage, operate based on quantum transitions between energy levels. Even the GPS that orients us daily depends on the precise synchronization of atomic clocks, whose operation is inherently quantum mechanical. In this sense, we already live in a “quantum world,” even if we rarely stop to think about it. Yet, despite these profound implications, most people have little familiarity with what quantum mechanics actually is.

This lack of understanding has also opened the door to widespread misconceptions and pseudoscientific interpretations. The term quantum has become a cultural symbol often stripped of its meaning, used to lend authority to ideas unrelated to physics. Concepts such as “quantum healing,” “quantum consciousness,” and “magnetic therapy” frequently borrow quantum language without any scientific basis. These popular distortions not only obscure what is truly remarkable about quantum theory but also create confusion when the term appears in legitimate research contexts, such as quantum biology. For this reason, it is crucial to distinguish between speculative or metaphoric uses of quantum ideas and the formal, experimentally grounded framework that defines modern physics.

In recent years, the field of quantum biology has attracted growing attention for exploring how genuine quantum phenomena—such as tunneling, coherence, and entanglement (Auletta et al., 2009) — might play functional roles in living systems. Unlike the pseudoscientific claims mentioned above, this field is grounded in established physical theory and empirical investigation. While the warm and noisy environment of biological systems might seem hostile to quantum effects, mounting theoretical and experimental evidence suggests that quantum mechanics can influence key biological processes at the molecular and cellular scales. These include electron tunneling in enzyme catalysis, coherent exciton transport in photosynthesis, and spin entanglement in avian magnetoreception (Hore and Mouritsen, 2016; Wiltschko, 2012). The scientific study of such effects does not invoke mystical forces but applies the same rigorous formalism used to describe atoms, molecules, and materials.

The goal of this work is therefore educational. It seeks to introduce the mathematical and conceptual foundations of quantum mechanics to readers without a strong background in physics and some knowledge on basic linear algebra, to illustrate how these principles can be applied to biological systems. By presenting the formalism of quantum theory in a clear and accessible way, and by discussing a concrete example where quantum effects may be relevant in biology, this material aims to provide newcomers with the tools to engage critically and scientifically with the field of quantum biology. Understanding these principles not only helps to appreciate the genuine intersections between physics and life sciences but also to separate scientific inquiry from pseudoscientific speculation, a distinction that is essential for the responsible development of this interdisciplinary area.

To introduce the concepts of quantum mechanics possibly present in biology, one biological mechanism will be explored in this work, the magnetic compass used by some species of night migratory songbirds (Hore and Mouritsen, 2016). This compass mechanism is inherently quantum in nature and contains a number of interactions which will be useful to introduce basic concepts of quantum mechanics (Schulten et al., 1978). In this work the avian compass mechanism will be explored in a step-by-step basis, where each quantum effect present will be explained in tandem with the theory necessary to describe it mathematically. The objective is to create a more intuitive understanding of how these effects appear in biology.

During seasonal migration small songbird species travel thousands of kilometers every year, from northern Europe to southern Europe or northern Africa. These animals are able to orient using a number of techniques, among them the position of the sun and stars, local landmarks and the Earth’s geomagnetic field. Behavioral experiments in the absence of other navigational queues suggests that songbirds use a radical pair based mechanism (RPM) to detect the geomagnetic field (Wiltschko, 2012; Wiltschko and Wiltschko, 1972; Mouritsen, , 2015; Leberecht et al., 2023; Emlen and Emlen, 1996). The RPM takes place in the animal’s retina and is activated by blue light. The mechanism functions as an inclination compass (Hore and Mouritsen, 2016; Alvarez et al., 2024). Instead of pointing towards north or south, an inclination compass detects the angle between the magnetic field and the compass, therefore giving information whether the animal is closer of farther from the equator. Let’s discuss the RPM step-by-step, following the reaction from the activation by blue light to the resulting reaction products. The first step to be explored is how the RPM is initialized.

2 Activation of the mechanism and energy quantization

The process for the avian compass starts with the activation of the mechanism. The RPM is activated by blue light reaching the animal’s retina and being absorbed by an electron in a molecule. In quantum mechanics, systems are described by states. Each state represents a possible arrangement of the physical system being studied.

2.1 Describing quantum states

To characterize a physical system is, roughly, to provide (i) a mathematical description of the system’s state and (ii) to predict how it changes as time passes. At the operational level, the state of a physical system boils down to a list of numbers. This list, often organized as a vector or matrix, must inform everything that can be known about the system. In particular, quantum mechanical states are vectors with complex components. We will see later that matrices are also useful in describing quantum systems in more general situations.

Since quantum mechanics is a probabilistic theory, these states inform us about the probabilities of the outcomes of any conceivable measurement one can effect on the system. As for (ii), the basic equation of motion describing the time evolution of quantum systems is the Schrödinger equation. This equation is supplemented by rules establishing how the states change when a measurement is carried out.

Let us start with the description of states. In quantum mechanics it is common to represent a vector through the notation | v 〉, instead of v ⃗. This is just a matter of convention, since | v 〉 is nothing but a string of numbers (coefficients):where, in the case of quantum states, the coefficients α j, j = 1 , … , N, generally, assume complex values. Also, valid states must be normalized, that is, ∑ j = 1 N | α j | 2 = 1. It is more economic to define the basis vectorsand rewrite | v 〉 as

We say that the previous relation is the representation of | v 〉 in the orthonormal basis { | u j 〉 }.

