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Complex number z ≡ r exp(iθ) multiplied by i gives z' = z×i = z exp(i π/2) (counter clockwise rotation over 90°). Division of z by i gives z". Division by i is multiplication by −i = exp(−i  π/2) (clockwise rotation over 90°).

Complex numbers in physics

Classical physics

Classical physics consists of classical mechanics, electromagnetic theory, and phenomenological thermodynamics. One can add Einstein's special and general theory of relativity to this list, although this theory, being formulated in the 20th century, is usually not referred to as "classical". In these four branches of physics the basic quantities and equations governing the behavior of the quantities are real.

Classical mechanics has three different, but equivalent, formulations. The oldest, due to Newton, deals with masses and position vectors of particles, which are real, as is time t. The first and second time derivatives of the position vectors enter Newton's equations and these are obviously real, too. The same is true for Lagrange's formulation of classical mechanics in terms of position vectors and velocities of particles and for Hamilton's formulation in terms of momenta and positions.

Maxwell equations, that constitute the basis of electromagnetic theory, are in terms of real vector operators (gradient, divergence, and curl) acting on real electric and magnetic fields.

Thermodynamics is concerned with concepts as internal energy, entropy, and work. Again, these properties are real.

The special theory of relativity is formulated in Minkowski space. Although this space is sometimes described as a 3-dimensional Euclidean space to which the axis ict (i is the imaginary unit, c is speed of light, t is time) is added as a fourth dimension, the role of i is non-essential. The imaginary unit is introduced as a pedestrian way to the computation of the indefinite, real, inner product that in Lorentz coordinates has the metric

${\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}},$

which obviously is real. In other words, Minkowski space is a space over the real field ℝ. The general theory of relativity is formulated over real differentiable manifolds that are locally Lorentzian. Further, the Einstein field equations contain mass distributions that are real.

So, although the classical branches of physics do not need complex numbers, this does not mean that these numbers cannot be useful. A very important mathematical technique, especially for those branches of physics where there is flow (of electricity, heat, or mass) is Fourier analysis. The Fourier series is most conveniently formulated in complex form. Although it would be possible to formulate it in real terms (expansion in terms of sines and cosines) this would be cumbersome, given the fact that the application of the usual trigonometric formulas for the multiplication of sines and cosines is so much more difficult than the corresponding multiplication of complex numbers. Especially electromagnetic theory makes heavy use of complex numbers, but it must be remembered that the final results, that are to be compared with observable quantities, are real.

Quantum physics

In quantum physics complex numbers are essential. In the oldest formulation, due to Heisenberg the imaginary unit appears in an essential way through the canonical commutation relation

$[p_{i},q_{j}]\equiv p_{i}q_{j}-q_{j}p_{i}=-i\hbar \delta _{ij},$

p_i and q_j are linear operators (matrices) representing the ith and jth component of the momentum and position of a particle, respectively,.

The time-dependent Schrödinger equation also contains i in an essential manner. For a free particle of mass m the equation reads

${\frac {\hbar }{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)=-i{\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t).$

This equation may be compared to the wave equation that appears in several branches of classical physics

$v^{2}\nabla ^{2}\Psi (\mathbf {r} ,t)={\frac {\partial ^{2}}{\partial t^{2}}}\Psi (\mathbf {r} ,t),$

where v is the velocity of the wave. It is clear from this similarity why Schrödinger's equation is sometimes called the wave equation of quantum mechanics. It is also clear that the essential difference between quantum physics and classical physics is the first-order time derivative including the imaginary unit. The classical equation is real and has on the right hand side a second derivative with respect to time.

