## 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.

The quoted values of physical constants are usually taken from NIST CODATA in 2012 or later. The final digit of the listed value is normally doubtful. (It corresponds to the first nonzero digit of the standard deviation). Numbers ending in triple dots are exact and could be written down to more digits than listed if needed.

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, ...).

Multiplication symbol. May indicate:
• An emphatic multiplication.
• Multiplication continued on the next line or from the previous line.
• A vectorial product between vectors. In index notation, the -th component of equals

where is the index following in the sequence 123123..., and the one preceding it (or second following). Alternatively, evaluate the determinant

Might be used to indicate a factorial. Example: 5! 1 2 3 4 5 120.

The function that generalizes to noninteger values of is called the gamma function; . The gamma function generalization is due to, who else, Euler. (However, the fact that instead of is due to the idiocy of Legendre.) In Legendre-resistant notation,

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

where the value indicated by the dots becomes negligibly small for large . The function can be extended further to any complex value of , except the negative integer values of , where is infinite, but is then no longer positive. Euler’s integral can be done for by making the change of variables , producing the integral , or , which equals and the integral under the square root can be done analytically using polar coordinates. The result is that

To get , multiply by , since .

A double exclamation mark may mean every second item is skipped, e.g. 5!! 1 3 5. In general, /. Of course, 5!! should logically mean (5!)!. Logic would indicate that 5 3 1 should be indicated by something like 5!’. But what is logic in physics?

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.

A ket is used to indicate some state. For example, indicates an angular momentum state with azimuthal quantum number and magnetic quantum number . Similarly, is the spin-down state of a particle with spin . Other common ones are for the position eigenfunction , i.e. , for the 1s or hydrogen state, for the 2p or state, etcetera. In short, whatever can indicate some state can be pushed into a ket.

A bra is like a ket , but appears in the left side of inner products, instead of the right one.

Indicates the spin up state. Mathematically, equals the function which is by definition equal to 1 at and equal to 0 at . A spatial wave function multiplied by is a particle in that spatial state with its spin up. For multiple particles, the spins are listed with particle 1 first.

Indicates the spin down state. Mathematically, equals the function which is by definition equal to 0 at and equal to 1 at . A spatial wave function multiplied by is a particle in that spatial state with its spin down. For multiple particles, the spins are listed with particle 1 first.

Summation symbol. Example: if in three dimensional space a vector has components 2, 1, 4, then stands for 7.

One important thing to remember: the symbol used for the summation index does not make a difference: is exactly the same as . 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, is not something that depends on an index . It is just a combined simple number. Like 7 in the example above. It is commonly said that the summation index sums away.

(Not to be confused with further down.) Multiplication symbol. Example: if in three dimensional space a vector has components 2, 1, 4, then stands for 6.

One important thing to remember: the symbol used for the multiplications index does not make a difference: is exactly the same as . 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, is not something that depends on an index . It is just a combined simple number. Like 6 in the example above. It is commonly said that the multiplication index factors away. (By who?)

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

is the summation of over all infinitesimally small fragments that make up the entire -range. For example, equals 3 2 6; the average value of between 0 and 2 is 3, and the sum of all the infinitesimally small segments gives the total length 2 of the range in from 0 to 2.

One important thing to remember: the symbol used for the integration variable does not make a difference: is exactly the same as . 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 is not something that depends on a variable . It is just a combined number. Like 6 in the example above. It is commonly said that the integration variable integrates away.

May indicate:
• An approaching process. indicates for practical purposes the value of the expression following the when is extremely small. Similarly, indicates the value of the following expression when is extremely large.
• The fact that the left side leads to, or implies, the right-hand side.

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 , velocity , linear momentum , acceleration , force , angular momentum , etcetera.

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: 0, 1, , , .
• A small or modified quantity.
• A quantity per unit length.

The spatial differentiation operator nabla. In Cartesian coordinates:

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

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:

or in index notation

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 -th component of this vector is

where is the index following in the sequence 123123..., and the one preceding it (or the second following it).

