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Notes on Quantum Mechanics


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PDF version: Notes on Quantum Mechanics – By Logan Thrasher Collins

The Schrödinger equation and wave functions

Overview of the Schrödinger equation and wave functions

Quantum mechanical systems are described in terms of wave functions Ψ(x,y,z,t). Unlike classical functions of motion, wave functions determine the probability that a given particle may occur in some region. The way that this is achieved involves integration and will be discussed later in these notes.

To find a wave function, one must solve the Schrödinger equation for the system in question. There are time-dependent and time-independent versions of the Schrödinger equation. The time-dependent version is given in 1D and 3D by the first pair of equations below and the time-independent version is given in 1D and 3D by the second pair of equations below. Here, ћ is h/2π (and h is Planck’s constant), V is the particle’s potential energy, E is the particle’s total energy, Ψ is a time dependent wave function, ψ is a time-independent wave function, and m is the mass of the particle. After this point, these notes will focus on 1D cases unless otherwise specified (it will often be relatively straightforward to extrapolate to the 3D case).

For a wave function to make physical sense, it needs to satisfy the constraint that its integral from –∞ to ∞ must equal 1. This reflects the probabilistic nature of quantum mechanics; the probability that a particle may be found anywhere in space must be 1. For this reason, one must usually find a (possibly complex) normalization constant A after finding the wave function solution to the Schrödinger equation. This is accomplished by solving the following integral for A. Here, Ψ* is the complex conjugate of the wave function without the normalization constant and Ψ is the wave function without the normalization constant.

To obtain solutions to the time-dependent Schrödinger equation, one must first solve the time-independent Schrödinger equation to get ψ(x). The general solution for the time-dependent Schrödinger equation is any linear combination of the product of ψ(x) with an exponential term (see below). The coefficients cn can be real or complex.

Physically, |cn|2 represents the probability that a measurement of the system’s energy would return a value of En. As such, an infinite sum of all the |cn|2 values is equal to 1. In addition, note that each Ψn(x,t) = ψn(x)e–iEnt/ is known as a stationary state. The reason these solutions are called stationary states is because the expectation values of measurable quantities are independent of time when the system is in a stationary state (as a result of the time-dependent term canceling out).

Using wave functions

Once a wave function is known, it can be used to learn about the given quantum mechanical system. Though wave functions specify the state of a quantum mechanical system, this state usually cannot undergo measurement without altering the system, so the wave function must be interpreted probabilistically. The way the probabilistic interpretation is achieved will be explained over the course of this section.

Before going further, it will be useful to understand some methods from probability. First, the expectation value is the average of all the possible outcomes of a measurement as weighted by their likelihood (it is not the most likely outcome as the name might suggest). Next, the standard deviation σ describes the spread of a distribution about an average value. Note that the square of the standard deviation is called the variance.

Equations for the expectation value and standard deviation are given as follows. The first equation computes the expectation value for a discrete variable j. Here, P(j) is the probability of measurement f(j) for a given j. The second equation is a convenient way to compute the standard deviation σ associated with the expectation value for j. The third equation computes the expectation value for a continuous function f(x). Here, ρ(x) is the probability density of x. When ρ(x) is integrated over an interval a to b, it gives the probability that measurement x will be found over that interval. The fourth equation the same as the second equation, but finds the standard deviation σ for the continuous variable x.

In quantum mechanics, operators are employed in place of measurable quantities such as position, momentum, and energy. These operators play a special role in the probabilistic interpretation of wave functions since they help one to compute an expectation value for the corresponding measurable quantity.

To compute the expectation value for a measurable quantity Q in quantum mechanics, the following equation is used. Here, Ψ is the time-dependent wave function, Ψ* is the complex conjugate of the time-dependent wave function, and Q̂ is the operator corresponding to Q.

Any quantum operator which corresponds to a classical dynamical variable can be expressed in terms of the momentum operator –iℏ(∂/∂x). By rewriting a given classical expression in terms of momentum p and then replacing every p within the expression by –iℏ(∂/∂x), the corresponding quantum operator is obtained. Below, a table of common quantum mechanical operators in 1D and 3D is given.

Heisenberg uncertainty principle

The Heisenberg uncertainty principle explains why quantum mechanics requires a probabilistic interpretation. According to the Heisenberg uncertainty principle, the more precisely the position of a particle is determined via some measurement, the less precisely its momentum can be known (and vice versa). The Heisenberg uncertainty principle is quantified by the following equation.

The reason for the Heisenberg uncertainty principle comes from the wave nature of matter (and not from the observer effect). For a sinusoidal wave, the wave itself is not really located at any particular site, it is instead spread out across the cycles of the sinusoid. For a pulse wave, the wave can be localized to the site of the pulse, but it does not really have a wavelength. There are also intermediate cases where the wavelength is somewhat poorly defined and the location is somewhat well-defined or vice-versa. Since the wavelength of a particle is related to the momentum by the de Broglie formula p = h/λ = 2πℏ/λ, this means that the interplay between the wavelength and the position applies to momentum and position as well. The Heisenberg uncertainty principle quantifies this interplay.

Some simple quantum mechanical systems

Infinite square well

The infinite square well is a system for which a particle’s V(x) = 0 when 0 ≤ x ≤ a and its V(x) = ∞ otherwise. Because the potential energy is infinite outside of the well, the probability of finding the particle there is zero. Inside the well, the time-independent Schrödinger equation is given as follows. This equation is the same as the classical simple harmonic oscillator.

For the infinite square well, certain boundary conditions apply. In order for the wave function to be continuous, the wave function must equal zero once it reaches the walls, so ψ(0) = ψ(a) = 0. The general solution to the infinite square well differential equation is given as the first equation below. The boundary condition ψ(0) = 0 is employed in the second equation below. Since the coefficient B = 0, there are only sine solutions to the equation. Furthermore, if ψ(a) = 0, then Asin(ka) = 0. This means that k = nπ/a (where n = 1, 2, 3…) as given by the third equation below. The fourth equation below shows that this set of values for k leads to a set of possible discrete energy levels for the system

To find the constant A, the wave function ψ = Asin(nπx/a) must undergo normalization. As mentioned earlier, normalization is achieved by setting the normalization integral equal to 1 and solving for the constant A. Note that the time-independent Schrödinger equation can be utilized in the normalization integral since the exponential component of the time-dependent Schrödinger equation would cancel anyways.

Using this information, the wave functions for the infinite square well particle system are obtained. The time-independent and time-dependent wave functions are both displayed below at left and right respectively.

This infinite set of wave functions has some important properties. They possess discrete energies that increase by a factor of n2 with each level (and n = 1 is the ground state). The wave functions are also orthonormal. This property is described by the following equation. Here, δmn is the Kronecker delta and is defined below.

Another important property of these wave functions is completeness. This means that any function can be expressed as a linear combination of the time-independent wave functions ψn. The reason for this remarkable property is that the general solution (see below) is equivalent to a Fourier series.

The first equation below can be employed to compute the nth coefficient cn. Here, f(x) = Ψ(x,0) which is an initial wave function. Note that the initial wave function can be any function Ψ(x,0) and the result will generate coefficients for that starting point. This first equation is derived using the orthonormality of the solution set. Note that the formula applies to most quantum mechanical systems since the properties of orthonormality and completeness hold for most quantum mechanical systems (though there are some exceptions). The second equation below computes the cn coefficients specifically for the infinite square well system.

