Notations

The below are the simplest possible descriptions of various symbols, just to help you keep reading if you do not remember/know what they stand for. Don't cite them on a math test and then blame this book for your grade.

Watch it. There are so many ad hoc usages of symbols, some will have been overlooked here. Always use common sense first in guessing what a symbol means in a given context.

$\cdot$

A dot might indicate

• A dot product between vectors, if in between them.
• A time derivative of a quantity, if on top of it.

And also many more prosaic things (punctuation signs, decimal points, ...).

$\times$

Multiplication symbol. May indicate:

• An emphatic multiplication.
• Multiplication continued on the next line / from the previous line.
• A vectorial product between vectors. In index notation, the $i$-th component of $\vec{v}\times\vec{w}$ equals
  
  $$(\vec{v}\times\vec{w})_i= v_{\overline{\imath}}w_{\overlin... ...th}}}- v_{\overline{\overline{\imath}}}w_{\overline{\imath}}$$
  
  
  where ${\overline{\imath}}$ is the index following $i$ in the sequence 123123..., and ${\overline{\overline{\imath}}}$ the one preceding it (or second following). Alternatively, evaluate the determinant
  
  $$\vec{v}\times\vec{w} = \left\vert \begin{array}{ccc} ... ...t k}\\ v_x&v_y&v_z\\ w_x&w_y&w_z \end{array} \right\vert$$
  
  

$!$

Might be used to indicate a factorial. Example: $5!=1\times2\times3\times4\times5=120$.

The function that generalizes $n!$ to noninteger values of $n$ is called the gamma function; $n!=\Gamma(n+1)$. The gamma function generalization is due to, who else, Euler. (However, the fact that $n!=\Gamma(n+1)$ instead of $n!=\Gamma(n)$ is due to the idiocy of Legendre.) In Legendre-resistant notation,

$$n!=\int_0^{\infty}t^ne^{-t}{\,\rm d}{t}$$

Straightforward integration shows that $0!$ is 1 as it should, and integration by parts shows that $(n+1)!=(n+1)n!$, which ensures that the integral also produces the correct value of $n!$ for any higher integer value of $n$ than 0. The integral, however, exists for any real value of $n$ above $-1$, not just integers. The values of the integral are always positive, tending to positive infinity for both $n\downarrow-1$, (because the integral then blows up at small values of $t$), and for $n\uparrow\infty$, (because the integral then blows up at medium-large values of $t$). In particular, Stirling’s formula says that for large positive $n$, $n!$ can be approximated as

$$n! \sim \sqrt{2\pi n} n^n e^{-n} \left[1 + \ldots\right]$$

where the value indicated by the dots becomes negligibly small for large $n$. The function $n!$ can be extended further to any complex value of $n$, except the negative integer values of $n$, where $n!$ is infinite, but is then no longer positive. Euler’s integral can be done for $n=-\frac12$ by making the change of variables $\sqrt{t}=u$, producing the integral $\int_0^\infty2e^{-u^2}{\,\rm d}{u}$, or $\int_{-\infty}^{\infty}e^{-u^2}{\,\rm d}{u}$, which equals $\sqrt{\int_{-\infty}^{\infty}e^{-x^2}{\,\rm d}{x}\int_{-\infty}^{\infty}e^{-y^2}{\,\rm d}{y}}$ and the integral under the square root can be done analytically using polar coordinates. The result is that

$$-\frac12! = \int_{-\infty}^{\infty}e^{-u^2}{\,\rm d}{u} = \sqrt{\pi}$$

To get $\frac12!$, multiply by $\frac12$, since $n!=n(n-1)!$.

A double exclamation mark may mean every second item is skipped, e.g. $5!!=1\times3\times5$. In general, $(2n+1)!!=(2n+1)!/2^nn!$. Of course, $5!!$ should logically mean $(5!)!$. Logic would indicate that $5\times3\times1$ should be indicated by something like $5!'$. But what is logic in physics?

$\vert$

May indicate:

• The magnitude or absolute value of the number or vector, if enclosed between a pair of them.
• The determinant of a matrix, if enclosed between a pair of them.
• The norm of the function, if enclosed between two pairs of them.
• The end of a bra or start of a ket.
• A visual separator in inner products.

$\vert\ldots\rangle$

A “ket” is used to indicate some state. For example, $\big\vert l\:m\big\rangle$ indicates an angular momentum state with azimuthal quantum number $l$ and magnetic quantum number $m$. Similarly, $\big\vert\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\... ...\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em\big\rangle$ is the spin-down state of a particle with spin $\frac12$. Other common ones are $\big\vert{\underline x}\big\rangle$ for the position eigenfunction ${\underline x}$, i.e.  $\delta(x-{\underline x})$, $\big\vert 1{\rm {s}}\big\rangle$ for the 1s or $\psi_{100}$ hydrogen state, $\big\vert 2{\rm {p}}_z\big\rangle$ for the 2p$_z$ or $\psi_{210}$ state, etcetera. In short, whatever can indicate some state can be pushed into a ket.

$\langle\ldots\vert$

A “bra” is like a ket $\big\vert\ldots\big\rangle$, but appears in the left side of inner products, instead of the right one.

$\uparrow$

Indicates the “spin up” state. Mathematically, equals the function ${\uparrow}(S_z)$ which is by definition equal to 1 at $S_z=\frac12\hbar$ and equal to 0 at $S_z=-\frac12\hbar$. A spatial wave function multiplied by $\uparrow$ is a particle in that spatial state with its spin up. For multiple particles, the spins are listed with particle 1 first.

$\downarrow$

Indicates the “spin down” state. Mathematically, equals the function ${\downarrow}(S_z)$ which is by definition equal to 0 at $S_z=\frac12\hbar$ and equal to 1 at $S_z=-\frac12\hbar$. A spatial wave function multiplied by $\downarrow$ is a particle in that spatial state with its spin down. For multiple particles, the spins are listed with particle 1 first.

$\sum$

(Not to be confused with ${\mit\Sigma}$ further down.) Summation symbol. Example: if in three dimensional space a vector $\vec f$ has components $f_1=2$, $f_2=1$, $f_3=4$, then $\sum_{{\rm {all~}}i}f_i$ stands for $2+1+4=7$.

One important thing to remember: the symbol used for the summation index does not make a difference: $\sum_{{\rm {all~}}j}f_j$ is exactly the same as $\sum_{{\rm {all~}}i}f_i$. So freely rename the index, but always make sure that the new name is not already used for something else in the part that it appears in. If you use the same name for two different things, it becomes a mess.

Related to that $\sum_{{\rm {all~}}i}f_i$ is not something that depends on an index $i$. It is just a combined simple number. Like 7 in the example above. It is commonly said that the summation index “sums away.”

$\int$

Integration symbol, the continuous version of the summation symbol. For example,

$$\int_{\mbox{\scriptsize all }x} f(x){\,\rm d}x$$

is the summation of $f(x){\,\rm d}{x}$ over all infinitesimally small fragments ${\rm d}{x}$ that make up the entire $x$-range. For example, $\int_{x=0}^2(2+x){\,\rm d}{x}$ equals $3\times2=6$; the average value of $2+x$ between $x$ = 0 and $x$ = 2 is 3, and the sum of all the infinitesimally small segments ${\rm d}{x}$ gives the total length 2 of the range in $x$ from 0 to 2.

One important thing to remember: the symbol used for the integration variable does not make a difference: $\int_{{\rm {all~}}y}f(y){\,\rm d}{y}$ is exactly the same as $\int_{{\rm {all~}}x}f(x){\,\rm d}{x}$. So freely rename the integration variable, but always make sure that the new name is not already used for something else in the part it appears in. If you use the same name for two different things, it becomes a mess.

Related to that $\int_{{\rm {all~}}x}f(x){\,\rm d}{x}$ is not something that depends on a variable $x$. It is just a combined number. Like 6 in the example above. It is commonly said that the integration variable “integrates away.”

$\to$

May indicate:

• An approaching process. $\lim_{\varepsilon\to 0}$ indicates for practical purposes the value of the expression following the $\lim$ when $\varepsilon$ is extremely small, $\lim_{r\to\infty}$ the value of the following expression when $r$ is extremely large.
• The fact that the left side leads to, or implies, the right-hand side.

$\vec{\phantom{a}}$

Vector symbol. An arrow above a letter indicates it is a vector. A vector is a quantity that requires more than one number to be characterized. Typical vectors in physics include position $\vec r$, velocity $\vec v$, linear momentum $\vec p$, acceleration $\vec a$, force $\vec F$, angular momentum $\vec L$, etcetera.

$\widehat{\phantom{a}}$

A hat over a letter in this book indicates that it is the operator, turning functions into other functions.

$'$

May indicate:

• A derivative of a function. Examples: $1'=0$, $x'=1$, $\sin'(x)=\cos(x)$, $\cos'(x)=-\sin(x)$, $(e^x)'=e^x$.
• A small or modified quantity.
• A quantity per unit length.