The operator is called the Laplacian. In Cartesian coordinates:

Sometimes the Laplacian is indicated as . In relativistic index notation it is equal to , with maybe a minus sign depending on who you talk with.

In non Cartesian coordinates, don’t guess; look these operators up in a table book, [40, pp. 124-126]: . For example, in spherical coordinates,

 (N.2)

That allows the gradient of a scalar function , i.e. , to be found immediately. But if you apply on a vector, you have to be very careful because you also need to differentiate , , and . In particular, the correct divergence of a vector is
 (N.3)

The curl of the vector is
 (N.4)

Finally the Laplacian is:
 (N.5)

See also spherical coordinates.

Cylindrical coordinates are usually indicated as , and . Here is the Cartesian coordinate, while is the distance from the -axis and the angle around the axis. In two dimensions, i.e. without the terms, they are usually called polar coordinates. In cylindrical coordinates:

 (N.6)

 (N.7)

 (N.8)

 (N.9)

The D'Alembertian is defined as

where is a constant called the wave speed. In relativistic index notation, is equal to .

A superscript star normally indicates a complex conjugate. In the complex conjugate of a number, every is changed into a .

Less than.

Less than or equal.

May indicate:
• An inner product.
• An expectation value.

Greater than.

Greater than or equal.

May indicate:
• A grouping of terms in a formula.
• A commutator. For example, .

Equals sign. The quantity to the left is the same as the one to the right.

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

Indicates approximately equal. Read it as “is approximately equal to.”

Indicates approximately equal. Often used when the approximation applies only when something is small or large. Read it as is approximately equal to or as “is asymptotically equal to.”

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

(alpha) May indicate:
• The fine structure constant, /, equal to 7.297,352,570 10, or about 1/137, 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 summation index.
• Component index of a vector.

(beta) May indicate:
• A nuclear decay mode in which an electron () or positron () is emitted. Sometimes 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 .
• A summation index.

(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) 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.

(capital delta) May indicate:
• An increment in the quantity following it.
• A delta particle.
• Often used to indicate the Laplacian .

(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 .

One often important way to create a three-dimensional delta function in spherical coordinates is to take the Laplacian of the function 1/. Chapter 13.3 explains why. In two dimensions, take the Laplacian of / 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 , and that are too much to go in here.

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

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

Also used in relativistic index notation, chapter 1.2.5.

(epsilon) May indicate:
• is the permittivity of space. Equal to 8.854,187,817... 10 C/J m. The exact value is 1 10 C/J m, because of the exact SI definitions of ampere and speed of light.
• Scaled energy.
• Orbital energy.
• Lagrangian multiplier.
• A small quantity, if symbol is not available.

(variant of epsilon) May indicate:
• A very small quantity.
• The slop in energy conservation during a decay process.

(eta) May be used to indicate a -position of a particle.

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

(theta) May indicate:
• In spherical coordinates, the angle from the chosen axis, with apex at the origin.
• -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.

(variant of theta) An alternate symbol for .

(kappa) May indicate:
• A constant that physically corresponds to some wave number.
• A summation index.
• Thermal conductivity.

(Lambda) May indicate:
• Lorentz transformation matrix.

(lambda) May indicate:
• Wave length.
• Decay constant.
• A generic eigenvalue.
• Entry of a Lorentz transformation.
• Scaled square momentum.
• Some multiple of something.

(mu) May indicate:
• Magnetic dipole moment. Alpha particle: 0 (spin is zero). Deuteron: 0.433,073,49 10 J/T or 0.857,438,231 . Electron: 9.284,764,3 10 J/T or 1.001,159,652,180,8 . Helion: 1.074,617,49 10 J/T or 2.127,625,306 . Neutron: 0.966,236,5 10 J/T or 1.913,042,7 . Proton: 1.410,606,74 10 J/T or 2.792,847,36 . Triton: 1.504,609,45 10 J/T or 2.978,962,45 .
• / 9.274,009,7 10 J/T or 5.788,381,807 10 eV/T is the Bohr magneton.
• / 5.050,783,5 10 J/T or 3.152,451,261 10 eV/T is the nuclear magneton.
• A summation index.
• Chemical potential/molar Gibbs free energy.