Quantum harmonic oscillator

For the quantum harmonic oscillator, the potential energy in the Schrödinger equation is given by V(x) = 0.5kx2 = 0.5mω2x2. This means that the following time-independent Schrödinger equation needs to be solved.

There are two main methods for solving this differential equation. These include a ladder operator approach and a power series approach. Both of these methods are quite complicated and will not be covered here. The solutions for n = 0, 1, 2, 3, 4, 5 are given below. Here, Hn(y) is the nth Hermite polynomial. The first five Hermite polynomials and the corresponding energies for the system are given in the table. Note that the discrete energy levels for the quantum harmonic oscillator follow the form (n + 0.5)ћω.

As with any quantum mechanical system, the quantum harmonic oscillator is further described by the general time-dependent solution. To identify the coefficients cn for this general solution, Fourier’s trick is employed (see previous section) where f(x) is once again any initial wave function Ψ(x,0).

Quantum free particle

Though the classical free particle is a simple problem, there are some nuances which arise in the case of the quantum mechanical free particle which greatly complicate the system.

To start, the Schrödinger equation for the quantum free particle is given in the first equation below. Here, k = (2mE)0.5/ћ. Note that V(x) = 0 since there is no external potential acting on the particle. The second equation below is a general time-independent solution to the system in exponential form. The third equation below is the time-dependent solution to the system where the terms are multiplied by e–iEt/ћ. Realize that this general solution can be written as a single term by redefining k as ±(2mE)0.5/ћ. When k > 0, the solution is a wave propagating to the right. When k < 0, the solution is a wave propagating to the left.

The speed of these propagating waves can be found by dividing the coefficient of t (which is ћk2/2m) by the coefficient of x (which is k). Since this is speed, the direction of the wave does not matter, so one can take the absolute value of k. By contrast, the speed of a classical particle is found by solving E = 0.5mv2, which gives a puzzling result that is twice as fast as the quantum particle.

Another challenge associated with the quantum free particle is that its wave function is non-normalizable (as shown below). Because of this, one can conclude that free particles cannot exist in stationary states. Equivalently, free particles never exhibit definite energies.

To resolve these issues with the quantum free particle, it has been found that the wave function of a quantum free particle actually carries a range of energies and speeds known as a wave packet. The solution for this wave packet involves the integral given by the first equation below and a function ϕ(k) given by the second equation below. This second equation allows one to determine ϕ(k) to fit a desired initial wave function Ψ(x,0). It was obtained using a mathematical tool called Plancherel’s theorem.

The above solution to the quantum free particle is now normalizable. Furthermore, the issue with the speed of the quantum free particle having a value twice as large as the speed of the classical free particle is fixed by considering a phenomenon known as group velocity. The waveform of the particle is an oscillating sinusoid (see image). This waveform includes an envelope, which represents the overall shape of the oscillations rather than the individual ripples. The group velocity vg is the speed of this envelope while the phase velocity vp is the speed of the ripples. It can be shown using the definitions of phase velocity and group velocity (see below) that the group velocity is twice the phase velocity, resolving the problem with the particle speed. The group velocity of the envelope is thus what actually corresponds to the speed of the particle.

Interlude on bound states and scattering states

To review, the solutions to the Schrödinger equation for the infinite square well and quantum harmonic oscillator were normalizable and labeled by a discrete index n while the solution to the Schrödinger equation for the free particle was not normalizable and was labeled by a continuous variable k.

The solutions which are normalizable and labeled by a discrete index are known as bound states. The solutions which are not normalizable and are labeled by a continuous variable are known scattering states.

Bound states and scattering states are related to certain classical mechanical phenomena. Bound states correspond to a classical particle in a potential well where the energy is not large enough for the particle to escape the well. Scattering states correspond to a particle which might be influenced by a potential but has a large enough energy to pass through the potential without getting trapped.

In quantum mechanics, bound states occur when E < V(∞) and E < V(–∞) since the phenomenon of quantum tunneling allows quantum particles to leak through any finite potential barrier. Scattering states occur when E > V(∞) or E > V(–∞). Since most potentials go to zero at infinity or negative infinity, this simplifies to bound states happening when E < 0 and scattering states happening when E > 0.

The infinite square well and the quantum harmonic oscillator represent bound states since V(x) goes to ∞ when x → ±∞. By contrast, the quantum free particle represents a scattering state since V(x) = 0 everywhere. However, there are also potentials which can result in both bound and scattering states. These kinds of potentials will be explored in the following sections.

Delta-function well

Recall that the Dirac delta function δ(x) is an infinitely high and infinitely narrow spike at the origin with an area equal to 1 (the area is obtained by integrating). The spike appears at the point a along the x axis when δ(x – a) is used. One important property of the Dirac delta function is that f(x)δ(x – a) = f(a)δ(x – a). By integrating both sides of the equation of this property, one can obtain the following useful expression. Note that a ± ϵ is used as the bounds since any positive value ϵ will then allow the bounds to encompass the Dirac delta function spike.

The delta-function well is a potential of the form –αδ(x) where α is a positive constant. As a result, the time-independent Schrödinger equation for the delta-function well system is given as follows. This equation has solutions that yield bound states when E < 0 and scattering states when E > 0.

For the bound states where E < 0, the general solutions are given by equations below. The substitution κ is defined by the first equation below, the second equation below is the general solution for x < 0, and the third equation below is the general solution for x > 0. (Since E is assumed to have a negative value, κ is real and positive). Note that V(x) = 0 for x < 0 and x > 0. In the solution for x < 0, the Ae–κx term explodes as x → –∞, so A must equal zero. In the solution for x > 0, the Feκx term explodes as x → ∞, so F must equal zero.

To combine these equations, one must use appropriate boundary conditions at x = 0. For any quantum system, ψ is continuous and dψ/dt is continuous except at points where the potential is infinite. The requirement for ψ to exhibit continuity means that F = B at x = 0. As a result, the solution for the bound states can be concisely stated as follows. In addition, a plot of the delta-function well’s bound state time-independent wave function is given below.

The presence of the delta function influences the energy E. To find the energy, one can integrate the time-independent Schrödinger equation for the delta-function well system. By making the bounds of integration ±ϵ and then taking the limit as ϵ approaches zero, the integral works only on the negative spike of the delta function at x = 0. The result for the energy is at the end of the following set of equations.

As seen above, the delta-function well only exhibits a single bound state energy E. By normalizing the wave function ψ(x) = Be–κ|x|, the constant B is found (as seen in the first equation below). The second equation below describes the single bound state wave function and reiterates the single bound state energy associated with this wave function.

For the scattering states where E > 0, the general solutions are given by equations below. The substitution k is defined by the first equation below, the second equation below is the general solution for x < 0, and the third equation below is the general solution for x > 0. (Since E is assumed to have a positive value, k is real and positive). Note that V(x) = 0 for x < 0 and x > 0. None of the terms explode this time, so none of the terms can be ruled out as equal to zero.

As a consequence of the requirement for ψ(x) to be continuous at x = 0, the following equation involving the constants A, B, F, and G must hold true. This is the first boundary condition.