Given another arbitrary vector | w 〉 = ∑ j = 1 N β j | u j 〉, the scalar product (or inner product) between | v 〉 and | w 〉 is a complex number, denoted by ⟨ v | w ⟩, and readswhere α j * represents the complex conjugation of α j. Note, therefore that, ⟨ v | w ⟩ = ( ⟨ w | v ⟩ ) *. A vector space equipped with an inner product is referred to as a Hilbert space and denoted by H.

⟨ v | w ⟩ = ∑ j = 1 N α j * β j ,

Physical states are modified by operators. An operator is a mathematical object that acts on the vectors of a Hilbert space taking them to other vectors within the same Hilbert space. Mathematically, an operator O is such that O | v 〉 = | v ′ 〉, with | v 〉 , | v ′ 〉 ∈ H.

To make quantum theory operational in practice, it is often useful to represent operators as matrices. This is possible whenever we choose any (convenient for the case at hand) orthonormal basis of the Hilbert space. Denoting again this set of orthonormal kets by { | u j 〉 } one can represent any other kets and arbitrary operators employing the identity operator, which reads 1 = ∑ j | u j 〉 〈 u j |. So, for instance, one may write | v 〉 = 1 | v 〉 = ∑ j | u j 〉 ⟨ u j | v ⟩, from which we see that the coefficients α j in (1) are α j = ⟨ u j | v ⟩.

For an arbitrary operator O we may analogously write O = 1 O 1 = ∑ j , k | u j 〉 ⟨ u j | O | u k ⟩ ⟨ u k | = ∑ j , k | u j ⟩ O j k 〈 u k |. We say that the matrix with entries O j k represents O in the basis { | u j 〉 }. The operator adjunct to O, and denoted by O † is obtained by transposing and taking the complex conjugate of the matrix entries of O, that is, ⟨ u j | O † | u k ⟩ = ⟨ u k | O | u j ⟩ *. There are particularly important operators in quantum theory such that O = O † ( O j k = O k j * ). Operators satisfying this condition are termed Hermitian and will be associated with measurable quantities in quantum mechanics. To exercise this new knowledge let us describe a simple system of one electron being excited by light using the state description. Firstly a basis needs to be defined, let us define two basis states which represent the electron energy, one state with lowest energy, | G 〉, usually called ground state, and one state with higher energy, | E 〉 usually called excited state. If represented as vectors the states would be

The process of light exciting the electron can be represented by an operator O e x = | E 〉 〈 G |, which when applied to the state | G 〉 transforms it in | E 〉 viasince G | G = 1. This way one can describe a very simple system and interaction. Using the state formalism we will explore more of the radical pair mechanism in the next sections.

O e x | G 〉 = | E 〉 G | G = | E 〉 ,

The change in state of a particle can be illustrated by an energy diagram (1), which describes how interactions change the system step by step, illustrating the relative energies of different states.

One important feature of quantum mechanics is that the results of a measurement may be quantized, meaning that they are described by discrete values instead of continuous ones. For example, if one measures the energy of an object in classical mechanics, which describes our everyday lives, then one can find any value for that energy. But when measuring the energy of bound systems in quantum mechanics the result can only be a certain set of values. This is the reason why the RPM is only activated by blue light. It has the right amount of energy to excite the electron. If it had any more, or any less energy it would not excite the electron (within some narrow “window of tolerance”).

Another important consequence of the quantization of energy is the photon. These are like little packages, or quanta, of luminous energy. A photon having a certain frequency, or being of a certain color, represents how much energy that package has, and it can only be absorbed by electrons that can reach a specific state with the energy the photon has. The process by which a photon of light gives energy to an electron and excites it into a higher energy state is called photoexcitation.

3 The radical pair and indistinguishable particles

The next step in the RPM is the formation of the titular radical pair. How this process happens is a bit complex, but can be easily understood by the use of energy diagrams as seen in Figure 1. Before getting into the full description of how a radical pair forms, one needs to understand a few more features of quantum mechanics, so this is a good point to introduce its rules.

3.1 The rules of quantum mechanics

Below we list the rather abstract rules (or postulates) of quantum mechanics (Cohen-Tannoudji et al., 2005; Ballentine, 1998). They are the primitive notions of the theory and, as such, cannot be mathematically demonstrated (much like the notions of “point” and “line” in Euclidean geometry). These postulates, however, have been confirmed with astonishing success for more than a century. Here they are:

P1: The state of a physical system is completely specified by a vector (or ket) | ψ 〉 belonging to a Hilbert space H.

P2: Every measurable quantity A is represented by a Hermitian operator A acting on H. We say that A is an observable. Hermiticity ensures that all eigenvalues (possible measurement outcomes) are real-valued.

P3: The only possible results of a measurement of A over a system whose normalized state is | ψ 〉 are the eigenvalues of the corresponding observable A.

P4: Whenever the physical quantity A is measured in a system whose state is the normalized ket | ψ 〉, the probability p ( a n ) of obtaining the eigenvalue a n is:where g n is the degeneracy of a n and | u n k 〉 is a set of orthonormal kets that spam the degenerate subspace. Note that we can write:where P n is the projector:

p a n = ∑ k = 1 g n | ⟨ ψ | u n k ⟩ | 2 ,

p a n = ⟨ ψ | P n | ψ ⟩

P n = ∑ k = 1 g n | u n k 〉 〈 u n k | .