(nu) May indicate:
• Electron neutrino.
• Scaled energy eigenfunction number in solids.
• A summation index.
• Strength of a delta function potential.

(xi) May indicate:
• Scaled argument of the one-dimensional harmonic oscillator eigenfunctions.
• -position of a particle.
• A summation or integration index.

(Oblique Pi) (Not to be confused with described higher up.) Parity operator. Replaces by . That is equivalent to a mirroring in a mirror through the origin, followed by a 180 rotation around the axis normal to the mirror.

(pi) May indicate:
• A geometrical constant with value 3.141,592,653,589,793,238,462... The area of a circle of radius is and its perimeter is . The volume of a sphere of radius is and its surface is . A 180 angle expressed in radians is . Note also that 1 and 1.
• A chemical bond that looks from the side like a p state.
• A particle involved in the forces keeping the nuclei of atoms together (-meson or pion for short).
• Parity.

Canonical momentum density.

(rho) May indicate:
• Electric charge per unit volume.
• Scaled radial coordinate.
• Radial coordinate.
• Eigenfunction of a rotation operator .
• Mass-base density.
• Energy density of electromagnetic radiation.

(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.
• 5.670,37 W/m K is the Stefan-Boltzmann

(tau) May indicate:
• A time or time interval.
• Life time or half life.
• Some coefficient.

(capital phi) May indicate:
• Some function of to be determined.
• The momentum-space wave function.
• Relativistic electromagnetic potential.

(phi) May indicate:
• In spherical coordinates, the angle around the chosen axis. Increasing by encircles the -axis exactly once.
• A phase angle.
• Something equivalent to an angle.
• Field operator annihilates a particle at position while creates one at that position.

(variant of phi) May indicate:
• A change in angle .
• An alternate symbol for .
• An electrostatic potential.
• An electrostatic quantum field.
• A hypothetical selectostatic quantum field.

(chi) May indicate
• Spinor component.
• Gauge function of electromagnetic field.

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

(psi) Typically used to indicate an energy eigenfunction. Depending on the system, indices may be added to distinguish different ones. In some cases might be used instead of 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 , not , with a constant of magnitude 1.

(Omega) May indicate:
• Solid angle. See angle and “spherical coordinates.”

(omega) May indicate:
• Angular frequency of the classical harmonic oscillator. Equal to where is the spring constant and the mass.
• Angular frequency of a system.
• Angular frequency of light waves.
• Perturbation frequency,
• Any quantity having units of frequency, 1/s.

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

Å
Ångstrom. Equal to 10 m.

May indicate:
• The value of a generic physical quantity with operator
• 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 or creation operator .
• Bohr radius of helium ion.

May indicate:
• Bohr radius, / or 0.529,177,210,9 Å, with Å 10 m. Comparable in size to atoms, and a good size to use to simplify various formulae.
• The initial value of a coefficient .

absolute
May indicate:
• The absolute value of a real number is indicated by . It equals is is positive or zero and if is negative.
• The absolute value of a complex number is indicated by . It equals the length of the number plotted as a vector in the complex plane. This simplifies to above definition if 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 C in degrees Centrigrade (Celsius). The SI absolute temperature scale is degrees Kelvin, K. Absolute zero temperature is 0 K, while 0 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 or 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-Hermitianoperators just change sign. Unitary operatorschange 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 rad is taken to be 360. 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 conical surface. This angle is defined in the following way: draw a sphere of unit radius centered at the apex of the conical surface. Then the solid angle is the area of the spherical surface that is inside the cone. Solid angles are in steradians. The cone does not need to be a circular one, (i.e. have a circular cross section), for this to apply. In fact, the most common case is the solid angle corresponding to an infinitesimal element of spherical coordinate system angles. In that case the surface of the unit sphere inside the conical surface is is approximately rectangular, with sides and . That makes the enclosed solid angle equal to .