There is also a second boundary condition which involves dψ/dx. Recall the following step (see first equation below) from the process of integrating the Schrödinger equation. To implement this step, the derivatives of ψ(x) (see second equation below) are found and then the limits of these derivatives from the left and right directions are taken (see third equation below). Since ψ(0) = A + B as seen in the equation above, the second boundary condition can be given as the final equation below.

By rearranging the final equation above and substituting in a parameter β = mα/ћ2k, the following expression is obtained. This expression is a compact way of writing the second boundary condition.

These two boundary conditions provide two equations, but there are four unknowns in these equations (five unknowns if k is included). Despite this, the physical significance of the unknown constants can be helpful. When eikx is multiplied by the factor for time-dependence e–iEt/ћ, it gives rise to a wave propagating to the right. When e–ikx is multiplied by the factor for time-dependence e–iEt/ћ, it gives rise to a wave propagating to the left. As a result, the constants describe the amplitudes of various waves. A is the amplitude of a wave moving to the right on the x < 0 side of the delta-function potential, B is the amplitude of a wave moving to the left on the x < 0 side of the delta-function potential, F is the amplitude of a wave moving to the right on the x > 0 side of the delta-function potential, and G is the amplitude of a wave moving to the left on the x > 0 side of the delta-function potential.

In a typical experiment on this type of system, particles are fired from one side of the delta-function potential, the left or the right. If the particles are coming from the left (moving to the right), the term with G will equal zero. If the particles are coming from the right (moving to the left), the term with A will equal zero. This can be understood intuitively by examining the figure above.

As an example, for the case of particles fired from the left (moving to the right), A is the amplitude of the incident wave, B is the amplitude of the reflected wave, and F is the amplitude of the transmitted wave. The equations of the two boundary conditions are reiterated in the first line below. By solving these equations, the second line of expressions is found. Since the probability of finding a particle at a certain location is |Ψ|2, the relative probability R of an incident particle undergoing reflection and the relative probability T of an incident particle undergoing transmission are given by the third line of expressions below. 

Also for the example case of particles fired from the left (moving to the right), by substituting back from β = mα/ћ2k and k = (2mE)0.5/ћ to get the expressions in terms of energy, the following equations are obtained for the reflection and transmission relative probabilities.

By performing the same process, but with A = 0 instead of G = 0, corresponding equations can be found for the case of particles fired from the right (moving towards the left).

It is important to note that, since these scattering wave functions are not normalizable, they do not actually represent possible particle states. To solve this problem, one must construct normalizable linear combinations of the stationary states in a manner similar to that performed with the quantum free particle system. In this way, wave packets will occur and the actual particles will be described by the range of energies of the wave packets. Because the actual normalizable system exhibits a range of energies, the probabilities R and T should be thought of as approximate measures of reflection and transmission for particles with energies in the vicinity of E.

Finite square well

The finite square well is a system for which a particle’s V(x) = –V0 when –a ≤ x ≤ a and its V(x) = 0 otherwise. For this system, the Schrödinger equation is given as follows for the conditions x < –a, –a ≤ x ≤ a, and x > a. Note that the equations for x < –a and x > a are the same since V(x) = 0 in both cases (but the boundary conditions will differ as will be explained soon). As with the Delta-function potential well, the finite square well has both bound states (with E < 0) and scattering states (with E > 0). First, the bound states with E < 0 will be considered. In this case, the Schrödinger equations for the finite square well are as follows.

For the cases of x < –a and x > a where V(x) = 0, the general solutions to the Schrödinger equation are respectively Ae–κx + Beκx and Fe–κx + Geκx where A, B, F, and G are arbitrary constants. In the x < –a case, the Ae–κx term blows up as x → –∞, making this term physically invalid. As a result, the physically admissible solution is ψ(x) = Beκx. In the x > a case, the Geκx term blows up as as x → ∞, making this term physically invalid. As a result, the physically admissible solution is ψ(x) = Fe–κx. For the case of –a ≤ x ≤ a, the general solution to the Schrödinger equation is ψ(x) = Csin(lx) + Dcos(lx). Note that, because E must be greater than the minimum potential energy Vmin = –V0, the value of l ends up real and positive (even though E is also negative). These solutions are summarized by the following equations.

Since the potential V(x) = –V0 is an even function (symmetric about the y axis), one can choose to write the solutions to the wave function as either even or odd. This comes from some properties of the time-independent Schrödinger equation. Next, it is again important to constrain these solutions using the boundary conditions which require the continuity of ψ(x) and dψ/dx at ±a.

For the even solutions, the constant C in ψ(x) = Csin(lx) + Dcos(lx) is zero. Because C = 0, the remaining equation is the even function ψ(x) = Dcos(lx) for –a ≤ x ≤ a. So, the continuity of ψ(x) and dψ/dx at +a necessitates the following two equations to hold true. The third equation comes from dividing the second equation by the first equation to solve for κ.

For the odd solutions, the constant D in ψ(x) = Csin(lx) + Dcos(lx) is zero. Because D = 0, the remaining equation is the odd function ψ(x) = Dsin(lx) for –a ≤ x ≤ a. So, the continuity of ψ(x) and dψ/dx at +a necessitates the following two equations to hold true. The third equation comes from dividing the second equation by the first equation to solve for κ.

As κ and l are both functions of E, the κ = ltan(la) and κ = –lcot(la) equations can be solved for E. To do this, it is convenient to use the notation z = la and z0 = (a/ћ)(2mV0)0.5. Simplifying the κ = ltan(la) and κ = –lcot(la) equations using this notation gives the following results. These equations can be solved numerically for z or graphically for z by looking for points of intersection (after obtaining z, E is easily computed).

Let us consider the tan(z) equation. There are two limiting cases of interest. These include a well which is wide and deep and a well which is shallow and narrow. Though not included in these notes, similar calculations can be performed for the –cot(z) equation.

For a wide and deep well, the value of z0 is large. Intersections between the curves of tan(zn) and ((z0/zn)2 – 1)0.5 occur at nπ/2 for odd n and at nπ for even n. This leads to the following equations which describe values of En. From this outcome, it can be seen that infinite V0 results in the infinite square well case with an infinite number of bound states. However, for any finite square well, there are only a finite number of bound states.

For a shallow and narrow well, the value of z0 is small. As the value of z0 decreases, fewer and fewer bound states exist. Once z0 is smaller than π/2, there is only one bound state (which is an even bound state). Interestingly, no matter how small the well, this one bound state always persists.

The scattering states, which occur when E > 0, will now be considered. In this case, the Schrödinger equations for the finite square well are as follows.

The general solutions to the Schrödinger equation for the finite square well’s scattering states are as follows.

But recall that in a typical scattering experiment, particles are fired from one side of the delta-function potential, the left or the right. Here it will be assumed that the particles are fired from the left side of the well (moving towards the right). Note that similar calculations could be performed for the opposite case. With this assumption, one can realize that the coefficient A represents the incident (from the left) wave’s amplitude, the coefficient B represents the reflected wave’s amplitude, and the coefficient F represents the transmitted (to the right) wave’s amplitude. Finally, the coefficient G = 0 since there is not an incident wave from the right moving towards the left.