$\nabla$

The spatial differentiation operator nabla. In Cartesian coordinates:

$$\nabla \equiv \left( \frac{\partial}{\partial x}, \f... ...partial}{\partial y} + {\hat k}\frac{\partial}{\partial z}$$

Nabla can be applied to a scalar function $f$ in which case it gives a vector of partial derivatives called the gradient of the function:

$$\mathop{\rm grad}\nolimits f = \nabla f = {\hat\imath}\f... ...al f}{\partial y} + {\hat k}\frac{\partial f}{\partial z}.$$

Nabla can be applied to a vector in a dot product multiplication, in which case it gives a scalar function called the divergence of the vector:

$$\div\vec v = \nabla\cdot\vec v = \frac{\partial v_x}{\pa... ...partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$$

or in index notation

$$\div\vec v = \nabla\cdot\vec v = \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i}$$

Nabla can also be applied to a vector in a vectorial product multiplication, in which case it gives a vector function called the curl or rot of the vector. In index notation, the $i$-th component of this vector is

$$\left(\mathop{\rm curl}\nolimits \vec v\right)_i = \left... ...line{\imath}}}}{\partial x_{{\overline{\overline{\imath}}}}}$$

where ${\overline{\imath}}$ is the index following $i$ in the sequence 123123..., and ${\overline{\overline{\imath}}}$ the one preceding it (or second following).

The operator $\nabla^2$ is called the Laplacian. In Cartesian coordinates:

$$\nabla^2 \equiv \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2}$$

Sometimes the Laplacian is indicated as $\Delta$. In relativistic index notation equal to $-\partial_i\partial^i$.

In non Cartesian coordinates, don’t guess; look these operators up in a table book.

$\mathop{\sqcup\kern-7.8pt\sqcap}\nolimits$

The D'Alembertian

$$\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\pa... ...partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}$$

In relativistic index notation equal to $-\partial_\mu\partial^\mu$.

$^*$

A superscript star normally indicates a complex conjugate. In the complex conjugate of a number, every ${\rm i}$ is changed into a $-{\rm i}$.

$<$

Less than.

$\langle\ldots\rangle$

May indicate:

• An inner product.
• An expectation value.

$>$

Greater than.

$[\ldots]$

May indicate:

• A grouping of terms in a formula.
• A commutator. For example, $[A,B]=AB-BA$.

$\equiv$

Emphatic equals sign. Typically means “by definition equal” or “everywhere equal.”

$\sim$

Indicates approximately equal. Normally the approximation applies when something is small or large. Read it as “is approximately equal to.”

$\propto$

Proportional to. The two sides are equal except for some unknown constant factor.

$\alpha$

(alpha) May indicate:

• The fine structure constant, $e^2/4\pi\epsilon_0\hbar c$, about 1/137.036 in value.
• A Dirac equation matrix.
• A nuclear decay mode in which a helium-4 nucleus is emitted.
• Internal conversion rate as fraction of the gamma decay rate.
• Some constant.
• Some angle.
• An eigenfunction of a generic operator $A$.
• A summation index.
• Component index of a vector.

$\beta$

(beta) May indicate:

• A nuclear decay mode in which an electron ($\beta^-$) or positron ($\beta^+$) is emitted. Sometimes $\beta^+$ is taken to also include electron capture.
• A nuclear vibrational mode that maintains the axial symmetry of the nucleus.
• Some constant.
• Some angle.
• An eigenfunction of a generic operator $B$.
• A summation index.

$\Gamma$

(Gamma) May indicate:

• The Gamma function. Look under “!” for details.
• The “width” or uncertainty in energy of an approximate energy eigenstate.
• Origin in wave number space.

$\gamma$

(gamma) May indicate:

• Gyromagnetic ratio.
• Standard symbol for a photon of electromagnetic radiation.
• A nuclear de-excitation mode in which a photon is emitted.
• A nuclear vibrational mode that messes up the axial symmetry of the nucleus.
• Summation index.
• Integral in the tunneling WKB approximation.

$\Delta$

(capital delta) May indicate:

• An increment in the quantity following it.
• A delta particle.
• Often used to indicate the Laplacian $\nabla^2$.

$\delta$

(delta) May indicate:

• With two subscripts, the “Kronecker delta”, which by definition is equal to one if its two subscripts are equal, and zero in all other cases.

• Without two subscripts, the “Dirac delta function”, which is infinite when its argument is zero, and zero if it is not. In addition the infinity is such that the integral of the delta function over its single nonzero point is unity. The delta function is not a normal function, but a distribution. It is best to think of it as the approximate function shown in the right hand side of figure 7.10 for a very, very, small positive value of $\varepsilon$.

  One often important way to create a three-dimensional delta function in spherical coordinates is to take the Laplacian of the function $-1/4{\pi}r$. Chapter 13.3 explains why. In two dimensions, take the Laplacian of $\ln(r)/2\pi$ to get a delta function.

• Often used to indicate a small amount of the following quantity, or of a small change in the following quantity. There are nuanced differences in the usage of $\delta$, $\partial$ and ${\rm d}$ that are too much to go in here.

• Often used to indicate a second small quantity in addition to $\varepsilon$.

$\partial$

(partial) Indicates a vanishingly small change or interval of the following variable. For example, $\partial f/\partial x$ is the ratio of a vanishingly small change in function $f$ divided by the vanishingly small change in variable $x$ that causes this change in $f$. Such ratios define derivatives, in this case the partial derivative of $f$ with respect to $x$.

Also used in relativistic index notation, chapter 1.2.5.

$\epsilon$

(epsilon) May indicate:

• $\epsilon_0$ is the permittivity of space. Equal to 8.854,19 10$\POW9,{-12}$ C$\POW9,{2}$/J m
• Scaled energy.
• Orbital energy.
• Lagrangian multiplier.
• A small quantity, if symbol $\varepsilon$ is not available.

$\varepsilon$

(variant of epsilon) May indicate:

• A very small quantity.
• The slop in energy conservation during a decay process.

$\eta$

(eta) $y$-position of a particle.

$\Theta$

(capital theta) Used in this book to indicate some function of $\theta$ to be determined.

$\theta$

(theta) May indicate:

• In spherical coordinates, the angle from the chosen $z$ axis, with apex at the origin.
• $z$-position of a particle.
• A generic angle, like the one between the vectors in a cross or dot product.
• Integral acting as an angle in the classical WKB approximation.
• Integral acting as an angle in the adiabatic approximation.

$\vartheta$

(variant of theta) An alternate symbol for $\theta$.

$\kappa$

(kappa) May indicate:

• A constant that physically corresponds to some wave number.
• A summation index.
• Thermal conductivity.

$\Lambda$

(Lambda) May indicate:

• Lorentz transformation matrix.

$\lambda$

(lambda) May indicate:

• Wave length.
• Decay constant.
• A generic eigenvalue.
• Entry of a Lorentz transformation.
• Scaled square momentum.
• Some multiple of something.

$\mu$

(mu) May indicate:

• Magnetic dipole moment.
• $\mu_{\rm B}$ = $e\hbar/2m_{\rm e}$ = 9.27 10$\POW9,{-24}$ J/T or 5.788 10$\POW9,{-5}$ eV/T is the Bohr magneton.
• $\mu_{\rm N}$ = $e\hbar/2m_{\rm p}$ = 5.050,78 10$\POW9,{-27}$ J/T is the nuclear magneton.
• A summation index.
• Chemical potential/molar Gibbs free energy.

$\nu$

(nu) May indicate:

• Electron neutrino.
• Scaled energy eigenfunction number in solids.
• A summation index.
• Strength of a delta function potential.

$\xi$

(xi) May indicate:

• Scaled argument of the one-dimensional harmonic oscillator eigenfunctions.
• $x$-position of a particle.
• A summation or integration index.

${\mit\Pi}$

(Pi) Peltier coefficient.

$\pi$

(pi) May indicate:

• The area of a circle of unit radius. Value 3.141,592...
• Half the perimeter of a circle of unit radius. Value 3.141,592...
• A 180$\POW9,{\circ}$ angle expressed in radians. Note that $e^{\pm{\rm i}\pi}=-1$. Value 3.141,592...
• A bond that looks from the side like a p state.
• A particle involved in the forces keeping the nuclei of atoms together ($\pi$-meson or pion for short).
• Parity.

$\rho$

(rho) May indicate:

• Electric charge per unit volume.
• Scaled radial coordinate.
• Radial coordinate.
• Eigenfunction of a rotation operator $R$.
• Mass-base density.
• Energy density of electromagnetic radiation.

${\mit\Sigma}$

(Oblique Sigma) (Not to be confused with $\sum$ described higher up.) Seebeck coefficient.

$\sigma$

(sigma) May indicate:

• A standard deviation of a value.
• A chemical bond that looks like an s state when seen from the side.
• Pauli spin matrix.
• Surface tension.
• Electrical conductivity.