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

May indicate:
• Magnetic field strength.

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 , , and are a basis for normal three-dimensional space. Every three-dimensional vector can be written as a linear combination of the three.

May indicate:
• A third matrix or operator.
• A variety of different constants.

C
Degrees Centigrade. A commonly used temperature scale that has the value 273.15 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.

May indicate:
• The speed of light, 299,792,458 m/s exactly (by definition of the velocity unit).
• 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 and ,

In words, the magnitude of an inner product is at most the magnitude (i.e. the length or norm) of times the one of . For example, if and are real vectors, the inner product is the dot product and you have , where is the length of vector and the one of , and 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 and are functions. To prove it, first recognize that may in general be a complex number, which according to (2.6) must take the form where is some real number whose value is not important, and that is its complex conjugate . Now, (yes, this is going to be some convoluted reasoning), look at

where is any real number. The above dot product gives the square magnitude of , so it can never be negative. But if you multiply out, you get

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

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.

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

curl
The curl of a vector is defined as .

May indicate:
• Difference in wave number values.

Primitive (translation) vector of a reciprocal lattice.

Density of states.

D
Often used to indicate a state with two units of orbital angular momentum.

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

Primitive (translation) vector of a crystal lattice.

Indicates a vanishingly small change or interval of the following variable. For example, can be thought of as a small segment of the -axis.

In three dimensions, is an infinitesimal volume element. The symbol means that you sum over all such infinitesimal volume elements.

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 with respect to is written as /, or also simply as . Note that the derivative of function is again a function of : a ratio can be found at every point . The derivative of a function with respect to is written as / to indicate that there are other variables, and , that do not vary.

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

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 1 matrix consisting of a single number, the determinant is simply that number:

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

so the determinant is in short. For 3 3 matrices, you have

and you already know how to work out those 2 2 determinants, so you now know how to do 3 3 determinants. Written out fully:

For 4 4 determinants,

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. You are allowed to use the following manipulations without seriously affecting the computed determinant:

1. You can transposethe matrix, i.e. change its columns into its rows.
2. You can create zeros in a row by subtracting a suitable multiple of another row.
3. You can also swap rows, as long as you remember that each time that you swap two rows, it will flip over the sign of the computed determinant.
4. You 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”, you can eliminate all matrix coefficients for which is greater than , and that makes evaluating the determinant pretty much trivial.

div(ergence)
The divergence of a vector is defined as .

May indicate:
• The total energy. Possible values are the eigenvalues of the Hamiltonian.
• / / / / may indicate the nonrelativistic (Bohr) energy levels of the hydrogen atom. The ground state energy equals -13.605,692,5 eV. This does not include relativistic and proton motion corrections.
• Internal energy of a substance.

May indicate:
• Electric field strength.

May indicate:
• The basis for the natural logarithms. Equal to 2.718,281,828,459... This number produces the exponential function , or , or in words to the power , whose derivative with respect to is again . If is a constant, then the derivative of is . Also, if is an ordinary real number, then is a complex number with magnitude 1.
• The magnitude of the charge of an electron or proton, equal to 1.602,176,57 10 C.
• Emissivity of electromagnetic radiation.
• Often used to indicate a unit vector.
• A superscript may indicate a single-electron quantity.
• Specific internal energy of a substance.

e
May indicate
• Subscript e may indicate an electron.

Assuming that is an ordinary real number, and a real variable, is a complex function of magnitude one. The derivative of with respect to is

eigenvector
A concept from linear algebra. A vector is an eigenvector of a matrix if is nonzero and for some number 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

has eigenvalues that are the two roots of the quadratic equation

The corresponding eigenvectors are

On occasion you may have to swap and to use these formulae. If and 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 1.602,176,57 10 J.

exponential function
A function of the form , also written as . See function and .

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 .
• Some function.
• Helmholtz free energy.

Fock operator.