There are four boundary conditions, continuity of ψ(x) at ±a and continuity of dψ/dx at ±a. These boundary conditions yield the following equations.

With the above equations, one can eliminate C and D and subsequently solve the system for B and F. This yields the equations below for B and F.

As with the delta-function well, a transmission coefficient T = |F|2/|A|2 can be computed across the finite square well. Recall that T represents the probability of the particle undergoing transmission across the well (in this case when moving from the right side to the left side). The probability of the particle undergoing reflection is R = 1 – T.

Since 1/T equals the equation below, whenever the sine squared term is zero, the probability of transmission T = 1.

Recall that a sine (or sine squared) term is zero when the function inside of it equals nπ such that n is any integer.

Remarkably, the above equation is the same as the one which describes the infinite square well’s energies. But realize that, for the finite square well, this only holds in the case of T = 1.

Reference: Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press. https://doi.org/DOI: 10.1017/9781316995433

Cover image source: wikimedia.org

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The Future of Biotechnology: Confluence of Next-Generation Experiment, Software, and Hardware for Deciphering and Rewriting Biological Systems


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PDF version: The Future of Biotechnology

Less than 250 years after the conclusion of the Enlightenment, we have reached a point in human history where science has given us seemingly mystical abilities. We interact across thousands of kilometers nigh-instantaneously, we hold millions of libraries of knowledge in the palms of our hands, and hosts of shining buildings tower into the sky. Despite popular conceptions of doom and gloom, we are healthier, more peaceful, and less impoverished than ever before (Pinker, 2018). Our medicines can perform miracles such as making the blind see (Kumar et al., 2016; Lu et al., 2020), repairing damaged organs (Attanasio et al., 2016; Fioretta et al., 2018), and eradicating smallpox and rinderpest (Njeumi et al., 2012; Willis, 1997). When reflecting on all that is possible today, Arthur C. Clarke’s famous statement that “any sufficiently advanced technology is indistinguishable from magic” takes on more truth now than ever. But the next revolution, the revolution where we decipher biological complexity and rewrite biology itself for the better, has only just begun.

The convergence of new experimental methods, software, and hardware may act as a driving force for deciphering complex biological systems at a vastly deeper level than ever before. Enormously data-intensive experimental techniques in areas such as spatial transcriptomics and high-resolution volume and video microscopy will provide the foundation for advancing our understanding of biological systems (Liao et al., 2021; McDole et al., 2018; Titze & Genoud, 2016; Vogt, 2020; Wan et al., 2019). Robotic laboratory automation may further enhance the throughput of such methods (Angelone et al., 2021; HamediRad et al., 2019; Holland & Davies, 2020). In the realm of software, artificial intelligence (AI) advances will facilitate interpretation of patterns in massive amounts of biological data (Motta et al., 2019; Scheffer et al., 2020; Topol, 2019). At its heart, AI is a technology which extracts patterns from data. This means that AI can automate the process of sifting through oceans of complex multidimensional data and isolating a manageable number of insights with relevance to human affairs. In addition to AI, detailed integrative simulation techniques will aid prediction and description of biological mechanisms (Bezaire et al., 2016; Billeh et al., 2020; Karr et al., 2012; Markram et al., 2015; Singharoy et al., 2019; Yu et al., 2016). Some examples of these include large-scale molecular dynamics (MD) simulations (Singharoy et al., 2019; Yu et al., 2016), kinetic simulations of whole cells (Karr et al., 2012), and neurobiological simulations with tens of thousands of detailed virtual neurons (Bezaire et al., 2016; Billeh et al., 2020; Markram et al., 2015). As essential supporting technologies for these software innovations, key hardware advances may take the forms of quantum computing architectures (Cao et al., 2019; Outeiral et al., 2021), neuroscience-optimized neuromorphic computing architectures (Brown et al., 2018; Indiveri et al., 2011; Schemmel et al., 2017), and neuromorphic tensor processing unit architectures (Bains, 2020). Quantum computing may support quantum mechanical MD simulations as well as MD simulations with more particles and longer timescales (Cao et al., 2019; Outeiral et al., 2021), neuroscience-optimized neuromorphic computing may support realistic brain simulations (Brown et al., 2018; Indiveri et al., 2011; Schemmel et al., 2017), and neuromorphic tensor processing unit architectures may support much more powerful AI (Bains, 2020). The advent of exascale supercomputing will also play a central role in aiding the outlined software methods for the biological sciences (Lee & Amaro, 2018; Service, 2018). These changes will facilitate massive enhancement of our ability to make accurate predictions of how biological systems behave.

The convergence of experimental methods, software, and hardware may further act as a driving force for rewriting complex biological systems in a scalable and reproducible manner. The previously mentioned hardware advances could enable a surge in computer-aided design (CAD) software for engineering biology with nanoscale precision. To design new biology, these CAD innovations particularly may leverage AI (Kriegman et al., 2020; Zielinski et al., 2020), in silico directed evolution (Benson et al., 2019; Kriegman et al., 2020), kinetic modeling of cellular signaling and metabolic networks (Karr et al., 2012; Zielinski et al., 2020), and molecular dynamics (Benson et al., 2019; Shi et al., 2017) as well as improved graphical user interfaces (Grun et al., 2015). On the experimental side, laboratory automation and novel experimental tools may align to rapidly synthesize, validate, and iteratively improve biological inventions (Angelone et al., 2021; Chao et al., 2015; HamediRad et al., 2019; Schneider, 2018). These changes will facilitate tremendous strides in our collective capacity to create entirely new biology and to interface this new biology with existing biology.

Advances in our capacity to decipher and rewrite biology will dramatically advance the biomedical sciences. For instance, immunotherapies have the potential to eventually cure most or all cancers (Eggermont et al., 2013; ‘Mac’ Cheever, 2008; Yong et al., 2017). Medical nanorobots, some of which will consist of an exciting material known as DNA origami (Jiang et al., 2019), may also contribute to cancer treatment (Tregubov et al., 2018) and treatment of other diseases. In the case of DNA origami especially, CAD and MD will likely play a significant role (Benson et al., 2019; Douglas et al., 2009; Shi et al., 2017). AI, classical MD, and quantum MD will also enable the creation of numerous protein-based nanomachines with diverse applications by enabling rational design of proteins which have sophisticated dynamics (Kuhlman & Bradley, 2019; Melo et al., 2018; Pirro et al., 2020). Experimental automation and computational methods involving AI and integrative simulations could enable extremely rapid responses in the form of treatments, vaccines, and diagnostics to future outbreaks of infectious disease (Angelone et al., 2021; Chao et al., 2015; Schneider, 2018; Singh et al., 2020). While the threat of antibiotic resistance is concerning, phage therapy and synthetic biology treatments may further combat future forms of bacterial infection (Collins et al., 2019; Kortright et al., 2019). AI may automate a large portion of biomedical image analysis in the clinical setting (Topol, 2019). Donor organ shortages may end with the advent of bioprinted replacement organs (Cui et al., 2017; Mir & Nakamura, 2017). CAD methods may help improve the quality of bioprinted organs (Fay, 2020). AI and integrative simulations might help unlock the secrets of aging, allowing development of treatments for aging as a disease. This could both greatly increase human longevity and greatly decrease the incidence of aging-related illnesses (Fontana et al., 2014; Zhavoronkov et al., 2019). Wearable medical devices such as electronic tattoos could monitor health and prevent tragedies by giving people early warnings before physiological dysfunctions occur (Jeong & Lu, 2019). These represent some of the many possible biomedical technologies which may make us happier and healthier in the relatively near future.