P5: If the result of the referred measurement turns out to be a n, then, the state of the system, immediately after the measurement becomes

P6: Given that, at some instant t 0, the state of the system is given by | ψ ( t 0 ) 〉, the future state, | ψ ( t ) 〉, is determined by Schrödinger’s Equation:where H ( t ) is the observable corresponding to the system’s Hamiltonian (in many situations of interest it coincides with the system energy) and ℏ ≈ 1,05 × 1 0 − 34 J⋅s is the rationalized Planck constant. The smallness of this fundamental constant, when expressed in units tailored for macroscopic phenomena (Joule and second), relates to the fact that quantum phenomena usually appear at the microscopic domain.

i ℏ d d t | ψ t 〉 = H t | ψ t 〉 ,

P7: Whenever a system is comprised of two or more indistinguishable particles, the corresponding state | ψ 〉 must be either completely symmetric or completely antisymmetric under permutation of particles.

These are the postulates. The best, perhaps the only way to understand them is to use them.

An important point about P1 is that it implicitly prescribes the way one should deal with composite systems (systems that can be seen as a set of subsystems). This is done via the tensor product, which is the standard way to combine Hilbert spaces. Since the RPM involves two electrons, this will be important for our understanding of the mechanism.

3.2 Composite systems and tensor product

The question arises, how to describe two such systems? Consider, for definiteness, that the state | R 〉 corresponds to the right-hand configuration and | L 〉 to the left-hand configuration of a certain molecule, e.g., a glyceraldehyde molecule.

Suppose that we are interested in two such molecules, which not necessarily are in the same laboratory. In this context, assume that the first molecule is in Alice’s laboratory (A) and the second molecule is in Bob’s laboratory (B).

If the molecules have been synthesized in these distinct laboratories and never interacted, they have no relation with each other whatsoever. It seems a superfluous task to describe the two molecules in a unified way (it is for the case at hand), since we can simply writewhere | ψ A 〉 ( | ψ B 〉 ) refers to Alice’s (Bob’s) molecule and | R 〉 and | L 〉 denote right-handed and left-handed, respectively. Thus, by measuring the molecule chirality, Alice has a probability | a | 2 ( | b | 2 ) of finding R ( L ). Analogously, Bob has a probability | c | 2 ( | d | 2 ) of finding R ( L ). Since the two molecules are completely independent, the joint probabilities must satisfy:

Table 1.

where p ( R , R ) denotes the joint probability of the event in which Alice measures R and Bob measures R. We used the fact that joint probabilities related to uncorrelated events must be given by the product of the individual probabilities, e.g., p ( R , R ) = p A ( R ) ⋅ p B ( R ) = | a | 2 ⋅ | c | 2.

Suppose Alice and Bob insist in having a unified description of the two molecules, whose arbitrary individual states are expressed as:

Now, note that the probability amplitudes referring to the previous table naturally arise if we multiply the amplitudes of both states:where the operation denoted by ⊗ is the tensor product. The resulting vector inhabits a new Hilbert space H, such that dim H = dim H A ⋅ dim H B = 4, for which we write H = H A ⊗ H B. Consistently, an orthonormal basis of this new space must comprise four kets. One can write:

Back to the definition of tensor product it is reassuring to note that

1 0 0 0 = 1 0 ⊗ 1 0 = | R 〉 ⊗ | R 〉 , 0 1 0 0 = 1 0 ⊗ 0 1 = | R 〉 ⊗ | L 〉 , ⋯

Therefore:where we employed the simplified notation | R R 〉 ≡ | R 〉 ⊗ | R 〉, and so on.

This is a quite consistent representation, since, according to P4 the squared modulus of a c, | a c | 2 = | a | 2 | c | 2, should be the probability of measuring R in both laboratories, with analogous expressions being valid for the other amplitudes. This is in perfect accordance with the table we built when considering the molecules separately.

In a more general scenario one can proceed analogously. Let H 1 and H 2 be Hilbert spaces with dimensions d 1 and d 2, respectively, and | u j 〉 and | v k 〉 orthonormal bases of H 1 and H 2, j = 0 , … , d 1 − 1 and k = 0 , … , d 2 − 1. Consider arbitrary states

In this case, the tensor product | ψ ( 1 ) 〉 ⊗ | ϕ ( 2 ) 〉 is given by

A seemingly innocent, but crucial step, is to acknowledge H = H 1 ⊗ H 2 as the space that encompass all states of the form

The d 1 × d 2 orthonormal vectors | u j 〉 ⊗ | v k 〉 constitute a basis of H. Before addressing the distinctions between (2) and (3), let us define the action of operators on H.

Given that A 1 ( B 2 ) is an operator acting on H 1 ( H 2 ), one can immediately define the corresponding operator C acting on H:

C | Ψ 〉 = A 1 ⊗ B 2 ∑ j , k γ j k | u j 〉 ⊗ | v k 〉 ≡ ∑ j , k γ j k A 1 | u j 〉 ⊗ B 2 | v k 〉 .

In general, an operator C acting on H can be written as C = ∑ ℓ ( A 1 ( ℓ ) ⊗ B 2 ( ℓ ) ). The extension of an operator A 1 acting on H 1 to the space H is simply A 1 ⊗ 1 2. Note, in particular, that 1 = 1 1 ⊗ 1 2, where 1 stands for the identity operator, which can be expressed as 1 = ∑ n | n 〉 〈 n |, where { | n 〉 } is an arbitrary orthonormal basis of H.