$\tau$

(tau) May indicate:

• Life time or half life.
• Some coefficient.

$\Phi$

(capital phi) May indicate:

• Some function of $\phi$ to be determined.
• The momentum-space wave function.
• Relativistic electromagnetic potential.

$\phi$

(phi) May indicate:

• In spherical coordinates, the angle around the chosen $z$ axis. Increasing $\phi$ by $2\pi$ encircles the $z$-axis exactly once.
• A phase angle.
• Something equivalent to an angle.
• Field operator $\phi({\skew0\vec r})$ annihilates a particle at position ${\skew0\vec r}$ while $\phi^\dagger({\skew0\vec r})$ creates one at that position.

$\varphi$

(variant of phi) May indicate:

• A change in angle $\phi$.
• An alternate symbol for $\phi$.
• An electrostatic potential.
• An electrostatic quantum field.
• A hypothetical selectostatic quantum field.

$\chi$

(chi) May indicate

• Spinor component.
• Gauge function of electromagnetic field.

$\Psi$

(capital psi) Upper case psi is used for the wave function.

$\psi$

(psi) Typically used to indicate an energy eigenfunction. Depending on the system, indices may be added to distinguish different ones. In some cases $\psi$ might be used instead of $\Psi$ to indicate a system in an energy eigenstate. Let me know and I will change it. A system in an energy eigenstate should be written as $\Psi=c\psi$, not $\psi$, with $c$ a constant of magnitude 1.

$\Omega$

(Omega) May indicate:

• Solid angle. See the entry on angle.

$\omega$

(omega) May indicate:

• Angular frequency of the classical harmonic oscillator. Equal to $\sqrt{c/m}$ where $c$ is the spring constant and $m$ the mass.
• Angular frequency of a system.
• Angular frequency of light waves.
• Perturbation frequency,
• Any quantity having units of frequency, 1/s.

$A$

May indicate:

• Repeatedly used to indicate the operator for a generic physical quantity $a$, with eigenfunctions $\alpha$.
• Electromagnetic vector potential, or four vector potential.
• Einstein $A$ coefficient.
• Some generic matrix.
• Some constant.
• Area.

Å

Ångstrom. Equal to 10$\POW9,{-10}$ m.

$a$

May indicate:

• The value of a generic physical quantity with operator $A$
• The amplitude of the spin-up state
• The amplitude of the first state in a two-state system.
• Acceleration.
• Start point of an integration interval.
• The first of a pair of particles.
• Some coefficient.
• Some constant.
• Absorptivity of electromagnetic radiation.
• Annihilation operator $\widehat a$ or creation operator $\widehat a^\dagger$.
• Bohr radius of helium ion.

$a_0$

May indicate:

• Bohr radius, $4\pi\epsilon_0\hbar^2/{m_{\rm e}}e^2$ or 0.529,177 Å. Comparable in size to atoms, and a good size to use to simplify various formulae.
• The initial value of a coefficient $a$.

absolute

May indicate:

• The absolute value of a real number $a$ is indicated by $\vert a\vert$. It equals $a$ is $a$ is positive or zero and $-a$ if $a$ is negative.
• The absolute value of a complex number $a$ is indicated by $\vert a\vert$. It equals the length of the number plotted as a vector in the complex plane. This simplifies to above definition if $a$ is real.
• An absolute temperature is a temperature measured from absolute zero. At absolute zero all systems are in their ground state. Absolute zero is $-$273.15 $\POW9,{\circ}$C in degrees Centrigrade (Celsius). The SI absolute temperature scale is degrees Kelvin, K. Absolute zero temperature is 0 K, while 0 $\POW9,{\circ}$C is 273.15 K.

adiabatic

An adiabatic process is a process in which there is no heat transfer with the surroundings. If the process is also reversible, it is called isentropic. Typically, these processes are fairly quick, in order not to give heat conduction enough time to do its stuff, but not so excessively quick that they become irreversible.

Adiabatic processes in quantum mechanics are defined quite differently to keep students on their toes. See chapter 7.1.5. These processes are very slow, to keep the system all possible time to adjust to its surroundings. Of course, quantum physicist were not aware that the same term had already been used for a hundred years or so for relatively fast processes. They assumed they had just invented a great new term!

adjoint

The adjoint $A^H$ or $A^\dagger$ of an operator is the one you get if you take it to the other side of an inner product. (While keeping the value of the inner product the same regardless of whatever two vectors or functions may be involved.) Hermitian operators are “self-adjoint;”they do not change if you take them to the other side of an inner product. “Skew-Hermitian”operators just change sign. “Unitary operators”change into their inverse when taken to the other side of an inner product. Unitary operators generalize rotations of vectors: an inner product of vectors is the same whether you rotate the first vector one way, or the second vector the opposite way. Unitary operators preserve inner products (when applied to both vectors or functions). Fourier transforms are unitary operators on account of the Parseval equality that says that inner products are preserved.

amplitude

Everything in quantum mechanics is an amplitude. However, most importantly, the “quantum amplitude” gives the coefficient of a state in a wave function. For example, the usual quantum wave function gives the quantum amplitude that the particle is at the given position.

angle

Consider two semi-infinite lines extending from a common intersection point. Then the angle between these lines is defined in the following way: draw a unit circle in the plane of the lines and centered at their intersection point. The angle is then the length of the circular arc that is in between the lines. More precisely, this gives the angle in radians, rad. Sometimes an angle is expressed in degrees, where $2\pi$ rad is taken to be 360$\POW9,{\circ}$. However, using degrees is usually a very bad idea in science.

In three dimensions, you may be interested in the so-called “solid angle” inside a cone. This angle is defined in the following way: draw a sphere of unit radius centered at the apex of the cone. Then the solid angle is the area of the spherical surface that is inside the cone. The solid angle is in steradians. The cone does not need to be a circular one, (i.e. have a circular cross section), for this to apply.

$B$

May indicate:

• Repeatedly used to indicate a generic second operator or matrix.
• Einstein $B$ coefficient.
• Some constant.

${\cal B}$

May indicate:

• Magnetic field strength.

$b$

May indicate:

• Repeatedly used to indicate the amplitude of the spin-down state
• Repeatedly used to indicate the amplitude of the second state in a two-state system.
• End point of an integration interval.
• The second of a pair of particles.
• Some coefficient.
• Some constant.

basis

A basis is a minimal set of vectors or functions that you can write all other vectors or functions in terms of. For example, the unit vectors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ are a basis for normal three-dimensional space. Every three-dimensional vector can be written as a linear combination of the three.

$\POW9,{\circ}$C

Degrees Centigrade. A commonly used temperature scale that has the value $-$273.15 $\POW9,{\circ}$C instead of zero when systems are in their ground state. Recommendation: use degrees Kelvin (K) instead. However, differences in temperature are the same in Centigrade as in Kelvin.

$C$

May indicate:

• A third operator.
• A variety of different constants.

$c$

May indicate:

• The speed of light, about 2.997,92 10$\POW9,{8}$ m/s.
• Speed of sound.
• Spring constant.
• A variety of different constants.

Cauchy-Schwartz inequality

The Cauchy-Schwartz inequality describes a limitation on the magnitude of inner products. In particular, it says that for any $f$ and $g$,

$$\vert\langle f\vert g \rangle\vert \le \sqrt{\langle f\vert f\rangle}\sqrt{\langle g\vert g\rangle}$$

In words, the magnitude of an inner product $\langle f\vert g\rangle$ is at most the magnitude (i.e. the length or norm) of $f$ times the one of $g$. For example, if $f$ and $g$ are real vectors, the inner product is the dot product and we have $f\cdot g=\vert f\vert\vert g\vert\cos\theta$, where $\vert f\vert$ is the length of vector $f$ and $\vert g\vert$ the one of $g$, and $\theta$ is the angle in between the two vectors. Since a cosine is less than one in magnitude, the Cauchy-Schwartz inequality is therefore true for vectors.

But it is true even if $f$ and $g$ are functions. To prove it, first recognize that $\langle f\vert g\rangle$ may in general be a complex number, which according to (2.6) must take the form $e^{{\rm i}\alpha}\vert\langle f\vert g\rangle\vert$ where $\alpha$ is some real number whose value is not important, and that $\langle g\vert f\rangle$ is its complex conjugate $e^{-{\rm i}\alpha}\vert\langle f\vert g\rangle\vert$. Now, (yes, this is going to be some convoluted reasoning), look at

$$\langle f + \lambda e^{-{\rm i}\alpha} g\vert f + \lambda e^{-{\rm i}\alpha} g\rangle$$

where $\lambda$ is any real number. The above dot product gives the square magnitude of $f+\lambda e^{-{\rm i}\alpha}g$, so it can never be negative. But if we multiply out, we get

$$\langle f\vert f\rangle + 2 \vert\langle f\vert g\rangle\vert \lambda + \langle g\vert g\rangle \lambda^2$$

and if this quadratic form in $\lambda$ is never negative, its discriminant must be less or equal to zero:

$$\vert\langle f\vert g\rangle\vert \le \langle f\vert f\rangle \langle g\vert g\rangle$$

and taking square roots gives the Cauchy-Schwartz inequality.