May indicate:
• A generic function.
• A generic vector.
• A fraction.
• The resonance factor.
• Specific Helmholtz free energy.
• Frequency.

function
A mathematical object that associates values with other values. A function associates every value of with a value . For example, the function associates 0 with 0, with , 1 with 1, 2 with 4, 3 with 9, and more generally, any arbitrary value of with the square of that value . Similarly, function associates any arbitrary with its cube , associates any arbitrary with the sine of that value, etcetera.

One way of thinking of a function is as a procedure that allows you, whenever given a number, to compute another number.

A wave function associates each spatial position 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.

May indicate:
• Gibbs free energy.
• Newton’s constant of gravitation, 6.673,8 10 m/kg s.

May indicate:
• A second generic function or a second generic vector.
• The strength of gravity, by definition equal to 9.806,65 m/s 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 2.002,319,304,361,5. For the proton 5.585,694,71. For the neutron, based on the mass and charge of the proton, 3.826,085,5.
• 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 ,

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; 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 is defined as .

May indicate:
• The Hamiltonian, or total energy, operator. Its eigenvalues are indicated by .
• stands for the -th order Hermite polynomial.
• Enthalpy.

May indicate:
• The original Planck constant .
• is a one-dimensional harmonic oscillator eigenfunction.
• Single-electron Hamiltonian.
• Specific enthalpy.

The reduced Planck constant, equal to 1.054,571,73 10 J s. A measure of the uncertainty of nature in quantum mechanics. Multiply by to get the original Planck constant . For nuclear physics, a frequently helpful value is 197.326,972 MeV fm.

hypersphere
A hypersphere is the generalization of the normal three-di­men­sion­al sphere to -di­men­sion­al space. A sphere of radius in three-di­men­sion­al space consists of all points satisfying

where , , and are Cartesian coordinates with origin at the center of the sphere. Similarly a hypersphere in -di­men­sion­al space is defined as all points satisfying

So a two-di­men­sion­al hypersphere of radius is really just a circle of radius . A one-di­men­sion­al hypersphere is really just the line segment .

The volume” and surface “area of an -di­men­sion­al hypersphere is given by

(This is readily derived recursively. For a sphere of unit radius, note that the -di­men­sion­al volume is an integration of -di­men­sion­al volumes with respect to . Then renotate as and look up the resulting integral in a table book. The formula for the area follows because where is the distance from the origin.) In three dimensions, according to the above formula. That makes the three-di­men­sion­al volume ​3 equal to the actual volume of the sphere, and the three-di­men­sion­al area equal to the actual surface area. On the other hand in two dimensions, . That makes the two-di­men­sion­al volume really the area of the circle. Similarly the two-di­men­sion­al surface area is really the perimeter of the circle. In one dimensions and the volume is really the length of the interval, and the area 2 is really its number of end points.

Often the infinitesimal -di­men­sion­al volume element is needed. This is the infinitesimal integration element for integration over all coordinates. It is:

Specifically, in two dimensions:

while in three dimensions:

The expressions in parentheses are in polar coordinates, respectively in spherical coordinates.

May indicate:
• The number of electrons or particles.
• Electrical current.
• Unit matrix or operator, which does not do anything. See matrix.
• is Avogadro’s number, 6.022,141,3 10 particles per kmol. (More standard symbols are or , but they are incompatible with the general notations in this book.)

The imaginary part of a complex number. If with and real numbers, then . Note that .

May indicate:
• is radiation energy intensity.
• is moment of inertia.

May indicate:
• The number of a particle.
• A summation index.
• A generic index or counter.
Not to be confused with .

The unit vector in the -direction.

The standard square root of minus one: , 1, 1/ , .

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, you might number the coordinates as 1, as 2, and as 3. In that case, a sum like can be more concisely written as . And a statement like 0, 0, 0 can be more compactly written as 0. 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: .

inverse
(Of matrices or operators.) If an operator converts a vector or function into a vector or function , then the inverse of the operator converts back into . For example, the operator 2 converts vectors or functions into two times themselves, and its inverse operator 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 is irrotational if its curl 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.

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

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

The unit vector in the -direction.