The more general form of the Schrödinger equation is

$H\Psi (t)=i\hbar {\frac {\partial }{\partial t}}\Psi (t),$

where H is the operator representing the energy of the quantum system under consideration. If this energy is time-independent (no time-dependent external fields interact with the system), the equation can be separated, and the imaginary unit enters fairly trivially through a so-called phase factor,

$\Psi (t)=e^{-iEt/\hbar }\Phi \quad {\hbox{with}}\quad H\Phi =E\Phi .$

The second equation has the form of an operator eigenvalue equation. The eigenvalue E (one of the possible observable values of the energy) is real, which is a fairly deep consequence of the quantum laws. The time-independent function Φ can very often be chosen to be real. The exception being the case that H is not invariant under time-reversal. Indeed, since the time-reversal operator θ is anti-unitary, it follows that

$\theta H\theta ^{\dagger }{\bar {\Phi }}=E{\bar {\Phi }}$

where the bar indicates complex conjugation. Now, if H is invariant,

$\theta H\theta ^{\dagger }=H\Longrightarrow H{\bar {\Phi }}=E{\bar {\Phi }}\quad {\hbox{and}}\quad H\Phi =E\Phi ,$

then also the real linear combination $\Phi +{\bar {\Phi }}$ is an eigenfunction belonging to E, which means that the wave function may be chosen real. If H is not invariant, it usually is transformed into minus itself. Then $\Phi \;$ and ${\bar {\Phi }}$ belong to E and −E, respectively, so that they are essentially different and cannot be combined to real form. Time-reversal symmetry is usually broken by magnetic fields, which give rise to interactions linear in spin or orbital angular momentum.

Virial theorem

In mechanics, a virial of a stable system of n particles is a quantity proposed by Rudolf Clausius in 1870. The virial is defined by

${\tfrac {1}{2}}\sum _{i=1}^{n}\langle \mathbf {F} _{i}\cdot \mathbf {r} _{i}\rangle ,$

where F_i is the total force acting on the i th particle and r_i is the position of the i th particle; the dot stands for an inner product between the two 3-vectors. The angular brackets indicate a long-time average. The importance of the virial arises from the virial theorem, which connects the virial to the long-time average ⟨ T ⟩ of the total kinetic energy T of the n-particle system,

$2\langle T\rangle =-\sum _{i=1}^{n}\langle \mathbf {F} _{i}\cdot \mathbf {r} _{i}\rangle .$

Proof of the virial theorem

Consider the quantity G defined by

$G\equiv \sum _{i=1}^{n}\mathbf {p} _{i}\cdot \mathbf {r} _{i}\quad {\hbox{with}}\quad \mathbf {p} _{i}=m_{i}{\frac {d\mathbf {r} _{i}}{dt}}.$

The vector p_i is the momentum of particle i. Differentiate G with respect to time:

${\frac {dG}{dt}}=\sum _{i=1}^{n}\left[{\frac {d\mathbf {p} _{i}}{dt}}\cdot \mathbf {r} _{i}+\mathbf {p} _{i}\cdot {\frac {d\mathbf {r} _{i}}{dt}}\right]$

Use Newtons's second law and the definition of kinetic energy:

$\mathbf {F} _{i}={\frac {d\mathbf {p} _{i}}{dt}}\quad {\hbox{and}}\quad 2T_{i}=\mathbf {p} _{i}\cdot {\frac {d\mathbf {r} _{i}}{dt}}$

and it follows that

${\frac {dG}{dt}}=2T+\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \mathbf {r} _{i}\quad {\hbox{with}}\quad T\equiv \sum _{i=1}^{n}T_{i}.$

Averaging over time gives:

$\left\langle {\frac {dG}{dt}}\right\rangle \equiv {\frac {1}{T}}\int _{0}^{T}{\frac {dG}{dt}}dt={\frac {1}{T}}\left[G(T)-G(0)\right].$

If the system is stable, G at time t = 0 and at time t = T is finite. Hence, if T goes to infinity, the quantity on the right hand side goes to zero. Alternatively, if the system is periodic with period T, G(T) = G(0) and the right hand side will also vanish. Whatever the cause, we assume that the time average of the time derivative of G is zero, and hence

$2\langle T\rangle +\sum _{i=1}^{n}\langle \mathbf {F} _{i}\cdot \mathbf {r} _{i}\rangle =0,$

which proves the virial theorem.