Classical

Can mean any older theory. In this work, most of the time it either means “nonquantum,” or “nonrelativistic.”

$\cos$

The cosine function, a periodic function oscillating between 1 and -1 as shown in [39, pp. 40-...].

curl

The curl of a vector $\vec{v}$ is defined as $\mathop{\rm curl}\nolimits \vec{v}=\mathop{\rm {rot}}\vec{v}=\nabla\times\vec{v}$.

$D$

May indicate:

• Difference in wave number values.

$Dp$

Density of states.

$\vec D$

Primitive (translation) vector of a reciprocal lattice.

${\rm d}$

Indicates a vanishingly small change or interval of the following variable. For example, ${\rm d}x$ can be thought of as a small segment of the $x$-axis.

In three dimensions, ${\rm d}^3{\skew0\vec r}\equiv{\rm d}x{\rm d}y{\rm d}z$ is an infinitesimal volume element. The symbol $\int$ means that you sum over all such infinitesimal volume elements.

$d$

May indicate:

• The distance between the protons of a hydrogen molecule.
• The distance between the atoms or lattice points in a crystal.
• A constant.

$\vec d$

Primitive (translation) vector of a crystal lattice.

derivative

A derivative of a function is the ratio of a vanishingly small change in a function divided by the vanishingly small change in the independent variable that causes the change in the function. The derivative of $f(x)$ with respect to $x$ is written as ${\rm d}f/{\rm d}x$, or also simply as $f'$. Note that the derivative of function $f(x)$ is again a function of $x$: a ratio $f'$ can be found at every point $x$. The derivative of a function $f(x,y,z)$ with respect to $x$ is written as $\partial{f}/\partial{x}$ to indicate that there are other variables, $y$ and $z$, that do not vary.

determinant

The determinant of a square matrix $A$ is a single number indicated by $\vert A\vert$. If this number is nonzero, $A\vec v$ can be any vector $\vec w$ for the right choice of $\vec v$. Conversely, if the determinant is zero, $A\vec v$ can only produce a very limited set of vectors, though if it can produce a vector $w$, it can do so for multiple vectors $\vec v$.

There is a recursive algorithm that allows you to compute determinants from increasingly bigger matrices in terms of determinants of smaller matrices. For a $1\times 1$ matrix consisting of a single number, the determinant is simply that number:

$$\left\vert a_{11} \right\vert = a_{11}$$

(This determinant should not be confused with the absolute value of the number, which is written the same way. Since we normally do not deal with $1\times 1$ matrices, there is normally no confusion.) For $2\times 2$ matrices, the determinant can be written in terms of $1\times 1$ determinants:

$$\left\vert \begin{array}{ll} a_{11} & a_{12} \\ a_{... ... \\ a_{21} & \phantom{a_{22}} \end{array} \right\vert$$

so the determinant is $a_{11}a_{22}-a_{12}a_{21}$ in short. For $3\times 3$ matrices, we have

$$\begin{eqnarray*} \lefteqn{ \left\vert \begin{array}{lll} a_{11} & a_{12... ...a_{31} & a_{32} & \phantom{a_{33}} \end{array} \right\vert \end{eqnarray*}$$

and we already know how to work out those $2\times 2$ determinants, so we now know how to do $3\times 3$ determinants. Written out fully:

$$a_{11}(a_{22}a_{33}-a_{23}a_{32}) -a_{12}(a_{21}a_{33}-a_{23}a_{31}) +a_{13}(a_{21}a_{32}-a_{22}a_{31})$$

For $4\times 4$ determinants,

$$\begin{eqnarray*} \lefteqn{ \left\vert \begin{array}{llll} a_{11} & a_{1... ...a_{42} & a_{43} & \phantom{a_{44}} \end{array} \right\vert \end{eqnarray*}$$

Etcetera. Note the alternating sign pattern of the terms.

As you might infer from the above, computing a good size determinant takes a large amount of work. Fortunately, it is possible to simplify the matrix to put zeros in suitable locations, and that can cut down the work of finding the determinant greatly. We are allowed to use the following manipulations without seriously affecting the computed determinant:

1. We may “transpose”the matrix, i.e. change its columns into its rows.
2. We can create zeros in a row by subtracting a suitable multiple of another row.
3. We may also swap rows, as long as we remember that each time that we swap two rows, it will flip over the sign of the computed determinant.
4. We can also multiply an entire row by a constant, but that will multiply the computed determinant by the same constant.

Applying these tricks in a systematic way, called “Gaussian elimination” or “reduction to lower triangular form”, we can eliminate all matrix coefficients $a_{ij}$ for which $j$ is greater than $i$, and that makes evaluating the determinant pretty much trivial.

div(ergence)

The divergence of a vector $\vec{v}$ is defined as $\div\vec{v}=\nabla\cdot\vec{v}$.

$E$

May indicate:

• The total energy. Possible values are the eigenvalues of the Hamiltonian.
• $E_n=-\hbar^2/2{m_{\rm e}}a_0^2n^2=E_1/n^2$ may indicate the nonrelativistic (Bohr) energy levels of the hydrogen atom. The ground state energy $E_1$ equals -13.605,7 eV.
• Internal energy of a substance.

${\cal E}$

May indicate:

• Electric field strength.

e

May indicate

• Subscript e indicates an electron.

$e$

May indicate:

• The basis for the natural logarithms. Equal to 2.71,281,828,459... This number produces the “exponential function” $e^x$, or $\exp(x)$, or in words “$e$ to the power $x$”, whose derivative with respect to $x$ is again $e^x$. If $a$ is a constant, then the derivative of $e^{ax}$ is $ae^{ax}$. Also, if $a$ is an ordinary real number, then $e^{{\rm i}a}$ is a complex number with magnitude 1.
• The magnitude of the charge of an electron or proton, equal to 1.602,18 10$\POW9,{-19}$ C.
• Emissivity of electromagnetic radiation.
• Often used to indicate a unit vector.
• A superscript $e$ may indicate a single-electron quantity.
• Specific internal energy of a substance.

$e^{{\rm i}a x}$

Assuming that $a$ is an ordinary real number, and $x$ a real variable, $e^{{\rm i}a x}$ is a complex function of magnitude one. The derivative of $e^{{\rm i}a x}$ with respect to $x$ is ${\rm i}ae^{{\rm i}a x}$

eigenvector

A concept from linear algebra. A vector $\vec v$ is an eigenvector of a matrix $A$ if $\vec v$ is nonzero and $A\vec v=\lambda\vec v$ for some number $\lambda$ called the corresponding eigenvalue.

The basic quantum mechanics section of this book avoids linear algebra completely, and the advanced part almost completely. The few exceptions are almost all two-dimensional matrix eigenvalue problems. In case you did not have any linear algebra, here is the solution: the two-dimensional matrix eigenvalue problem

$$\left(\begin{array}{cc} a_{11}&a_{12} \\ a_{21}&a_{22} \end{array}\right) \vec v = \lambda \vec v$$

has eigenvalues that are the two roots of the quadratic equation

$$\lambda^2 - (a_{11}+a_{22})\lambda + a_{11}a_{22}-a_{12}a_{21} = 0$$

The corresponding eigenvectors are

$$\vec v_1 = \left(\begin{array}{c} a_{12} \\ \lambda_1-a_... ...begin{array}{c} \lambda_2-a_{22} \\ a_{21}\end{array}\right)$$

On occasion you may have to swap $\lambda_1$ and $\lambda_2$ to use these formulae. If $\lambda_1$ and $\lambda_2$ are equal, there might not be two eigenvectors that are not multiples of each other; then the matrix is called defective. However, Hermitian matrices are never defective.

See also “matrix” and “determinant.”

eV

The electron volt, a commonly used unit of energy equal to 1.602,18 10$\POW9,{-19}$ J.

exponential function

A function of the form $e^{\ldots}$, also written as $\exp(\ldots)$. See function and $e$.

$F$

May indicate:

• The force in Newtonian mechanics. Equal to the negative gradient of the potential. Quantum mechanics is formulated in terms of potentials, not forces.
• The anti-derivative of some function $f$.
• Some function.
• Helmholtz free energy.

${\cal F}$

Fock operator.