3.3 Spin and indistinguishable particles

Spin is a fundamental property of objects, similar to charge or mass. It functions as an intrinsic angular momentum. Spin describes how an object interacts with magnetic fields, and results in all objects with spin and charge generating a small magnetic field. Spin is quantized. Spin assumes values multiples of ℏ / 2, but for simplicity let’s ignore the ℏ for now and assume the spin as multiples of 1 / 2, which separates all particles in two groups. If a particle’s spin number is an integer, such as 0, 1, 2 … , then the particle is a boson. If a particle’s spin number is not an integer, such as 1/2, 3/2, 5/2 … , then the particle is a fermion. Particles with spin can assume spin states based on their spin number in steps of size 1. Take a particle with spin number 1, the particle can assume spin states −1, 0 and 1. A particle with spin 3/2 can assume spin states −3/2, −1/2, 1/2 and 3/2.

The main particle we will explore in this work is the electron, which has spin number 1/2, therefore it can assume the states −1/2 and 1/2 commonly called spin-down and spin-up states, respectively. The states can also be referred as | ↑ 〉 and | ↓ 〉. As a particle with spin number not integer, the electron is a fermion. Due to Pauli’s exclusion principle, two or more fermions cannot be in the same state. So if two electrons are in the same energy level, or have the same energy, they need to assume opposite spin states. This can be illustrated in an energy diagram as seen in Figure 1.

As defined by P7, some particles in quantum mechanics cannot be distinguished from each other. This is the case of electrons as seen in Figure 1. If a system has two electrons, then there is no way to mark one electron as electron A and the other as electron B. Any way we describe a system has to be either symmetric or anti-symmetric for any change of the labels on the particles. To understand this point let us look at the case of two electrons in the same energy level, also called paired electrons. We can describe the state of each electron as a vector | ↑ 〉 i or | ↓ 〉 i where i = A , B. As discussed above, the state of two electrons in the same energy level can be defined as | ↑ ↓ 〉, since the electrons have opposite spin states due to Pauli’s exclusion principle. The first part of the ket is the state of electron A and the second is the state of electron B. Since electrons are indistinguishable, the state has to be the same if we flip the labels, or the order in this case, but | ↑ ↓ 〉 ≠ | ↓ ↑ 〉. To fix the issue we can combine both states into a new state vector | ↑ ↓ 〉 − | ↓ ↑ 〉. The new state is anti-symmetric for any change of labels, therefore it is allowed. This new state is formally called the singlet state, and is represented by | S 〉 = 1 2 ( | ↑ ↓ 〉 − | ↓ ↑ 〉 ), where 1 2 is a normalization factor to guarantee the total probability is 1. This state represents two electrons with opposite spins, and is equivalent to having total spin 0.

With a better understanding of quantum mechanics we can discuss radical pair formation. Radicals are molecules with one unpaired electron, and radical pairs are pairs of radicals with correlated electrons. The unpaired electrons in each molecule are correlated to each other, meaning that their spin states are correlated and, to some extend, depend of each other. To form a radical pair one can start with two molecules, an acceptor (A) and a donor (D) as seen in Figure 1. Both with no unpaired electrons. A photon is then absorbed by one of the electrons in the acceptor molecule and excites it into a higher energy level. The excitation frees a lower energy site in the acceptor, which allows an electron from the donor to decay into the lower energy state and be transferred to the acceptor. The transferred electron is now paired with the electron in the acceptor molecule. Since electrons are indistinguishable, the correlation that existed between the electrons due to them being paired is transferred to the two unpaired electrons, causing them to be correlated and forming the radical pair.

After the radical pair is formed the electrons will maintain correlation and evolve in time, interacting with the environment due to a number of external influences which we will discuss later, among them the Earth’s geomagnetic field which allows for magnetoreception.

4 Electron transfer and tunneling

One part of the radical pair formation is the electron transfer between the donor and acceptor molecules. Usually an electron needs energy to become detached from a molecule, which can be acquired by either absorbing photons or by thermal energy from a heated environment. None of which are the case for the electron transfer in the radical pair formation. The electron transfer can happen thanks to the tunnel effect.

4.1 A qualitative description of the tunnel effect

Every student of elementary classical mechanics has met the concept of turning points. For instance, if a cart is going uphill, from the left, but does not have sufficient kinetic energy to reach the top, at some lower height it will momentarily stop (at the turning point) and reverse its motion. The upper part of the hill and its right side are forbidden regions. In the quantum realm, an analog problem would be an atom or molecule (the cart) going through a region with increasing potential (the hill). In this regime, surprisingly, there is a finite probability to find the atom or molecule in classically forbidden regions. This is due to the quantum phenomenon of tunneling, which, in turn, is related to the wave-like nature of the wave function.

An archetypal description of the tunnel effect is as follows. Consider, initially, a classical particle (with mass m) moving freely in a straight line (our x-axis) from left to right, with linear momentum p > 0, so that its kinetic energy is K = p 2 2 m. If the particle reaches a region where the potential energy goes abruptly from zero to V 0 > K, the particle is reflected. It is classically impossible for this particle to overcome the potential barrier.

Let us consider a quantum analog of this situation, see Figure 2. Suppose we have a step potential: V ( x ) = V 0 for − a < x < a, and V ( x ) = 0 elsewhere. Consider that at t = t 0 the particle’s wave function is non-vanishing only in the region x < − a and that the expectation value of its linear momentum is ⟨ ψ | P | ψ ⟩ = ⟨ P ⟩ = p (numerically the same as that of the classical case), so that its average kinetic energy is also K < V 0. As a consequence of the Schrödinger equation, when the wave function impinges on the barrier, part of it penetrates the barrier, which is classically forbidden region, while another part of ψ ( x ) is reflected. For longer times one finds that the probability that the particle can be found in the right side of the barrier is non-zero, a conclusion that has no parallel in classical physics.