One biomedical technology which may particularly make gains throughout the coming decades is gene therapy. Through synthetic biology manufacturing techniques (Le et al., 2019), gene therapies may shake off their currently prohibitive level of expense. Multiscale computational methods for understanding the human body at general and personalized levels (through AI and integrative simulations), CRISPR tools (Doudna, 2020), and superior nanobiotechnology delivery systems (Lundstrom, 2018; Wang et al., 2019) may allow gene therapy to start treating complex polygenic disorders (Carlson-Stevermer et al., 2020). These factors may even someday enable genetic modifications which make the human body more suited to space colonization (Norman & Reiss, 2020). If political polarization declines and the specter of genetic inequality loses its imminence, gene therapy could even enhance cognitive abilities and empathy in humans. While these prospects may seem frightening to some, it is important to realize that even a few more highly intelligent and empathetic people may make dramatic positive changes in our world (Rinn & Bishop, 2015). Gene therapy may also make major contributions to increasing human longevity (Bernardes de Jesus et al., 2012). Gene therapy could result in many positive transformations to our lives and even help to preserve the long-term future of humanity.

Neurotechnology may also soon come of age. Connectomics techniques, AI, and integrative simulations may give far better understanding of how to treat brain diseases in precisely targeted ways (Bullmore & Sporns, 2009; Markram, 2006; Markram et al., 2015; Mizutani et al., 2019). In particular, nanoscale connectomics might soon undergo a revolution as 4th generation synchrotrons (Pacchioni, 2019) and the relatively cheap miniature synchrotrons called Lyncean Compact Light Sources (Hornberger et al., 2019) facilitate rapid imaging of brains at nanoscale resolution (Kuan et al., 2020). On the neuroelectronics side, brain-machine interfaces and electronic neural prostheses could treat traumatic brain injuries and sensory and motor ailments as well as extend human abilities to interface with the cloud and the physical environment (Acarón Ledesma et al., 2019; Flesher et al., 2016; Gaillet et al., 2020; Hampson et al., 2018; Liu et al., 2015; Musk, 2019). Optogenetic methods, which enable control of genetically modified neurons with pulses of light, might synergize with gene therapy to create much more precise and complex brain-computer interfaces (Balasubramaniam et al., 2018; Chen et al., 2018). Though currently in its infancy, neurotechnology will likely grow rapidly into a mature discipline which grants us new abilities in neuromedicine and beyond.

Novel biotechnologies will also have great influence on manufacturing and environmental conservation. Biological CAD methods, integrative simulations of metabolism and gene regulation, and laboratory automation may allow synthetic biology to create a panoply of new microorganisms which can cheaply and rapidly produce medicines (Meng & Ellis, 2020), nanostructures (Bhaskar & Lim, 2017; Furubayashi et al., 2020), and even useful macroscale materials (Tang et al., 2020). Engineered microorganisms may also act to clean up pollutants and greenhouse gases (Gong et al., 2016). Molecular CAD methods, MD simulations, and laboratory automation may further revolutionize manufacturing through the creation of artificial molecular factories (Krause & Feringa, 2020). These molecular factories could involve immobilizing optically programmable supramolecular complexes such as certain rotaxanes and catenanes (Bruns & Stoddart, 2014) on metal-organic frameworks or similar crystalline structures (Krause & Feringa, 2020). With these miniscule factories, the dream of molecularly or even atomically precise construction at scale might be in reach. In addition, molecular factories which clean up pollutants and greenhouse gases could also make great contributions to combatting environmental degradation (Aithal & Aithal, 2020; Subramanian et al., 2020). Another suite of emerging technologies for ecoengineering are gene drives. These propagate gene editing tools which modulate the reproduction of populations of mosquitos and other disease vectors, potentially helping to stop illnesses like malaria (Gantz et al., 2015; Noble et al., 2017). Synthetic biology may also provide “off switches” for these gene drives, preventing them from causing environmental problems if they get out of control (Xu et al., 2020). In the realm of food production, gene edited plants can be made more suited to vertical farming (Kwon et al., 2020; O’Sullivan et al., 2020), indoor farming on the moon or Mars (Cannon & Britt, 2019), or ocean-based agriculture (Simke, 2020). In vitro meat may eventually transform meat production into a much more sustainable industry while decreasing the prevalence of animal cruelty (Bryant & Barnett, 2020; Zhang et al., 2020). These innovations and others could go a long way towards combatting global challenges such as hunger and climate change.

The confluence of advances in experiment, software, and hardware will enable many exciting biotechnological changes in the coming decades. Clever new experimental techniques will couple with automation to produce oceans of biological data. AI and integrative simulations extract meaningful insights from those otherwise unmanageable data point oceans. Hardware advances in neuromorphic computing, quantum computing, and exascale supercomputing could enable the titanic computations necessary to push software to its full potential. With this trinity of drivers of scientific progress, a plethora of new biotechnologies may enter common use and radically transform how we live. Some major areas of impact for these biotechnologies will include biomedicine, neurotechnology, gene therapy, manufacturing, agriculture, environmental remediation, and space colonization. Some may raise objections about the risks of such rapid technological changes. To answer these objections, consider that any kind of human progress, technological or social, must involve missteps. Yet human ingenuity and determination corrects these missteps in an ever-evolving trajectory, leading to an overall better world. Technology will synergize with the indomitable human spirit to build a bright and beautiful future.

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Want to learn biology? Recommended texts from beginner to advanced


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Preface:

I have seen a variety of online resources which recommend books for learning physics and mathematics (e.g. Chicago undergraduate mathematics bibliography, Susan Fowler’s So You Want To Learn Physics, How to Learn Math and Physics, etc.), yet there seems to be a paucity of similar resources for biological fields. To help fill this gap, I have compiled a handpicked list of textbooks which may aid those with a desire to learn biology.

I have also included books from fields such as mathematics, computer science, chemistry, physics, imaging, and nanotechnology which are important in biology. The books from adjacent fields which I recommend here are mostly targeted towards those readers who come from backgrounds which are not greatly quantitative. For this reason, books filled with lots of detailed mathematics are located in the advanced category. That said, I do assume some familiarity with mathematics and physics in the lower levels also.  

Though this page so far does not include resources beyond textbooks, there are many other useful tools for learning about biology. Video lectures, educational books which are not textbooks (e.g. Thieme FlexiBooks, Lippincott’s Illustrated Reviews, etc.), scientific journal articles (especially review papers), reputable scientific news articles (e.g. Nature News and Views, Science Daily, Neuroscience News, etc.), Wikipedia, other educational websites, and research experience come to mind.

While this list is certainly not comprehensive, I have tried to cover as much ground as possible for the interested autodidact. These books represent the ones that I personally feel are the best for the given subjects at the given levels (beginner, lower intermediate, upper intermediate, and advanced). There are a lot of texts related to microbiology, biochemistry, and neuroscience. This bias reflects my own background in synthetic biology, nanobiotechnology, and connectomics. My list is currently lacking in ecology and evolutionary biology texts. If anyone is interested in contributing their own recommendations for these or other missing topics, feel free to contact me and we can figure out how to incorporate your texts.