$f$

May indicate:

• A generic function.
• A generic vector.
• A fraction.
• The resonance factor.
• Specific Helmholtz free energy.

function

A mathematical object that associates values with other values. A function $f(x)$ associates every value of $x$ with a value $f$. For example, the function $f(x)=x^2$ associates $x=0$ with $f=0$, $x=\frac12$ with $f=\frac14$, $x=1$ with $f=1$, $x=2$ with $f=4$, $x=3$ with $f=9$, and more generally, any arbitrary value of $x$ with the square of that value $x^2$. Similarly, function $f(x)=x^3$ associates any arbitrary $x$ with its cube $x^3$, $f(x)=\sin(x)$ associates any arbitrary $x$ with the sine of that value, etcetera. A wave function $\Psi(x,y,z)$ associates each spatial position $(x,y,z)$ with a wave function value. Going beyond mathematics, its square magnitude associates any spatial position with the relative probability of finding the particle near there.

functional

A functional associates entire functions with single numbers. For example, the expectation energy is mathematically a functional: it associates any arbitrary wave function with a number: the value of the expectation energy if physics is described by that wave function.

$G$

Gibbs free energy.

$g$

May indicate:

• A second generic function or a second generic vector.
• The strength of gravity, 9.806,65 m/s$\POW9,{2}$ exactly under standard conditions on the surface of the earth.
• The g-factor, a nondimensional constant that indicates the gyromagnetic ratio relative to charge and mass. For the electron $g_e$ = 2.002,319,304,362. For the proton $g_p$ = 5.585,694,7. For the neutron, based on the mass and charge of the proton, $g_n$ = -3.826,085.
• Specific Gibbs free energy/chemical potential.

Gauss' Theorem

This theorem, also called divergence theorem or Gauss-Ostrogradsky theorem, says that for a continuously differentiable vector $\vec v$,

$$\int_V \nabla \cdot \vec v {\,\rm d}V = \int_A \vec v \cdot \vec n {\,\rm d}A$$

where the first integral is over the volume of an arbitrary region and the second integral is over all the surface area of that region; $\vec{n}$ is at each point found as the unit vector that is normal to the surface at that point.

grad(ient)

The gradient of a scalar $f$ is defined as $\mathop{\rm grad}\nolimits {f}=\nabla{f}$.

$H$

May indicate:

• The Hamiltonian, or total energy, operator. Its eigenvalues are indicated by $E$.
• $H_n$ stands for the $n$-th order Hermite polynomial.
• Enthalpy.

$h$

May indicate:

• The original Planck constant $h=2\pi\hbar$.
• $h_n$ is a one-dimensional harmonic oscillator eigenfunction.
• Single-electron Hamiltonian.
• Specific enthalpy.

$\hbar$

The reduced Planck constant, equal to 1.054,57 10$\POW9,{-34}$ Js. A measure of the uncertainty of nature in quantum mechanics. Multiply by $2\pi$ to get the original Planck constant $h$. For nuclear physics, a frequently helpful value is $\hbar{c}$ = 197.329 MeV fm.

$I$

May indicate:

• The number of electrons or particles.
• Electrical current.
• Unit matrix or operator, which does not do anything. See the entry on matrix.
• $I_{\rm A}$ is Avogadro’s number, 6.022,1 10$\POW9,{26}$ particles per kmol. (More standard symbols are $N_{\rm {A}}$ or $L$, but they are incompatible with the general notations in this book.)

$\Im$

The imaginary part of a complex number. If $c=c_r+{{\rm i}}c_i$ with $c_r$ and $c_i$ real numbers, then $\Im(c)=c_i$. Note that $c-c^*=2{\rm i}\Im(c)$.

${\cal I}$

May indicate:

• ${\cal I}$ is radiation energy intensity.
• ${\cal I}_{\rm {R}}$ is moment of inertia.

$i$

May indicate:

• The number of a particle.
• A summation index.
• A generic index or counter.

Not to be confused with ${\rm i}$.

${\rm i}$

The standard square root of minus one: ${\rm i}=\sqrt{-1}$, ${\rm i}^2 = -1$, $1/{\rm i}=-{\rm i}$, ${\rm i}^*=-{\rm i}$.

${\hat\imath}$

${\hat\imath}$ is the unit vector in the $x$-direction.

index notation

A more concise and powerful way of writing vector and matrix components by using a numerical index to indicate the components. For Cartesian coordinates, we might number the coordinates $x$ as 1, $y$ as 2, and $z$ as 3. In that case, a sum like $v_x+v_y+v_z$ can be more concisely written as $\sum_i v_i$. And a statement like $v_x\ne0,v_y\ne0,v_z\ne0$ can be more compactly written as $v_i\ne0$. To really see how it simplifies the notations, have a look at the matrix entry. (And that one shows only 2 by 2 matrices. Just imagine 100 by 100 matrices.)

iff

Emphatic “if.” Should be read as “if and only if.”

integer

Integer numbers are the whole numbers: $\ldots,-2,-1,0,1,2,3,4,\ldots$.

inverse

(Of matrices or operators.) If an operator $A$ converts a vector or function $f$ into a vector or function $g$, then the inverse of the operator $A^{-1}$ converts $g$ back into $f$. For example, the operator 2 converts vectors or functions into two times themselves, and its inverse operator $\frac12$ converts these back into the originals. Some operators do not have inverses. For example, the operator 0 converts all vectors or functions into zero. But given zero, there is no way to figure out what function or vector it came from; the inverse operator does not exist.

irrotational

A vector $\vec{v}$ is irrotational if its curl $\nabla\times\vec{v}$ is zero.

iso

Means “equal” or “constant.”

• Isenthalpic: constant enthalpy.
• Isentropic: constant entropy. This is a process that is both adiabatic and reversible.
• Isobaric: constant pressure.
• Isochoric: constant (specific) volume.
• Isospin: you don’t want to know.
• Isothermal: constant temperature.

isolated

An isolated system is one that does not interact with its surroundings in any way. No heat is transfered with the surroundings, no work is done on or by the surroundings.

$J$

May indicate:

• Total angular momentum.
• Number of nuclei in a quantum computation of electronic structure.

$j$

May indicate:

• The azimuthal quantum number of total angular momentum, including both orbital and spin contributions.
• $\vec j$ is electric current density.
• The number of a nucleus in a quantum computation.
• A summation index.
• A generic index or counter.

${\hat\jmath}$

${\hat\jmath}$ is the unit vector in the $y$-direction.

K

May indicate:

• The atomic states or orbitals with theoretical Bohr energy $E_1$
• Degrees Kelvin.

$K$

May indicate:

• An exchange integral in Hartree-Fock.
• Maximum wave number value.

${\cal K}$

Thomson (Kelvin) coefficient.

$k$

May indicate:

• A wave number. A wave number is a measure for how fast a periodic function oscillates with variations in spatial position. In quantum mechanics, $k$ is normally defined as $\sqrt{2m(E-V)}/\hbar$. The vector ${\vec k}$ is not to be confused with the unit vector in the $z$-direction ${\hat k}$.
• A summation index.

${\hat k}$

${\hat k}$ is the unit vector in the $z$-direction.

$k_{\rm B}$

Boltzmann constant. Equal to 1.380,65 10$\POW9,{-23}$ J/K. Relates absolute temperature to a typical unit of heat motion energy.

kmol

A kilo mole refers to 6.022,1 10$\POW9,{26}$ atoms or molecules. The weight of this many particles is about the number of protons and neutrons in the atom nucleus/molecule nuclei. So a kmol of hydrogen atoms has a mass of about 1 kg, and a kmol of hydrogen molecules about 2 kg. A kmol of helium atoms has a mass of about 4 kg, since helium has two protons and two neutrons in its nucleus. These numbers are not very accurate, not just because the electron masses are ignored, and the free neutron and proton masses are somewhat different, but also because of relativity effects that cause actual nuclear masses to deviate from the sum of the free proton and neutron masses.

L

The atomic states or orbitals with theoretical Bohr energy $E_2$

$L$

May indicate:

• Angular momentum.
• Orbital angular momentum.

${\cal L}$

Lagrangian.

$l$

May indicate:

• The azimuthal quantum number of angular momentum.
• The azimuthal quantum number of orbital angular momentum. Here $s$ is used for spin, and $j$ for combined angular momentum.)
• A generic summation index.

$\ell$

May indicate:

• The typical length in the harmonic oscillator problem.
• The dimensions of a solid block (with subscripts).
• A length.
• Multipole level in transitions.

$\lim$

Indicates the final result of an approaching process. $\lim_{\varepsilon\to 0}$ indicates for practical purposes the value of the following expression when $\varepsilon$ is extremely small.

linear combination

A very generic concept indicating sums of objects times coefficients. For example, a position vector ${\skew0\vec r}$ in basic physics is the linear combination $x{\hat\imath}+y{\hat\jmath}+z{\hat k}$ with the objects the unit vectors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ and the coefficients the position coordinates $x$, $y$, and $z$.

M

The atomic states or orbitals with theoretical Bohr energy $E_3$

$M$

May indicate:

• Molecular mass. See separate entry.
• Mirror operator.
• Figure of merit.