In the radical pair mechanism, tunneling plays a role during the electron transfer between the donor and the acceptor molecules. The transfer not only initiates the radical pair, but subsequent electron transfers can move the electron to other donor molecules which might be better adapted to magnetoreception (Hore and Mouritsen, 2016; Leberecht et al., 2023; Alvarez et al., 2024).

5 Spin dynamics and coherence

With the radical pair formed it can evolve in time. Precisely modeling the evolution of radical pairs is an ongoing question which depends on the physical system being studied. In this work we are exploring the evolution of a singlet-born radical pair theorized to play a role in animal magnetoreception (Hore and Mouritsen, 2016). As in previous sections, some theory needs to be established before discussing how it applies to the RPM.

5.1 Quantum coherence

The first postulate states that the state of an arbitrary physical system is given by a vector in a Hilbert space, that is, a vector space equipped with an inner product (scalar product). In the case of quantum mechanics, the vector components, in general, assume complex values. The simplest example is that of a space with dim H = 2, describing what we know as a two-level system. This is the case of the electron spin.

Let { | 0 〉 , | 1 〉 } be an orthonormal basis spanning H. An arbitrary state is expressed as:where a , b ∈ C are the probability amplitudes satisfying the normalization condition | a | 2 + | b | 2 = 1. This state may, for instance, represent the magnetic momentum of an electron, or two distinct microscopic configurations of a molecule. In quantum information science state (Equation 3) is the analog of a classical bit, being termed a quantum bit, or qubit.

Let us consider an observable (in the sense of P2), denoted by σ z and whose eigenstates are | 0 〉 and | 1 〉, such that σ z | 0 〉 = + | 0 〉 and σ z | 1 〉 = − | 1 〉. According to P3 the only possible outcomes of a measurement of σ z are the eigenvalues ± 1. In addition, according to P4, if we measure this observable in a system whose state is given by | ψ 〉, then we have probability | a | 2 of getting the result + 1 and the probability | b | 2 = 1 − | a | 2 of obtaining − 1. It is clear from this example that the normalization requirement corresponds to the fact that the probabilities associated with all possible outcomes must sum to unity.

Suppose, for definiteness, that we are interested in the configuration (chirality) of a certain molecule, a glyceraldehyde molecule, for example. It can assume the values R (for right-handed molecule) and L (for left-handed molecule). Therefore, with respect to this specific physical quantity, the general state of the molecule can be written as in (4) with | 0 〉 ≡ | R 〉 and | 1 〉 ≡ | L 〉.

This is a good point to refer to the concept of quantum coherence. We know that a classical bit can take either the value “0” or “1”. In contrast, the state (4) allows for a certain kind of coexistence of “0” and “1”. Analogously, in our macroscopic experience, we either find left-handed desks or right-handed desks (never a superposition of both), while in the quantum mechanical state | ψ 〉 = a | R 〉 + b | L 〉 we have what is termed a superposition of R and L, except when a = 1 and b = 0, or vice-versa. The possibility of coexistence of states that we would consider as macroscopically exclusionary is, arguably, the most fundamental feature of quantum systems.

Depending on the physical quantity one chooses to measure, superposition states give rise to interference phenomena. So they are at the root of the intriguing wave-like behavior of electrons, atoms, and molecules under certain conditions. Quantum coherence manifests itself routinely in a variety of quantum systems, and is employed in many present-day technologies.

From a dynamical perspective, the time evolution described by the Schrödinger equation leads to the following prescription. Consider the arbitrary state of a qubit, Equation 3 as describing the system at t = 0 and suppose, for simplicity, that { | 0 〉 , | 1 〉 } is the basis of eigenstates of the Hamiltonian H, with eigenvalues E 0 and E 1, respectively. Then, P 6 leads toas representing the system’s state for any time t > 0. In general, if, at t = 0, the state of the system is given by | ψ 〉 = ∑ i α i | i 〉, where { | i 〉 } is the set of eigenvectors of H with corresponding eigenvalues { E i }, then, the state of the system at some future time t is given by:

We address other aspects of quantum coherence within the discussion of density operators.

5.2 Entanglement

Let us resume our example of two molecules. The pivotal point, which leads to the concept of entanglement, is the following. Our unified approach, in terms of tensor products, would be a mere change of notation, if it were not for the superposition principle implied by P1. Consider, initially, that | ψ A 〉 = | R 〉 e | ψ B 〉 = | R 〉, so that the state of the composite system is | R 〉 ⊗ | R 〉. In a second situation A and B prepare the state | L 〉 ⊗ | L 〉.

The superposition principle, that arises from the linearity of the Hilbert space, states that if any two kets represent physical states, then, any normalized linear combination of these states is necessarily a plausible physical state. So, in particular,is a possible state of the system comprised by the two molecules, and a correlated one. Note that whenever one of the molecules is found to be in the state R ( L ), one is sure that the other molecule is also in the state R ( L ).

The relevant fact is that it is not possible to factorize this state as a single tensor product of the kind | ψ A 〉 ⊗ | ψ B 〉. Consider, e.g., α = β = 1 / 2 and suppose one can write

1 2 | R 〉 ⊗ | R 〉 + 1 2 | L 〉 ⊗ | L 〉 = ? | ψ A 〉 ⊗ | ψ B 〉 = a c | R 〉 ⊗ | R 〉 + a d | R 〉 ⊗ | L 〉 + b c | L 〉 ⊗ | R 〉 + b d | L 〉 ⊗ | L 〉

⇒ a c = 1 2 , a d = 0 b c = 0 , b d = 1 2

⇒ a = b = 0 or a = c = 0 or d = b = 0 or d = c = 0

In any case it is impossible to satisfy a c = b d = 1 2. Thus, our factorization hypothesis is incorrect. With this example we see that the state (Equation 1) is a very particular case of (Equation 2), for which the equations γ j k = α j β k admit a non-trivial solution.