One point that I would like to make is that you by no means need to read these books from cover to cover. It is much more efficient to learn biology by creating a curriculum for yourself and reading selected chapters and sections as they interest you. Over time, the knowledge will build up and you will start to see how it all connects. You will eventually begin to gain the ability to think critically about biological mechanisms and how perturbing them may influence the systems. I would recommend practicing this kind of thinking early on. You can begin to do thought experiments even when you are starting out. As you carry out these thought experiments, you can explore your books and the internet to try and figure out any missing pieces. This will exercise your ability to understand and make predictions about biological systems.

Biology is an expansive, interdisciplinary, and extremely exciting field. I hope that you enjoy your journey into the biological sciences!

Beginner:

These represent foundational texts which introduce biology and associated fields which are essential for understanding biology (i.e. chemistry, physics, and mathematics). They are at a high school or maybe college freshman level.

Biology

Campbell Biology – by Urry, Cain, Wasserman, Minorsky, Reece || An authoritative introduction to biology and its subdisciplines. It features clear explanations, good organization, and helpful illustrations. Though lengthy, you can often read desired subsections in any order. That said, I would recommend reading some molecular biology and genetics chapters before diving into physiology. It should be noted that this text is a primary source for the high school Biology Olympiad competition.

Chemistry

Chemistry – by Zumdahl, Zumdahl, and DeCoste || An introductory chemistry text which has good organization and illustrations. Though other general chemistry books could work just as well, I have a mild personal preference for this one.

Mathematics

Calculus – by Stewart || Though lengthy, this book is a good introduction to calculus. It explains single-variable calculus and multivariable calculus and even gives a small taste of differential equations. This is excellent since calculus and differential equations are so central to computational modeling of biological systems.

Physics

Physics for Scientists and Engineers: A Strategic Approach with Modern Physics – by Knight || While I have not used this book personally, I have heard good things with regards to its applicability for biology. As such, I picked out Knight’s text for this list entry because of its organization, its inclusion of modern physics, and its emphasis on practical applications.

Lower intermediate:

These books introduce a range of key subfields in biology. Though some of the texts are quite long (e.g. 800+ pages), I will say again that they do not need to be read cover to cover. These do not require greatly specialized knowledge to understand. They are typically used for first year or second year university courses. As with the previous section, I have included some non-biology texts covering fields adjacent to biology. Note that, because biology is an interdisciplinary enterprise, these adjacent fields are vitally important for understanding and applying biological knowledge. 

Biochemistry

Lehninger’s Biochemistry – by Nelson and Cox || Great textbook which discusses biochemistry with both depth and breadth. It is not as detailed as Voet’s book (see the upper intermediate section), but it is not a light treatment either. This text features beautiful illustrations which are very helpful for gaining a deeply visual appreciation of how biochemistry works. In my opinion, it also has well-written treatments of the mathematics of enzyme kinetics and related topics.

Computer science

MATLAB: A Practical Introduction to Programming and Problem Solving – by Attaway || Since computer science is an integral part of biology research, it is important to have at least some understanding of programming and modeling. For those who are not already familiar with programming, Attaway’s MATLAB book provides an excellent entry point. It instructs on how to use MATLAB in a clear and concise way and also discusses essential mathematics that come up in scientific computing. Another strength of this text is its clean organization, which allows one to jump around the different sections more easily as required by one’s explorations in MATLAB coding. MATLAB is one of the most user-friendly programming languages and so it is great for beginners. Though MATLAB is not as grounded in the fundamentals of computational logic as some languages, it is quite useful as a tool for many scientific computing applications such as modeling, image processing, and data analysis. It should be noted that MATLAB itself is not free, though if you are affiliated with a university, the school will probably pay for your license.

Python Programming: An Introduction to Computer Science – by Zelle || This text provides another excellent entry point into programming. Zelle acts as a well-organized reference for learning the basics of Python. It is clear and reasonably concise. By contrast to MATLAB, Python is freely available. Another benefit of Python is the wide array of user-created software packages that you can easily install into your Python infrastructure. Many of these packages provide tools that handle specific areas of computational biology such as nucleic acid sequence analysis or biologically realistic neuron simulation.

Genetics

Essentials of Genetics – by Klug, Cummings, Spencer, Palladino, Killian || A standard text which introduces the various branches of genetics. Though there is perhaps not enough focus on modern techniques for my personal taste, I do appreciate the clarity of this book’s molecular genetics sections.

Gene Cloning and DNA Analysis: An Introduction – by Brown || Excellent book which describes molecular genetics techniques. It is concise and clear and yet still covers a lot of important methods in sufficient detail to convey real understanding.

Immunology

Cellular and Molecular Immunology – by Abbas, Lichtman, Pillai || Explains immunological principles in a through yet digestible way. It features very consistent diagrams which carefully represent specific molecules and cell types with the same images throughout the book.

Basic Immunology: Functions and Disorders of the Immune System – by Abbas, Lichtman, Pillai || This text is essentially a more concise version of Cellular and Molecular Immunology. Since it is written by the same authors, it also features its sister text’s helpfully consistent diagrams. 

Mathematics

Fundamentals of Differential Equations and Boundary Value Problems – by Nagle, Saff, and Snider || Differential equations are vitally important for modeling and simulation in biology, so if you want to go into any kind of biotechnology-related field, you should learn about this branch of mathematics. This text covers differential equations in a clear manner, provides lots of good exercises, and focuses on application rather than theory.

Linear Algebra: Step by Step – by Kuldeep Singh || Linear algebra is another area of mathematics which is vitally important for modeling and simulation in biology and bioengineering fields. This book goes over linear algebra in a clear fashion, has some illustrations to aid intuitive understanding, includes many good exercises, and emphasizes application rather than theory.

Microbiology

Brock Biology of Microorganisms – by Madigan, Bender, Buckley, Sattley, Stahl || For those who want to explore infectious disease and/or synthetic biology, it can be valuable to get acquainted with microbiology. This authoritative text is friendly to beginners in biology and has strong illustrations.

Molecular and Cellular Biology of Viruses – by Lostroh || This is a good book for virology in general. It has very pretty illustrations which are quite helpful to the reader. I do think that the book meanders too much in its explanations. The organization of the book as a whole seems a little haphazard as well. Nonetheless, this text can serve as a good reference if you want to read up on a specific type of virus and are looking for intuitive comprehension of its mechanisms.

Molecular biology

Molecular Biology of the Cell – by Alberts, Johnson, Lewis, Raff, Roberts, Walter || A comprehensive and yet approachable book on molecular biology. It has numerous excellent illustrations, a crucial feature in any molecular biology text. It thoroughly covers a large array of important topics. There are even supplemental digital chapters on further topics in molecular biology for interested readers.  

Essential Cell Biology – by Alberts, Hopkin, Johnson, Morgan, Raff, Roberts, Walter || Though this book is somewhat less detailed and thorough than the Molecular Biology of the Cell, it provides a more concise introduction to cell biology, while still covering enough detail to grant a good understanding of the subject. It also has great illustrations.