$m$

May indicate:

• Mass.
  • $m_{\rm e}$: electron mass. Equal to 9.109,382 10$\POW9,{-31}$ kg. The rest mass energy is 0.510,998,910 MeV.
  • $m_{\rm p}$: proton mass. Equal to 1.672,621 10$\POW9,{-27}$ kg. The rest mass energy is 938.272,013 MeV.
  • $m_{\rm n}$: neutron mass. Equal to 1.674,927 10$\POW9,{-27}$ kg. The rest mass energy is 939.565,561 MeV.
  • $m$: particle mass.
• The magnetic quantum number of angular momentum. The type odf angular momentum may be indicated by a subscript $l$ for orbital, $s$ for spin, or $j$ for net (orbital plus spin).
• Number of a single-electron wave function.
• A generic summation index or generic integer.

matrix

A table of numbers.

As a simple example, a two-dimensional matrix $A$ is a table of four numbers called $a_{11}$, $a_{12}$, $a_{21}$, and $a_{22}$:

$$\left( \begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)$$

unlike a two-dimensional (ket) vector $\vec v$, which would consist of only two numbers $v_1$ and $v_2$ arranged in a column:

$$\left( \begin{array}{l} v_1 \\ v_2 \end{array} \right)$$

(Such a vector can be seen as a “rectangular matrix” of size $2\times1$, but let’s not get into that.)

In index notation, a matrix $A$ is a set of numbers $\{a_{ij}\}$ indexed by two indices. The first index $i$ is the row number, the second index $j$ is the column number. A matrix turns a vector $\vec v$ into another vector $\vec w$ according to the recipe

$$w_i = \sum_{\mbox{{\scriptsize all }}j} a_{ij} v_j \quad \mbox{for all $i$}$$

where $v_j$ stands for “the $j$-th component of vector $\vec v$,” and $w_i$ for “the $i$-th component of vector $\vec w$.”

As an example, the product of $A$ and $\vec v$ above is by definition

$$\left( \begin{array}{ll} a_{11} & a_{12} \\ a_{21} ... ...2} v_2 \\ a_{21} v_1 + a_{22} v_2 \end{array} \right)$$

which is another two-dimensional ket vector.

Note that in matrix multiplications like the example above, in geometric terms we take dot products between the rows of the first factor and the column of the second factor.

To multiply two matrices together, just think of the columns of the second matrix as separate vectors. For example:

$$\left( \begin{array}{ll} a_{11} & a_{12} \\ a_{21} ... ...{21} & a_{21} b_{12} + a_{22} b_{22} \end{array} \right)$$

which is another two-dimensional matrix. In index notation, the $ij$ component of the product matrix has value $\sum_k a_{ik}b_{kj}$.

The zero matrix is like the number zero; it does not change a matrix it is added to and turns whatever it is multiplied with into zero. A zero matrix is zero everywhere. In two dimensions:

$$\left( \begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array} \right)$$

A unit matrix is the equivalent of the number one for matrices; it does not change the quantity it is multiplied with. A unit matrix is one on its “main diagonal” and zero elsewhere. The 2 by 2 unit matrix is:

$$\left( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} \right)$$

More generally the coefficients, $\{\delta_{ij}\}$, of a unit matrix are one if $i=j$ and zero otherwise.

The transpose of a matrix $A$, $A^{\rm {T}}$, is what you get if you switch the two indices. Graphically, it turns its rows into its columns and vice versa. The Hermitian “adjoint”$A^\dagger$ is what you get if you switch the two indices and then take the complex conjugate of every element. If you want to take a matrix to the other side of an inner product, you will need to change it to its Hermitian adjoint. “Hermitian matrices”are equal to their Hermitian adjoint, so this does nothing for them.

See also “determinant” and “eigenvector.”

metric prefixes

In the metric system, the prefixes Y, Z, E, P, T, G, M, and k stand for 10$\POW9,{i}$ with $i$ = 24, 21, 18, 15, 12, 9, 6, and 3, respectively. Similarly, d, c, m, $\mu$, n, p, f, a, z, y stand for 10$\POW9,{-i}$ with $i$ = 1, 2, 3, 6, 9, 12, 15, 18, 21, and 24 respectively. For example, 1 ns is 10$\POW9,{-9}$ seconds. Corresponding names are yotta, zetta, exa, peta, tera, giga, mega, kilo, deci, centi, milli, micro, nano, pico, femto, atto, zepto, and yocto.

molecular mass

Typical thermodynamics books for engineers tabulate values of the “molecular mass,” as a nondimensional number. The bottom line first: these numbers should have been called the “molar mass” of the substance, for the naturally occurring isotope ratio on earth. And they should have been given units of kg/kmol. That is how you use these numbers in actual computations. So just ignore the fact that what these books really tabulate is officially called the “relative molecular mass” for the natural isotope ratio.

Don’t blame these textbooks too much for making a mess of things. Physicists have historically bandied about a zillion different names for what is essentially a single number. “Molecular mass,” “relative molecular mass,” “molecular weight,” “atomic mass,” “relative atomic mass,” “atomic weight,” “molar mass,” “relative molar mass,” etcetera are basically all the same thing.

All of these have values that equal the mass of a molecule relative to a reference value for a single nucleon. So these value are about equal to the number of nucleons (protons and neutrons) in the nuclei of a single molecule. (For an isotope ratio, that becomes the average number of nucleons. Do note that nuclei are sufficiently relativistic that a proton or neutron can be noticeably heavier in one nucleus than another, and that neutrons are a bit heavier than protons even in isolation.) The official reference nucleon weight is defined based on the most common carbon isotope carbon-12. Since carbon-12 has 6 protons plus 6 neutrons, the reference nucleon weight is taken to be one twelfth of the carbon-12 atomic weight. That is called the unified atomic mass unit (u) or Dalton (Da). The atomic mass unit (amu) is an older virtually identical unit, but physicists and chemists could never quite agree on what its value was. No kidding.

If you want to be politically correct, the deal is as follows. “Molecular mass” is just what the term says, the mass of a molecule, in mass units. (I found zero evidence in either the IUPAC Gold Book or NIST SP811 for the claim of Wikipedia that it must always be expressed in u.) “Molar mass” is just what the words says, the mass of a mole. Official SI units are kg/mol, but you will find it in g/mol, equivalent to kg/kmol. (You cannot expect enough brains from international committees to realize that if you define the kg and not the g as unit of mass, then it would be a smart idea to also define kmol instead of mol as unit of particle count.) Simply ignore relative atomic and molecular masses, you do not care about them. (I found zero evidence in either the IUPAC Gold Book or NIST SP811 for the claims of Wikipedia that the molecular mass cannot be an average over isotopes or that the molar mass must be for a natural isotope ratio. In fact, NIST uses “molar mass of carbon-12” and specifically includes the possibility of an average in the relative molecular mass.)

N

May indicate

• The atomic states or orbitals with theoretical Bohr energy $E_4$.
• Subscript N indicates a nucleus.

$N$

May indicate:

• Number of states.
• Number of single-particle states.
• Number of neutrons in a nucleus.

n

May indicate

• Subscript n indicates a neutron.

$n$

May indicate:

• The principal quantum number for hydrogen atom energy eigenfunctions.
• A quantum number for harmonic oscillator energy eigenfunctions.
• Number of a single-electron or single-particle wave function.
• Generic summation index over energy eigenfunctions.
• Generic summation index over other eigenfunctions.
• Integer factor in Fourier wave numbers.
• Probability density.
• A generic index.
• A natural number.
• $n_s$ is the number of spin states.

and maybe some other stuff.

natural

Natural numbers are the numbers: $1,2,3,4,\ldots$.

normal

A normal operator or matrix is one that has orthonormal eigenfunctions or eigenvectors. Since eigenvectors are not orthonormal in general, a normal operator or matrix is abnormal! Another example of a highly confusing term. Such a matrix should have been called orthodiagonalizable or something of the kind. To be fair, the author is not aware of any physicists being involved in this particular term; it may be the mathematicians that are to blame here.

For an operator or matrix $A$ to be “normal,” it must commute with its Hermitian adjoint, $[A,A^\dagger]=0$. Hermitian matrices are normal since they are equal to their Hermitian adjoint. Skew-Hermitian matrices are normal since they are equal to the negative of their Hermitian adjoint. Unitary matrices are normal because they are the inverse of their Hermitian adjoint.

opposite

The opposite of a number $a$ is $-a$. In other words, it is the additive inverse.

$P$

May indicate:

• The linear momentum eigenfunction.
• A power series solution.
• Probability.
• Pressure.
• Hermitian part of an annihilation operator.

${\cal P}$

Parity, or better inversion, operator that inverts the positive direction of all three Cartesian axes. It turns any position vector ${\skew0\vec r}$ into $-{\skew0\vec r}$. The parity operator mirrors nature and then rotates it 180$\POW9,{\circ}$ around the axis normal to the mirror. (Mathematicians might refer to any operator that performs a mirroring as a parity operator.)

p

May indicate

• Subscript p indicates a proton.