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

Thomson (Kelvin) coefficient.

K
May indicate:
• The atomic states or orbitals with theoretical Bohr energy
• Degrees Kelvin.

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, is normally defined as /. The vector is not to be confused with the unit vector in the -direction .
• A generic summation index.

The unit vector in the -direction.

Boltzmann constant. Equal to 1.380,649 10 J/K. Relates absolute temperature to a typical unit of heat motion energy.

kmol
A kilo mole refers to 6.022,141,3 10 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.

May indicate:
• Angular momentum.
• Orbital angular momentum.

Lagrangian.

L
The atomic states or orbitals with theoretical Bohr energy

May indicate:
• The azimuthal quantum number of angular momentum.
• The azimuthal quantum number of orbital angular momentum. Here is used for spin, and for combined angular momentum.)
• A generic summation index.

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

Lagrangian density. This is best understood in the UK.

Indicates the final result of an approaching process. indicates for practical purposes the value of the following expression when is extremely small.

linear combination
A very generic concept indicating sums of objects times coefficients. For example, a position vector in basic physics is the linear combination with the objects the unit vectors , , and and the coefficients the position coordinates , , and . A linear combination of a set of functions would be the function

where are constants, i.e. independent of .

linear dependence
A set of vectors or functions is linearly dependent if at least one of the set can be expressed in terms of the others. Consider the example of a set of functions . This set is linearly dependent if

where at least one of the constants is nonzero. To see why, suppose that say is nonzero. Then you can divide by and rearrange to get

That expresses in terms of the other functions.

linear independence
A set of vectors or functions is linearly independent if none of the set can be expressed in terms of the others. Consider the example of a set of functions . This set is linearly independent if

only if every one of the constants is zero. To see why, assume that say could be expressed in terms of the others,

Then taking 1, , , ... , the condition above would be violated. So cannot be expressed in terms of the others.

May indicate:
• Molecular mass. See molecular mass.
• Figure of merit.

Mirror operator.

M
The atomic states or orbitals with theoretical Bohr energy

May indicate:
• Mass.
• : electron mass. Equal to 9.109,382,9 10 kg. The rest mass energy is 0.510,998,93 MeV.
• : proton mass. Equal to 1.672,621,78 10 kg. The rest mass energy is 938.272,013 MeV.
• : neutron mass. Equal to 1.674,927 10 kg. The rest mass energy is 939.565,561 MeV.
• Alpha particle: 6.644,656,8 10 kg or 3,727.379,24 MeV. Deuteron: 3.343,583,5 10 kg or 1,875.612,86 MeV. Helion: 5.006,412,3 10 kg or 2,808.391,482 MeV.
• 1.660,538,92 10 kg is the atomic mass constant.
• : generic particle mass.
• The magnetic quantum number of angular momentum. The type odf angular momentum may be indicated by a subscript for orbital, for spin, or for net (orbital plus spin).
• Number of a single-electron wave function.
• Number of rows in a matrix.
• A generic summation index or generic integer.

matrix
A table of numbers.

As a simple example, a two-dimensional matrix is a table of four numbers called , , , and :

unlike a two-dimensional (ket) vector , which would consist of only two numbers and arranged in a column:

(Such a vector can be seen as a rectangular matrix of size 2 1, but let’s not get into that.)

In index notation, a matrix is a set of numbers indexed by two indices. The first index is the row number, the second index is the column number. A matrix turns a vector into another vector according to the recipe

where stands for “the -th component of vector ,” and for “the -th component of vector .”

As an example, the product of and above is by definition

which is another two-dimensional ket vector.

Note that in matrix multiplications like the example above, in geometric terms you 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:

which is another two-dimensional matrix. In index notation, the component of the product matrix has value .

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:

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:

More generally the coefficients, , of a unit matrix are one if and zero otherwise.