Neuroscience

Neuroscience: Exploring the Brain – by Bear, Connors, Paradiso || This book talks about a wide range of topics in neurobiology, so it is useful for introducing neuroscience as a broad field of study. I found the chapters on sensory neuroscience to be especially strong. In my admittedly biased opinion, the book neglects computational neuroscience and modern neuroscientific techniques. If you are coming from a highly mathematical background and/or wanting to go into a mathematically-focused field of neuroscience, you might want to supplement this text with some computational neuroscience books (see the intermediate and advanced sections of this page).

Organic chemistry

Organic Chemistry as a Second Language: First Semester Topics – by Klein || Klein’s short books on organic chemistry are amazing at helping the reader to understand the core principles of the subject. The first semester topics text is especially good for explaining the principles governing structure and mechanisms in organic chemistry.

Organic Chemistry as a Second Language: Second Semester Topics – by Klein || The second installment in Klein’s short texts on organic chemistry is similarly fantastic for gaining intuitive understanding. It goes into more depth on why certain reaction mechanisms happen as well as covering spectroscopy topics.

Organic Chemistry – by Klein || Klein’s full-length textbook provides further detail on organic chemistry while still emphasizing skills and principles rather than memorization.

Physiology

Principles of Anatomy and Physiology – by Tortora and Derrickson || Very long book, but wonderfully illustrated, clearly explained, and highly informative. I really appreciate how this text discusses molecular biology and biochemistry in the context of human physiology. It includes a wealth of fascinating details on how physiology works from the molecular level on up to the whole body. I especially enjoyed the chapter on endocrinology. For those who are medically inclined, there is also a lot of detail on the anatomical terminology (but this can easily be skimmed if you are not planning on going into medicine). Finally, there are numerous boxes which discuss specific diseases and other clinical subjects of special interest.

Plant biology

Raven Biology of Plants – by Evert and Eichhorn || An authoritative text on plant biology. Though I never got into this book much, I have heard great reviews from others. It covers a wide range of topics in botany and offers clear explanations as well as very nice illustrations and photographs. It spends a lot of time reviewing content from other areas of biology, which can be good or bad depending on your level of background.

Upper intermediate:

Books which cover more specialized topics in various subfields of biology or cover broader fields of biology in more depth. In contrast to the previous texts, these books tend to go into more detail and assume that the reader has more background. They are often employed in upper-level undergraduate elective courses. It should be noted that the degree of background required for my “lower intermediate” and “upper intermediate” categories is a matter of opinion. People may find certain texts more challenging or less challenging depending on their background and learning style. That said, I think that these categories can still serve as a rough guide for those seeking to expand their knowledge of the biological sciences.

Biochemistry

Introduction to Proteins: Structure, Function, and Motion – by Kessel and Ben-Tal || Discusses protein biochemistry and biophysics. This text does not go into great mathematical detail (it only includes relatively simple equations), but it does discuss the conceptual underpinnings of biophysical phenomena in a lot of detail. As an example, it contains some excellent biophysical explanations of why protein folding is such a challenging computational problem. The book also provides a wealth of information about how proteins operate in the larger cellular and physiological contexts. The illustrations are only moderately attractive, but still helpful from a practical perspective.

Biochemistry – by Voet and Voet || Though I have not personally used this book much, I have heard it is an excellent text from a number of sources, so I wanted to include it here. Voet’s textbook is known for going into a lot of detail, so it should serve you well if you are looking for a comprehensive discussion of general biochemistry. It also has very good illustrations.

An Introduction to Medicinal Chemistry – by Patrick || Beautiful book on drug design, drug development, and how drugs interact with the body. This textbook is really great because it clearly explains the fundamental principles of medicinal chemistry in a highly generalizable fashion. Its writing and diagrams really help the reader to understand the “why” underlying pharmacology. The text is also quite concise, direct, and practical in its presentation.

Developmental biology

Developmental Biology – by Gilbert and Barresi || This book contains impressive details on the development of various organisms. It has beautiful diagrams and describes complicated signaling pathways in an engaging and meaningful manner. When I read Gilbert’s text, I get excited about how the process of organismal development follows a gorgeously complex extrapolation of fundamental chemical logic.

Imaging

Fluorescence Microscopy: From Principles to Biological Applications – edited by Ulrich Kubitscheck || An excellent introduction to the engineering principles of fluorescence microscopy. This book provides background on optical physics, explains the physical mechanisms behind key types of modern fluorescence microscopy systems (e.g. confocal microscopy, light-sheet microscopy, etc.), and discusses how fluorescence itself works and is applied. While the text does not shy away from using the necessary mathematical tools to properly explain the subject, it is clear enough that even readers with relatively light backgrounds in physics should find it reasonably understandable.

Introduction to Medical Imaging: Physics, Engineering and Clinical Applications – by Barrie Smith and Webb || Clear and well-organized introduction to the main modalities of medical imaging. This text explains physical principles behind the operation of technologies such as magnetic resonance imaging, x-ray computed tomography, ultrasound, and more. It also discusses some important concepts in computational image processing. While mathematics certainly plays a key role in this book, it is overall fairly light on quantitative aspects. Depending on your goals, this can be advantageous or a drawback. The illustrations are helpful from a practical perspective, though not especially lush.

Microbiology

Bacterial Pathogenesis: A Molecular Approach – by Wilson, Winkler, Ho || Really nice book on the molecular mechanisms of bacterial pathogenesis. This book has a fair amount of detail on the subject but explains clearly. I own the 3rd edition rather than the more recent 4th edition, but I have had a chance to look through the 4th edition. It should be noted that the 4th edition has major updates including beautiful full-color illustrations which greatly enhance its explanatory power. The 3rd edition already had quite helpful diagrams, but the 4th edition appears to have taken this to a new level entirely.

Virology: Principles and Applications – by Carter and Saunders || This virology text is less comprehensive than many other virology books, but it makes up for this in that it explains viruses in a highly concise and pragmatic manner. The sections on bacteriophages and HIV are especially strong. For the reader who seeks to gain clear and direct understanding of the key molecular mechanisms used by viruses, this text is excellent.

Principles of Virology – by Flint, Racaniello, Rall, Skalka, Enquist || This book comes in two volumes. The first emphasizes molecular biology of viruses and the second emphasizes the pathogenesis and control of viruses. The diagrams are quite consistent, beautiful, and helpful. The text explains clearly and covers a lot of valuable topics. As a result of its thoroughness, this book may seem somewhat overwhelming, but it still is excellent as a reference and as a general source of virology knowledge.

Molecular biology and genetics

Molecular Biology of the Gene – by Watson, Baker, Bell, Gann, Levine, Losick || Classic text which discusses molecular genetics at a somewhat higher level than a typical introductory molecular biology book. Great illustrations and clear explanations aid the reader’s understanding of the intricate molecular machines which tirelessly carry out the myriad of tasks necessary to run the genome and transcriptome. The book is fairly long, but if you already know some molecular biology, you can certainly jump around to learn more details about specific areas of interest.

Molecular Genetics of Bacteria – by Snyder, Peters, Henkin, Champness || Similar to Watson’s text (above), but specifically covering bacterial molecular genetics rather than molecular genetics in general. In the 4th edition, the illustrations convey strong understanding of molecular mechanisms, though they are not as sumptuous as the diagrams in some biology books. In the 5th edition, the illustrations are both sumptuous and convey strong understanding of molecular mechanisms. There is a lot of great material here which can be especially useful for biohackers (and other researchers) who want to use the bacterial cell as a chassis for synthetic biology.