$p$

May indicate:

• Linear momentum.
• Linear momentum in the $x$-direction.
• Integration variable with units of linear momentum.

p

May indicate:

• Energy state with orbital azimuthal quantum number $l=1$.
• A superscript p may indicate a single-particle quantity.
• A subscript p may indicate a periodic function.

perpendicular bisector

For two given points $P$ and $Q$, the perpendicular bisector consists of all points $R$ that are equally far from $P$ as they are from $Q$. In two dimensions, the perpendicular bisector is the line that passes through the point exactly half way in between $P$ and $Q$, and that is orthogonal to the line connecting $P$ and $Q$. In three dimensions, the perpendicular bisector is the plane that passes through the point exactly half way in between $P$ and $Q$, and that is orthogonal to the line connecting $P$ and $Q$. In vector notation, the perpendicular bisector of points $P$ and $Q$ is all points $R$ whose radius vector ${\skew0\vec r}$ satisfies the equation:

$$({\skew0\vec r}-{\skew0\vec r}_P)\cdot({\skew0\vec r}_Q-{\... ..._Q-{\skew0\vec r}_P)\cdot({\skew0\vec r}_Q-{\skew0\vec r}_P)$$

(Note that the halfway point ${\skew0\vec r}-{\skew0\vec r}_P={\textstyle\frac{1}{2}}({\skew0\vec r}_Q-{\skew0\vec r}_P)$ is included in this formula, as is the half way point plus any vector that is normal to $({\skew0\vec r}_Q-{\skew0\vec r}_P)$.)

phase angle

Any complex number can be written in “polar form” as $c=\vert c\vert e^{{\rm i}\alpha}$ where both the magnitude $\vert c\vert$ and the phase angle $\alpha$ are real numbers. Note that when the phase angle varies from zero to $2\pi$, the complex number $c$ varies from positive real to positive imaginary to negative real to negative imaginary and back to positive real. When the complex number is plotted in the complex plane, the phase angle is the direction of the number relative to the origin. The phase angle $\alpha$ is often called the argument, but so is about everything else in mathematics, so that is not very helpful.

In complex time-dependent waves of the form $e^{{\rm i}({\omega}t-\phi)}$, and its real equivalent $\cos({\omega}t-\phi)$, the phase angle $\phi$ gives the angular argument of the wave at time zero.

photon

Unit of electromagnetic radiation (which includes light, x-rays, microwaves, etcetera). A photon has a energy $\hbar\omega$, where $\omega$ is its angular frequency, and a wave length $2\pi c/\omega$ where $c$ is the speed of light.

potential

In order to optimize confusion, pretty much everything in physics that is scalar is called potential. Potential energy is routinely concisely referred to as potential. It is the energy that a particle can pick up from a force field by changing its position. It is in Joule. But an electric potential is taken to be per unit charge, which gives it units of volts. Then there are thermodynamic potentials like the chemical potential.

$p_x$

Linear momentum in the $x$-direction. (In the one-dimensional cases at the end of the unsteady evolution chapter, the $x$ subscript is omitted.) Components in the $y$- and $z$-directions are $p_y$ and $p_z$. Classical Newtonian physics has $p_x=mu$ where $m$ is the mass and $u$ the velocity in the $x$-direction. In quantum mechanics, the possible values of $p_x$ are the eigenvalues of the operator $\widehat p_x$ which equals $\hbar\partial/{\rm i}\partial x$. (But which becomes canonical momentum in a magnetic field.)

$Q$

May indicate

• Number of energy eigenfunctions of a system of particles.
• Anti-Hermitian part of an annihilation operator divided by ${\rm i}$.
• Heat flow or heat.
• Charge.
• Electric quadrupole moment.
• Energy release.

$q$

May indicate:

• Charge.
• Heat flux density.
• The number of an energy eigenfunction of a system of particles.
• Generic index.

$R$

May indicate:

• Some function of $r$ to be determined.
• Some function of $(x,y,z)$ to be determined.
• Rotation operator.
• Ideal gas constant.
• Transition rate.
• Nuclear radius.
• $R_{nl}$ is a hydrogen radial wave function.
• $R_u$ = 8.314,472 kJ/kmol K is the universal gas constant, the equivalent of Boltzmann's constant for a kmol instead of a single atom or molecule.

$\Re$

The real part of a complex number. If $c=c_r+{{\rm i}}c_i$ with $c_r$ and $c_i$ real numbers, then $\Re(c)=c_r$. Note that $c+c^*=2\Re(c)$.

relativity

The special theory of relativity accounts for the experimental observation that the speed of light $c$ is the same in all local coordinate systems. It necessarily drops the basic concepts of absolute time and length that were corner stones in Newtonian physics.

Albert Einstein should be credited with the boldness to squarely face up to the unavoidable where others wavered. However, he should also be credited for the boldness of swiping the basic ideas from Lorentz and Poincaré without giving them proper, or any, credit. The evidence is very strong he was aware of both works, and his various arguments are almost carbon copies of those of Poincaré, but in his paper it looks like it all came from Einstein, with the existence of the earlier works not mentioned. (Note that the general theory of relativity, which is of no interest to this book, is almost surely properly credited to Einstein. But he was a lot less hungry then.)

Relativity implies that a length seen by an observer moving at a speed $v$ is shorter than the one seen by a stationary observer by a factor $\sqrt{1-(v/c)^2}$ assuming the length is in the direction of motion. This is called Lorentz-Fitzgerald contraction. It makes galactic travel somewhat more conceivable because the size of the galaxy will contract for an astronaut in a rocket ship moving close to the speed of light. Relativity also implies that the time that an event takes seems to be slower by a factor $1/\sqrt{1-(v/c)^2}$ if the event is seen by an observer in motion compared to the location where the event occurs. That is called time dilation. Some high-energy particles generated in space move so fast that they reach the surface of the earth though this takes much more time than the particles would last at rest in a laboratory. The decay time increases because of the motion of the particles. (Of course, as far as the particles themselves see it, the distance to travel is a lot shorter than it seems to be to earth. For them, it is a matter of length contraction.)

The following formulae give the relativistic mass, momentum, and kinetic energy of a particle in motion:

$$m= \frac{m_0}{\sqrt{1-(v/c)^2}} \qquad p = m v \qquad T = mc^2 - m_0c^2$$

where $m_0$ is the rest mass of the particle, i.e.  the mass as measured by an observer to whom the particle seems at rest. The formula for kinetic energy reflects the fact that even if a particle is at rest, it still has an amount of “build-in” energy equal to $m_0c^2$ left. The total energy of a particle in empty space, being kinetic and rest mass energy, is given by

$$E = m c^2 = \sqrt{(m_0c^2)^2 + c^2p^2}$$

as can be verified by substituting in the expression for the momentum, in terms of the rest mass, and then taking both terms inside the square root under a common denominator. For small linear momentum $p$, this can be approximated as $\frac12m_0v^2$.

Relativity seemed quite a dramatic departure of Newtonian physics when it developed. Then quantum mechanics started to emerge...

$r$

May indicate:

• The radial distance from the chosen origin of the coordinate system.
• $r_i$ typically indicates the $i$-th Cartesian component of the radius vector ${\skew0\vec r}$.
• Some ratio.

$\vec r$

The position vector. In Cartesian coordinates $(x,y,z)$ or $x{\hat\imath}+y{\hat\jmath}+z{\hat k}$. In spherical coordinates $r\hat\imath_r$. Its three Cartesian components may be indicated by $r_1,r_2,r_3$ or by $x,y,z$ or by $x_1,x_2,x_3$.

reciprocal

The reciprocal of a number $a$ is $1/a$. In other words, it is the multiplicative inverse.

rot

The rot of a vector $\vec{v}$ is defined as $\mathop{\rm curl}\nolimits \vec{v}=\mathop{\rm {rot}}\vec{v}=\nabla\times\vec{v}$.

$S$

May indicate:

• Number of states per unit volume.
• Number of states at a given energy level.
• Spin angular momentum (as an alternative to using $L$ or $J$ for generic angular momentum.)
• Entropy.
• $S_{12}$ is a factor in the so-called tensor potential of nucleons.

s

Energy state with orbital azimuthal quantum number $l=0$. Spherically symmetric.

$s$

May indicate:

• Spin value of a particle. Equals $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ for electrons, protons, and neutrons, is also half an odd natural number for other fermions, and is a nonnegative integer for bosons. It is the azimuthal quantum number $l$ due to spin.
• Specific entropy.
• As an index, shelf number.

scalar

A quantity that is not a vector, a quantity that is just a single number.

$\sin$

The sine function, a periodic function oscillating between 1 and -1 as shown in [39, pp. 40-]. Good to remember: $\cos^2 \alpha + \sin^2 \alpha=1$.

solenoidal

A vector $\vec{v}$ is solenoidal if its divergence $\nabla\cdot\vec{v}$ is zero.

spectrum

In this book, a spectrum normally means a plot of energy levels along the vertical axis. Often, the horizontal coordinate is used to indicate a second variable, such as the density of states or the particle velocity.