The transpose of a matrix , , is what you get if you switch the two indices. Graphically, it turns its rows into its columns and vice versa. The Hermitian adjoint 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 matricesare 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 with 24, 21, 18, 15, 12, 9, 6, and 3, respectively. Similarly, d, c, m, , n, p, f, a, z, y stand for 10 with 1, 2, 3, 6, 9, 12, 15, 18, 21, and 24 respectively. For example, 1 ns is 10 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. Like 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.)

See also the atomic mass constant .

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

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

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.
• Number of columns in a matrix.
• A generic summation index or generic integer.
• A natural number.
• is the number of spin states.
and maybe some other stuff.

n
May indicate
• A subscript n may indicate a neutron.

natural
Natural numbers are the numbers: .

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 to be normal, it must commute with its Hermitian adjoint, 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.

O
May indicate the origin of the coordinate system.

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

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

Particle exchange operator. Exchanges the positions and spins of two identical particles.

Peltier coefficient.

P
Often used to indicate a state with one unit of orbital angular momentum.

May indicate:
• Linear momentum.
• Linear momentum in the -direction.
• Integration variable with units of linear momentum.

p
May indicate
• An energy state with orbital azimuthal quantum number 1.
• A superscript p may indicate a single-particle quantity.
• A subscript p may indicate a periodic function.
• A subscript p may indicate a proton.

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

(Note that the halfway point is included in this formula, as is the half way point plus any vector that is normal to .)

phase angle
Any complex number can be written in polar form as where both the magnitude and the phase angle are real numbers. Note that when the phase angle varies from zero to , the complex number 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 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 , and its real equivalent , the phase angle 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 , where is its angular frequency, and a wave length / where 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.

Linear momentum in the -direction. (In the one-dimensional cases at the end of the unsteady evolution chapter, the subscript is omitted.) Components in the - and -directions are and . Classical Newtonian physics has where is the mass and the velocity in the -direction. In quantum mechanics, the possible values of are the eigenvalues of the operator which equals /. (But which becomes canonical momentum in a magnetic field.)

May indicate
• Number of energy eigenfunctions of a system of particles.
• Anti-Hermitian part of an annihilation operator divided by .
• Heat flow or heat.
• Charge.
• Electric quadrupole moment.
• Energy release.

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

May indicate:
• Ideal gas constant.
• Transition rate.
• Nuclear radius.
• Reflection coefficient.
• Some radius or typical radius (like in the Yukawa potential).
• Some function of to be determined.
• Some function of to be determined.
• is a hydrogen radial wave function.
• 8.314,462 kJ/kmol K is the universal gas constant. It is the equivalent of Boltzmann's constant, but for a kmol instead of a single atom or molecule.

Rotation operator.

The real part of a complex number. If with and real numbers, then . Note that .

May indicate:
• The radial distance from the chosen origin of the coordinate system.
• typically indicates the -th Cartesian component of the radius vector .
• Some ratio.

The position vector. In Cartesian coordinates or . In spherical coordinates . Its three Cartesian components may be indicated by or by or by .

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

relativity
The special theory of relativity accounts for the experimental observation that the speed of light 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 is shorter than the one seen by a stationary observer by a factor 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 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:

where 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 left. The total energy of a particle in empty space, being kinetic and rest mass energy, is given by

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 , this can be approximated as .

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

rot
The rot of a vector is defined as .

May indicate:
• Number of states per unit volume.
• Number of states at a given energy level.
• Spin angular momentum (as an alternative to using or for generic angular momentum.)
• Entropy.
• is a factor in the so-called tensor potential of nucleons.

The action integral of Lagrangian mechanics, {A.1}

Seebeck coefficient.

S
Often used to indicate a state of zero orbital angular momentum.

May indicate:
• Spin value of a particle. Equals 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 due to spin.
• Specific entropy.
• As an index, shelf number.

s
May indicate:
• An energy state with orbital azimuthal quantum number 0. Such a state is spherically symmetric.