Neuroscience

Cognition, Brain, and Consciousness: Introduction to Cognitive Neuroscience – by Baars and Gage || Discusses cognitive neuroscience from both neuropsychological and neurophysiological perspectives. This text goes over a lot of psychological experiments for those who are interested in behavioral neuroscience, but also discusses mechanisms for those who want to focus more on the underlying ways that the brain operates. In my opinion, the largest drawback of this book is that it is weak on cellular neurophysiology.

Fundamentals of Computational Neuroscience – by Trappenberg || An excellent introduction to computational neuroscience for someone coming into the area from a less quantitatively-focused background. You will still need to know calculus and maybe a small amount of differential equations, but the book is less mathematically intense than most other computational neuroscience texts. Furthermore, the book explains key ideas from areas of mathematics such as linear algebra and probability so that the reader does not necessarily have to already know these subjects. It is fairly concise yet still clearly explains a wide variety of topics from the field.

Physical chemistry

Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience – by Dill and Bromberg || This book features elegant explanations of how statistical thermodynamics and molecular physics apply to biology and nanotechnology. In my opinion, one of its strengths is its excellent organization. The text also features very clean formatting which makes it a smoother read. Though this text is mathematics-focused, it reviews key concepts in probability and multivariable calculus for readers who have less quantitative backgrounds. There are some great chapters on foundational topics (e.g. entropy, the Boltzmann distribution, electrostatics, etc.) as well as numerous chapters on exciting applications such as polymer physics, biochemical machines and nanomachines, and cooperative binding.

Physiology

The Biology of Cancer – by Weinberg || This book provides an amazing introduction to the molecular biology, genetics, biochemistry, and treatment of cancer. Lots of great content on tumor pathogenesis from perspectives of cell signaling, DNA repair and recombination, tissue microenvironment, immunobiological aspects, virology, and more. The book features a wealth of breathtaking diagrams and histological photographs which are colorful, detailed, and highly informative. Though some of the book goes through a lot of basic molecular biology review, readers who feel comfortable with that material can easily skip to more advanced sections.

Advanced:

Books that cover specialized topics in depth and books that involve somewhat complicated mathematics are listed here. These texts typically assume that you have a fair amount of background. They are usually employed at the senior undergraduate level or at the graduate level (but please do not let this discourage you from trying them out regardless). Note that a few of these might be called monographs rather than textbooks. Because of the breadth of the biological sciences, there are many thousands of possible titles to include in this section, so please realize that these texts represent a small sampling.

Biochemistry

Protein Actions: Principles and Modeling – by Bahar, Jernigan, Dill || Excellent text on the biophysics of proteins. This book goes through a lot of challenging content on physical chemistry and computational modeling, yet it is presented in a very understandable way. Full color illustrations, clearly organized equations, and elegant explanations contribute to its pedagogical strength.

Genetics

Epigenetics – by Allis, Caparros, Jenuwein, Reinberg || Very detailed but also very rewarding, this book goes over epigenetics in a series of engaging chapters written by expert authors. Despite having different authors for different chapters, the book uses consistent illustrations throughout. The illustrations are also of high quality and are in full color, which helps to motivate the reader and aids understanding. This text covers the epigenetics of a series of model organisms as well as a myriad of key topics in mammalian epigenetic research.

Mobile DNA III – edited by Craig, Chandler, Gellert, Lambowitz, Rice, Sandmeyer || Very long and highly technical, this monograph delves deep into research on topics such as transposons, recombination, and programmed DNA rearrangements. Despite its technical character, this book still includes a myriad of helpful (and colorful) diagrams and usually has good explanations. I especially enjoyed the chapter on integrons.  

Imaging

Fundamentals of Biomedical Optics || A good text on microscopy and other forms of imaging as well as the underlying optical physics involved in the engineering of imaging systems. The book is well-organized, engagingly illustrated, detailed, and emphasizes generalizable principles. Many parts of this text can be a struggle for a reader without a strong physics background, but this makes sense given the subject matter and level of depth.

Nanotechnology

Bioconjugate Techniques – by Hermanson || A great reference text for those interested in nanobiotechnology, drug delivery, contrast agents, and other areas involving bioconjugates. This book is filled with beautiful diagrams which aid understanding. The explanations are a less concise than would be ideal, though they are still effective. The text also provides lots of clear laboratory protocols for interested researchers.

The Nature of the Mechanical Bond: From Molecules to Machines – by Bruns and Stoddart || Beautiful and comprehensive text on supramolecular chemistry, an area which is highly relevant to bioengineering disciplines. It focuses on the synthesis and dynamics of supramolecular structures which perform desired mechanical actions. The book is somewhat long due to its high level of detail and coverage, but it is gorgeously illustrated and well-written. There is a fair amount of historical content included throughout and the first chapter discusses some connections between supramolecular chemistry and art. I would recommend having a strong understanding of your chemical thermodynamics, chemical kinetics, organic chemistry, and perhaps even some organometallic chemistry when reading this book. While this kind of background knowledge is not absolutely necessary, it can certainly help to get more out of the text.

Neuroscience

Dendrites – edited by Stuart, Spruston, Häusser || Beautiful text which goes through the biology of dendrites in a series of engaging chapters by expert authors. Exceptionally well-made diagrams (with full color also) help the reader to understand concepts and useful tables facilitate referencing of detailed information. One drawback of the book is that it is lacking in concision, though this is partly due to the need to discuss ambiguity in content at the frontiers of dendrite research.  

Handbook of Brain Microcircuits – edited by Shepherd and Grillner || This book provides a series of short reviews on the mechanistic workings of neuronal microcircuits in both vertebrate and invertebrate systems. Though brief, each chapter packs in a lot of interesting information. As with many of the texts I have chosen for this list, the text features many full color diagrams to aid the reader. If you want to see a myriad of examples of the precise mechanisms which produce cognition and behavior, this book is excellent. Of course, the book is far from comprehensive; there are many papers which examine other neural circuits and there remains a vast universe of neural circuits still waiting to be uncovered.

Neuronal Dynamics: From single neurons to networks and models of cognition – by Gerstner, Kistler, Naud, Paninski || An elegantly-written computational neuroscience book which has been made freely available by the authors online. Lots of mathematical modeling is discussed in this text, but it explains the mathematics clearly and does not muddle understanding through unnecessary digressions. Note that this book focuses much more on the mathematical models than on actual coding (depending on your goals, you may find this beneficial or detrimental). This textbook is great for facilitating deeper understanding of computational neuroscience.

Fundamentals of Brain Network Analysis – by Fornito, Zalesky, Bullmore || An excellent text on using graph theory in neuroscience. It is beautifully illustrated, well-organized, and clearly explained. The mathematical tools of graph theory and complex networks are made accessible to those coming from a biological background. My only complaint about this book is that it is somewhat lacking in conciseness. My personal view is that it would have been possible to explain the subject more concisely without losing out on the depth and other beneficial qualities. Nonetheless, the book can be very rewarding (and enjoyable).

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