A state of a composite system that does not admit the factorization | ψ ( 1 ) 〉 ⊗ | ϕ ( 2 ) 〉 is termed entangled (Horodecki et al., 2009). Whenever the factorization is possible we refer to the state as separable, or unentangled. So far we have addressed bipartite systems, that is, the ones comprised by two subsystems. The concept of entanglement can be immediately extended to systems with n parties, associated with the kets of the type | Ψ 〉 ∈ H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n.

Let | ψ i 〉 be arbitrary kets in H i. We say that | Ψ 〉 is entangled if a decomposition of the kind:is not possible. In general, it is not straightforward to know whether or not an arbitrary ket | Ψ 〉 is entangled. Consider the examples:

One can write the first state aswhile it is not possible to factorize | Φ 〉 in terms of a single tensor product. Therefore, | Φ 〉 is an entangled state.

Whenever entanglement is present one cannot trivially refer to the state of system 1 or system 2, separately, but only to the global state of system 1-2. One must be careful in interpreting this statement. It is not the subsystems themselves that loose their individuality, but rather, the state that describes them. Thus, even when the state is entangled it is perfectly consistent to refer to molecule A or molecule B as individual entities. What one cannot do is to refer to the state of molecule A without mentioning the other molecule. Consider, for instance, that the two molecules are described by the state ( | R R 〉 + | L L 〉 ) / 2. In this case, we have to say something like: When Alice measures the chirality of her molecule and finds it to be R ( L ), then the state of Bob’s molecule must be | R 〉 ( | L 〉 ). Note that, Bob knows this without the need of measuring his molecule, provided that Alice informs him about her result. This is due to the correlation that exists between the two molecules. It is important to realize this kind of phenomena is not exclusive of quantum mechanics (there are classical correlations). However, some quantum correlations, that may arise from entangled states, cannot be reproduced by any classical system. In this sense, quantum correlations may be stronger than their classical counterparts.

5.3 The density matrix

We have just seen that, in general, even knowing the full state of a system, one can only predict the probability of measurement outcomes. In this particular sense, quantum mechanics is a probabilistic theory. These probabilities are intrinsic of physical law, and, thus, cannot be eliminated by refining the experimental procedure, the precision of the employed equipments, etc. It is well known that probabilities can also appear, including in classical systems, when we are not able to describe the systems’s state with arbitrary certainty. In practice, it is not possible to eliminate this second source of indefiniteness. The question arises, how to describe quantum systems, also taking into account the incomplete knowledge about the state.

Let us consider a simple example. Assume that an apparatus is supposed to prepare states | ψ 〉 every time one pushes a certain button. However, due to imperfections, it produces the expected result in 90 % of the runs and another state, | ϕ 〉, in the remaining runs ( 10 % ). In order to understand how to mathematically describe such a system we resort to elementary probability theory. Suppose we intend to calculate the average value of an arbitrary observable A. The average of A when the state is | ψ 〉 is simply ⟨ ψ | A | ψ ⟩ and, analogously, ⟨ ϕ | A | ϕ ⟩ when the state is | ϕ 〉. It is, then, just natural to write the average in the imperfect situation as ⟨ A ⟩ = 0.9 ⟨ ψ | A | ψ ⟩ + 0.1 ⟨ ϕ | A | ϕ ⟩ = 0.9 ⟨ ψ | A 1 | ψ ⟩ + 0.1 ⟨ ϕ | A 1 | ϕ ⟩, that is.

〈 A 〉 = ∑ j 0.9 〈 ψ | A | u j 〉 〈 u j | ψ 〉 + 0.1 〈 ϕ | A | u j 〉 〈 u j | ϕ 〉

= ∑ j 0.9 〈 u j | ψ 〉 〈 ψ | A | u j 〉 + 0.1 〈 u j | ϕ 〉 〈 ϕ | A | u j 〉

= ∑ j 〈 u j | 0.9 | ψ 〉 〈 ψ | + 0.1 | ϕ 〉 〈 ϕ | A | u j 〉 .

Note that the quantity between square brackets, ρ = 0.9 | ψ 〉 ⟨ ψ | + 0.1 | ϕ ⟩ 〉 ϕ |, depends only on the system. Recalling that the trace of an arbitrary operator B is given by the sum of its diagonal elements, T r ( B ) ≡ ∑ j ⟨ u j | B | u j ⟩, we see that the above expression can be written as

⟨ A ⟩ = T r ρ A .

The quantity ρ depends on both the quantum probabilities (intrinsic to the states | ψ 〉 and | ϕ 〉) and the “classical” probabilities (0.9 and 0.1). In fact, ρ fully describes the state of the system in the situation we described, and is called the density operator. In general, for a system where we have several possible states { | ψ i 〉 } with corresponding probabilities { p i }, the state of the system readswith 0 ≤ p i ≤ 1 and ∑ i p i = 1. A detailed description of density operators is given in (Ballentine, 1998).

ρ = ∑ i p i | ψ i 〉 ⟨ ψ i | ,

The defining properties of a density operator are:

(i) T r ( ρ ) = 1, which is equivalent to ∑ i = 1 N ρ i i = 1.

(ii) ρ = ρ † [equivalently ρ i j * = ρ j i], i. e., the density operator is Hermitian.