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

The sine function, a periodic function oscillating between 1 and -1 as shown in [40, pp. 40-]. Good to remember: 1 and and .

solenoidal
A vector is solenoidal if its divergence 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.

spherical coordinates
The spherical coordinates , , and of an arbitrary point P are defined as

In Cartesian coordinates, the unit vectors in the , , and directions are called , , and . Similarly, in spherical coordinates, the unit vectors in the , , and directions are called , , and . Here, say, the direction is defined as the direction of the change in position if you increase by an infinitesimally small amount while keeping and the same. Note therefore in particular that the direction of is the same as that of ; radially outward.

An arbitrary vector can be decomposed in components , , and along these unit vectors. In particular

Recall from calculus that in spherical coordinates, a volume integral of an arbitrary function takes the form

In other words, the volume element in spherical coordinates is

Often it is convenient of think of volume integrations as a two-step process: first perform an integration over the angular coordinates and . Physically, that integrates over spherical surfaces. Then perform an integration over to integrate all the spherical surfaces together. The combined infinitesimal angular integration element

is called the infinitesimal solid angle . In two-dimensional polar coordinates and , the equivalent would be the infinitesimal polar angle . Recall that , (in proper radians of course), equals the arclength of an infinitesimal part of the circle of integration divided by the circle radius. Similarly is the surface of an infinitesimal part of the sphere of integration divided by the square sphere radius.

See the entry for the gradient operator and Laplacian in spherical coordinates.

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 ,

where the first integral is over any smooth surface area 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 how!

One important consequence of the Stokes theorem is for vector fields that are irrotational, i.e. that have 0. Such fields can be written as

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

symmetry
A symmetry is an operation 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.

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 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 /.
• 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.

Translation operator that translates a wave function through space. The amount of translation is usually indicated by a subscript.

May indicate:
• Time.
• 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 . In index notation,

where is the index following in the sequence 123123..., and the one preceding it. This triple product equals the determinant formed with the three vectors. Geometrically, it is plus or minus the volume of the parallelepiped that has vectors , , and as edges. Either way, as long as the vectors are normal vectors and not operators,

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 . In index notation, component number of this triple product is

which may be rewritten as

In particular, as long as the vectors are normal ones,

May indicate:
• 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.

The time shift operator: changes the wave function into . If the Hamiltonian is independent of time

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

u
May indicate the atomic mass constant, equivalent to 1.660,538,92 10 kg or 931.494,06 MeV/.

May indicate:
• The potential energy. is used interchangeably for the numerical values of the potential energy and for the operator that corresponds to multiplying by . In other words, is simply written as .

Volume.

May indicate:
• The second velocity component in a Cartesian coordinate system.
• Magnitude of a velocity (speed).
• is specific volume.
• A complex coordinate in the derivation of spherical harmonics.
• As , a single electron pair potential.

May indicate:
• Velocity vector.
• Generic vector.
• Summation index of a lattice potential.

vector
Simply put, a list of numbers. A vector in index notation is a set of numbers indexed by an index . In normal three-dimensional Cartesian space, takes the values 1, 2, and 3, making the vector a list of three numbers, , , and . These numbers are called the three components of . 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:

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

Formal definitions of vectors vary, but real mathematicians will tell you that vectors are objects that can be manipulated in certain ways (addition and multiplication by a scalar). Some physicists define vectors as objects that transform in a certain way under coordinate transformation (one-di­men­sion­al tensors); that is not the same thing.

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

it means in index notation that the -th component of vector is

where is the index following in the sequence 123123..., and the one preceding it. For example, will equal .

W
May indicate:
• Watt, the SI unit of power.
• The W are the charged carriers of the weak force. See also Z.
• W.u. stands for Weisskopf unit, a simple decay ballpark for gamma decay.

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

Generic vector.

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

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

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

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

May indicate:
• Second coordinate in a Cartesian coordinate system.
• A second generic argument of a function.
• A second unknown value.

May indicate:
• Atomic number (number of protons in the nucleus).
• Number of particles.
• Partition function.
• The Z is the uncharged carrier of the weak force. See also W.
• Used in this book to indicate a function of to be determined.

May indicate:
• Third coordinate in a Cartesian coordinate system.
• A third generic argument of a function.
• A third unknown value.