For light (photons), a spectrum can be obtained experimentally by sending the light through a prism. This separates the colors in the light, and each color means a particular energy of the photons.

The word spectrum is also often used in a more general mathematical sense, but not in this book as far as I can remember.

Stokes' Theorem

This theorem, first derived by Kelvin and first published by someone else I cannot recall, says that for any reasonably smoothly varying vector $\vec v$,

$$\int_A \left(\nabla \times \vec v\right) {\,\rm d}A = \oint \vec v \cdot {\rm d}\vec r$$

where the first integral is over any smooth surface area $A$ and the second integral is over the edge of that surface. How did Stokes get his name on it? He tortured his students with it, that’s why!

One important consequence of the Stokes theorem is for vector fields $\vec{v}$ that are “irrotational,” i.e. that have $\nabla\times\vec{v}$ = 0. Such fields can be written as

$$\vec v = \nabla f \qquad f({\skew0\vec r}) \equiv \int_{... ...erline{\skew0\vec r}})\cdot{\rm d}{\underline{\skew0\vec r}}$$

Here ${\skew0\vec r}_{\rm {ref}}$ is the position of an arbitrarily chosen reference point, usually the origin. The reason the field $\vec{v}$ can be written this way is the Stokes theorem. Because of the theorem, it does not make a difference along which path from ${\skew0\vec r}_{\rm {ref}}$ to ${\skew0\vec r}$ you integrate. (Any two paths give the same answer, as long as $\vec{v}$ is irrotational everywhere in between the paths.) So the definition of $f$ is unambiguous. And you can verify that the partial derivatives of $f$ give the components of $\vec{v}$ by approaching the final position ${\skew0\vec r}$ in the integration from the corresponding direction.

symmetry

Symmetries are operations under which an object does not change. For example, a human face is almost, but not completely, mirror symmetric: it looks almost the same in a mirror as when seen directly. The electrical field of a single point charge is spherically symmetric; it looks the same from whatever angle you look at it, just like a sphere does. A simple smooth glass (like a glass of water) is cylindrically symmetric; it looks the same whatever way you rotate it around its vertical axis.

$T$

May indicate:

• Absolute temperature. The absolute temperature in degrees K equals the temperature in centigrade plus 273.15. When the absolute temperature is zero, (i.e. at $-$273.15 $\POW9,{\circ}$C), nature is in the state of lowest possible energy.
• Kinetic energy. A hat indicates the associated operator. The operator is given by the Laplacian times $-\hbar^2/2m$.
• Isospin. A hat indicates the associated operator. A vector symbol or subscript distinguishes it from kinetic energy.
• Tesla. The unit of magnetic field strength, kg/C-s.

${\cal T}$

Translation operator that translates a wave function a given amount through space.

$t$

May indicate:

• Time.
• $t_t$ is the quantum number of square isospin.

temperature

A measure of the heat motion of the particles making up macroscopic objects. At absolute zero temperature, the particles are in the “ground state” of lowest possible energy.

triple product

A product of three vectors. There are two different versions:

• The scalar triple product $\vec{a}\cdot(\vec{b}\times\vec{c})$. In index notation,
  
  $$\vec a \cdot (\vec b\times\vec c) = \sum_i a_i(b_{{\over... ... b_{{\overline{\overline{\imath}}}} c_{{\overline{\imath}}})$$
  
  
  where ${\overline{\imath}}$ is the index following $i$ in the sequence 123123..., and ${\overline{\overline{\imath}}}$ the one preceding it. This triple product equals the determinant $\vert\vec{a}\vec{b}\vec{c}\vert$ formed with the three vectors. Geometrically, it is plus or minus the volume of the parallelepiped that has vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ as edges. Either way, as long as the vectors are normal vectors and not operators,
  
  $$\vec a \cdot (\vec b\times\vec c) = \vec b \cdot (\vec c\times\vec a) = \vec c \cdot (\vec a\times\vec b)$$
  
  
  and you can change the two sides of the dot product without changing the triple product, and/or you can change the sides in the vectorial product with a change of sign. If any of the vectors is an operator, use the index notation expression to work it out.
• The vectorial triple product $\vec{a}\times(\vec{b}\times\vec{c})$. In index notation, component number $i$ of this triple product is
  
  $$a_{{\overline{\imath}}} (b_ic_{{\overline{\imath}}}-b_{{\o... ...verline{\imath}}}}c_i-b_ic_{{\overline{\overline{\imath}}}})$$
  
  
  which may be rewritten as
  
  $$a_ib_ic_i + a_{{\overline{\imath}}}b_ic_{{\overline{\imath... ...e{\overline{\imath}}}} b_{{\overline{\overline{\imath}}}}c_i$$
  
  
  In particular, as long as the vectors are normal ones,
  
  $$\vec a \times(\vec b\times\vec c) = (\vec a\cdot\vec c)\vec b - (\vec a\cdot\vec b)\vec c$$
  
  

$U$

May indicate:

• The time shift operator: $U(\tau,t)$ changes the wave function $\Psi(\ldots;t)$ into $\Psi(\ldots;t+\tau)$. If the Hamiltonian is independent of time
  
  $$U(\tau,t) = U_\tau = e^{-{\rm i}H \tau/\hbar}$$
  
  

• A unitary operator, in other words one that does not change the magnitude of the wave function.
• Often used for energy, though not in this book.

u

May indicate the atomic mass unit, equivalent to 1.660,538,78 10$\POW9,{-27}$ kg or 931.494,03 MeV/$c^2$.

$u$

May indicate:

• The first velocity component in a Cartesian coordinate system.
• A complex coordinate in the derivation of spherical harmonics.
• An integration variable.

$V$

May indicate:

• The potential energy. $V$ is used interchangeably for the numerical values of the potential energy and for the operator that corresponds to multiplying by $V$. In other words, $\widehat V$ is simply written as $V$.

${\cal V}$

Volume.

$v$

May indicate:

• The second velocity component in a Cartesian coordinate system.
• Magnitude of a velocity (speed).
• $v$ is specific volume.
• A complex coordinate in the derivation of spherical harmonics.
• As $v^{\rm ee}$, a single electron pair potential.

$\vec v$

May indicate:

• Velocity vector.
• Generic vector.
• Summation index of a lattice potential.

vector

A list of numbers. A vector $\vec v$ in index notation is a set of numbers $\{v_i\}$ indexed by an index $i$. In normal three-dimensional Cartesian space, $i$ takes the values 1, 2, and 3, making the vector a list of three numbers, $v_1$, $v_2$, and $v_3$. These numbers are called the three components of $\vec v$. The list of numbers can be visualized as a column, and is then called a ket vector, or as a row, in which case it is called a bra vector. This convention indicates how multiplication should be conducted with them. A bra times a ket produces a single number, the dot product or inner product of the vectors:

$$\left(1,3,5\right)\left(\begin{array}{c}7\\ 11\\ 13\end{array}\right) = 1\;7 + 3\;11 + 5\;13 = 105$$

To turn a ket into a bra for purposes of taking inner products, write the complex conjugates of its components as a row.

vectorial product

An vectorial product, or cross product is a product of vectors that produces another vector. If

$$\vec c=\vec a\times\vec b,$$

it means in index notation that the $i$-th component of vector $\vec c$ is

$$c_i = a_{{\overline{\imath}}} b_{{\overline{\overline{\ima... ... - a_{{\overline{\overline{\imath}}}}b_{{\overline{\imath}}}$$

where ${\overline{\imath}}$ is the index following $i$ in the sequence 123123..., and ${\overline{\overline{\imath}}}$ the one preceding it. For example, $c_1$ will equal $a_2b_3-a_3b_2$.

W

May indicate:

• Watt, the SI unit of power.
• The W$\POW9,{\pm}$ are the charged carriers of the weak force. See also Z$\POW9,{0}$.
• W.u. stands for Weisskopf unit, a simple decay ballpark for gamma decay.

$w$

May indicate:

• The third velocity component in a Cartesian coordinate system.
• Weight factor.

$\vec w$

Generic vector.

$X$

Used in this book to indicate a function of $x$ to be determined.

$x$

May indicate:

• First coordinate in a Cartesian coordinate system.
• A generic argument of a function.
• An unknown value.

$Y$

Used in this book to indicate a function of $y$ to be determined.

$Y_l^m$

Spherical harmonic. Eigenfunction of both angular momentum in the $z$-direction and of total square angular momentum.

$y$

May indicate:

• Second coordinate in a Cartesian coordinate system.
• A generic argument of a function.

$Z$

May indicate:

• Atomic number (number of protons in the nucleus).
• Number of particles.
• Partition function.
• The Z$\POW9,{0}$ is the uncharged carrier of the weak force. See also W$\POW9,{\pm}$.
• Used in this book to indicate a function of $z$ to be determined.

$z$

May indicate:

• Third coordinate in a Cartesian coordinate system.
• A generic argument of a function.