(iii) A density operator must be semi-positive definite: ⟨ φ | ρ | φ ⟩ ≥ 0 for all | φ 〉 ∈ H. This ensures that all eigenvalues of ρ are non negative.

The diagonal elements ρ i i are referred to as populations, whereas the off-diagonal entries ρ i j are called coherences. The populations are real numbers that sum to unity (i) and, as such, can be interpreted as probabilities. The coherences are associated with quantum superpositions: if all off-diagonal elements vanish we say that the state is incoherent in the chosen basis. This would lead to a classical behavior that could be reproduced by sampling a box with N balls, N p 1 being red, N p 2 being blue, and so on. So, coherences are essential to quantum phenomena.

Note that any pure state described by the ket | ψ 〉 can also be expressed as a density matrix ρ = | ψ 〉 ⟨ ψ |, corresponding to p 1 = 1 and p i = 0 for i > 1. Of course, the converse is not true: given an arbitrary density matrix ρ it is generally not possible to express it in terms of a single ket.

When we apply postulate P 6 to the dynamics of states described by density matrices, we obtain the following equation of motion:where [ H ( t ) , ρ ( t ) ] = H ( t ) ρ ( t ) − ρ ( t ) H ( t ) is the commutator between H and ρ.

i ℏ d d t ρ t = H t , ρ t , (4)

One should comment on more general forms of time evolution in quantum systems. The Schrödinger equation and its natural extension (9) refer to completely isolated systems (closed systems). In several relevant situations these dynamical descriptions are useful and provide a sufficiently precise information on the dynamics. However, the fact is that no system can be perfectly isolated from its surroundings. This is particularly critical in quantum-biology systems. Although an account of the techniques necessary to describe the dynamics of open quantum systems (for an elementary introduction see chapter 9 of (Schumac et al., 2010)) is outside the scope of the present work, we will provide a qualitative description of an important and ubiquitous effect: decoherence (Schlosshauer, 2019).

Suppose again, that we are interested in a simple system with state | ψ 〉 = a | 0 〉 + b | 1 〉, the corresponding density operator being ρ = | ψ 〉 ⟨ ψ | = | a | 2 | 0 ⟩ ⟨ 0 | + a b * | 0 ⟩ ⟨ 1 | + a * b | 1 ⟩ ⟨ 0 | + | b | 2 | 1 ⟩ ⟨ 1 |, whose matrix representation in the same basis is

ρ = | a | 2 a b * a * b | b | 2 .

One of the most common effects of the surroundings (or environment) on the state of a quantum system is a fast decrease in its level of coherence. More specifically, a usual effect of the environment on the previous state is an exponential decay in the off-diagonal elements (or coherences):where Γ is a constant (depending on the coupling between system and environment) with dimension of frequency. Therefore, typically, open quantum systems tend to loose their coherence and, thus, its “quantumness”, at least with respect to the employed basis. An analogous description is valid for systems with more than two levels.

ρ = | a | 2 a b * a * b | b | 2 → ρ = | a | 2 a b * e − Γ t a * b e − Γ t | b | 2 ,

In spite of this ubiquitous effect, there is evidence that several effects in quantum biology are due to the partial “survival” of quantum coherence in some key processes (Arndt et al., 2009; Huelga and Plenio, 2013; Fuller et al., 2014; Purchase and de Groot, 2015).

5.4 The evolution of radical pairs

The two electrons present in the radical pair are entangled and can assume four possible states depending on how their spins align.

• Singlet State (S = 0): In this anti-symmetric configuration, the spins of the two electrons are antiparallel, leading to a total spin of zero. Mathematically, the singlet state is represented as:

• Triplet State (S = 1): In this symmetric configuration, the spins of the two electrons are parallel, resulting in a total spin of one. The triplet state has three possible sublevels, represented as:

The behavior of radical pairs in magnetic fields is significantly influenced by two key interactions: the Zeeman interaction and hyperfine interactions (Hore and Mouritsen, 2016; Griffiths and Schroeter, 2018). The Zeeman interaction describes the effect of an external magnetic field on the energy levels of the electron spins in a radical pair. When a magnetic field B is applied, it interacts with the magnetic moment of the electron spin, leading to a shift in the energy levels of the spin states. The Zeeman interaction is defined by the Hamiltonian (Griffiths and Schroeter, 2018; Ritz et al., 2000):where g is the g-factor (a dimensionless quantity that characterizes the magnetic moment and angular momentum of the electron), and μ B is the Bohr magneton. The Zeeman interaction causes the degenerate spin states in the absence of a magnetic field to split into distinct energy levels, depending on the orientation of the spins relative to the magnetic field (Griffiths and Schroeter, 2018).

H Z = g μ B B ⋅ S

The hyperfine interaction is the interaction between the magnetic moment of an electron and the magnetic moments of nearby atomic nuclei (Hore and Mouritsen, 2016). In a radical pair, the unpaired electron’s spin interacts with the nuclear spins of the atoms within the radical, leading to a local magnetic field that further influences the electron’s spin state. The hyperfine interaction is described by the Hamiltonian (Hore and Mouritsen, 2016; Ritz et al., 2000):where A i is the hyperfine coupling tensor for nucleus i, and I i is the nuclear spin of the i-th nucleus. This interaction leads to a complex splitting of energy levels, which depends on the number and type of surrounding nuclei, their magnetic moments, and their spatial distribution (Hore and Mouritsen, 2016).

H hf = ∑ i S ⋅ A i ⋅ I i

The combined effect of Zeeman and hyperfine interactions determines the spin dynamics and, consequently, the magnetic sensitivity of the radical pair. The Zeeman interaction aligns the electron spins relative to the external magnetic field, while the hyperfine interaction introduces additional energy level splitting due to the influence of nearby nuclear spins. These interactions can induce transitions between the singlet and triplet states of the radical pair (Hore and Mouritsen, 2016).

The direction of the external magnetic field plays a critical role in the Radical-Pair Mechanism due to the anisotropic nature of hyperfine interactions (Hore and Mouritsen, 2016). This means that the strength and effect of these interactions vary depending on the orientation of the external magnetic field relative to the molecular orbitals of the radicals. Anisotropy in hyperfine interactions allows the radical pair to act as a magnetic-direction sensor, rather than merely a magnetic-intensity sensor (Hore and Mouritsen, 2016; Schulten et al., 1978; Mohseni, 2014).

These interactions collectively influence the coherent oscillations between the singlet and triplet states of a radical pair, a process known as singlet-triplet interconversion (Bagryansky et al., 2007). The rate and extent of these oscillations play a significant role in determining the outcome of radical-pair reactions, making it a central focus in the study of magnetic field effects in biological systems (Hore and Mouritsen, 2016; Schulten et al., 1978; Mohseni, 2014).

The dynamics of interconversion can be described mathematically using quantum mechanical principles. The Hamiltonian that governs the spin dynamics of a radical pair includes contributions from both the Zeeman interaction with an external magnetic field and the hyperfine interactions with nearby nuclear spins (Bagryansky et al., 2007; Jones and Hore, 2010; Fay et al., 2020):

H = H Z + H hf = g μ B B ⋅ S + ∑ i S ⋅ A i ⋅ I i

The time evolution of the spin state can be represented by a density matrix ρ ( t ), which evolves according to Equation 4 (Fay et al., 2020; Sakurai and Napolitano, 2020). Given the density matrix ρ S corresponding to the singlet state, the probability of finding the radical pair in this state at any given time is (Jones and Hore, 2010; Fay et al., 2020; Sakurai and Napolitano, 2020):

p S t = Tr ρ t ⋅ ρ S

The fine balance between the singlet and triplet states, modulated by hyperfine and Zeeman interactions, enables a radical pair to sense magnetic fields. Even small changes in the strength and direction of these fields can lead to significant shifts in the populations of singlet and triplet states (Sakurai and Napolitano, 2020). In a biochemical context, the interconversion rate between singlet and triplet states can determine the fate of some reactions.

6 Recombination and conservation of spin

The final step in the RPM is the recombination, where the system relaxes into reaction products depending on the spin states of the electrons. During recombination the total spin of the system needs to be conserved. This conservation is akin to conservation of angular momentum in classical mechanics. If the electron spins are in the singlet state, then the product needs to conserve the total spin 0. The same is true for the triplet states, where the product needs to conserve total spin 1. The recombination can be modeled in a number of ways either by quantum or semi-classical methods (Alvarez, 2023). The most common method is the Haberkorn operators (Ritz et al., 2000; Ritz et al., 2001), where non-unitary operators are added to Equation 4 representing the fraction of the system that produces either singlet or triplet products. The Haberkorn equation has the formwhere { k S 2 P S , ρ ( t ) } = k S 2 P S ρ ( t ) + ρ ( t ) k S 2 P S is the anti-commutator. k S and k T are the recombination rates to the singlet and triplet products respectively. P S = | S 〉 〈 S | is the singlet projector and P T = 1 − P S is the triplet projector. The three triplet states are equivalent and recombine into the same product. The total yield of singlet or triplet products, Φ i with i = { S , T }, can be calculated by integrating the probability of the system creating a given product in time.

d d t ρ t = 1 i ℏ H t , ρ t − k S 2 P S , ρ t − k T 2 P T , ρ t ,

Φ i = ∫ 0 ∞ Tr P i ⋅ ρ t d t .

The ratio between singlet and triplet yields is what carries the directional information from the geomagnetic field. As the animal is traveling, if it turns it’s head or moves to an area where the magnetic field is slightly different, the change in the alignment between the Earth’s geomagnetic field and the RPM affects the dynamics of the mechanism via the Zeeman interaction. The change in the dynamics results in a change in the product yields from the RPM’s chemical reaction, which informs the animal of the change in magnetic field conditions.

7 Conclusion

In this work, we attempted to introduce the readers - assumed to have some previous knowledge on linear algebra - to the basic theory of quantum mechanics and its formalism via a case study of the radical pair mechanism. The main objective was to give non-physicist readers the basic tools to be able to understand the possible role of quantum mechanics in biology. Quantum mechanics is infamous for being non-intuitive, and its role in biological systems started being explored recently, so the reader should use the information presented here as a first step into understanding quantum mechanics. Deeper study through traditional textbooks (Griffiths and Schroeter, 2018) is recommended for readers interested in learning more.

The radical pair mechanism is the biological system most well studied under the lens of quantum biology. Other effects in biology are known and have been studied for some time now, but the research of their role as quantum effects in biology is relatively new. More study is needed in both the theoretical and experimental aspects of quantum biology, specially with the development of experimental tools sensitive enough to detect quantum effects in living cells.

In conclusion, this work serves as a first step into the theory of quantum mechanics, and its role in biology, aimed at other researchers with little experience on quantum mechanics, or interested in learning the role of quantum mechanics in biology.

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Footnotes

Edited by: Shuo Chen, Northeastern University, China

Reviewed by: Prajay Patel, University of Dallas, United States