Application

An interesting application arises when each particle experiences a potential V of the form

$V=Ar^{k}\quad {\hbox{with}}\quad r=(x^{2}+y^{2}+z^{2})^{1/2},$

where A is some constant (independent of space and time).

An example of such potential is given by Hooke's law with k = 2 and Coulomb's law with k = −1. The force derived from a potential is

$\mathbf {F} =-{\boldsymbol {\nabla }}V\equiv -\left({\frac {\partial V}{\partial x}},\;{\frac {\partial V}{\partial y}},\;{\frac {\partial V}{\partial z}}\right)$

Consider

${\frac {\partial V}{\partial x}}=a{\frac {\partial r^{k}}{\partial x}}=akr^{k-1}{\frac {\partial r}{\partial x}}=akr^{k-1}(x/r)=k{\frac {x}{r^{2}}}V\quad \Longrightarrow \mathbf {F} \quad =-kV{\frac {\mathbf {r} }{r^{2}}}.$

Then applying this for i = 1, … n,

$2\langle T\rangle =k\sum _{i=1}^{n}\left\langle V(\mathbf {r} _{i})\cdot {\frac {\mathbf {r} _{i}\cdot \mathbf {r} _{i}}{r_{i}^{2}}}\right\rangle =k\langle V\rangle \quad {\hbox{with}}\quad V=\sum _{i=1}V(\mathbf {r} _{i}).$

For instance, for a system of charged particles interacting through a Coulomb interaction:

$2\langle T\rangle =-\langle V\rangle .$

Quantum mechanics

The virial theorem holds also in quantum mechanics. Quantum mechanically the angular brackets indicate not a time-average, but an expectation value with respect to an exact stationary eigenstate of the Hamiltonian of the system. In order to prove it we consider an exact time-dependent wave function Ψ(t) and an arbitrary operator G(t).

$i{\frac {d}{dt}}\langle \Psi (t)|G|\Psi (t)\rangle =\langle -H\Psi (t)|G|\Psi (t)\rangle +\langle \Psi (t)|G|H\Psi (t)\rangle +\langle \Psi (t)|i{\frac {\partial G}{\partial t}}|\Psi (t)\rangle =\langle \Psi (t)|[G,H]|\Psi (t)\rangle +\langle \Psi (t)|i{\frac {\partial G}{\partial t}}|\Psi (t)\rangle$

Here the time-dependent Schrödinger equation (with ℏ = 1), the self-adjointness of H, and the definition of a commutator is used,

$i{\frac {d}{dt}}|\Psi (t)\rangle =H|\Psi (t)\rangle ,\quad \langle \Psi |H|\Psi \rangle =\langle H\Psi |\Psi \rangle ,\quad [G,H]=GH-HG.$

Consider next the case that G does not depend on time, and that Ψ(t) is a stationary state,

${\frac {\partial G}{\partial t}}=0,\quad |\Psi (t)\rangle =exp(-iEt)|\Psi (0)\rangle .$

Then

$0=\langle \Psi (0)|[G,H]|\Psi (0)\rangle \quad {\hbox{with}}\quad H|\Psi (0)\rangle =E|\Psi (0)\rangle .$

So far the result is valid for an arbitrary time-independent operator G. In order to obtain the virial theorem, we consider

$G=\sum _{i=1}^{n}\mathbf {p} _{i}\cdot \mathbf {r} _{i}\quad {\hbox{with}}\quad \mathbf {p} =-i{\boldsymbol {\nabla }}.$

If V is of the form

$V=\sum _{i=1}^{n}a_{i}r_{i}^{k}$

it folows that

$[\mathbf {p} _{j},V]=-ia_{j}k\mathbf {r} _{j}r_{j}^{k-1}+\sum _{i\neq j}a_{i}r_{i}^{k}\quad {\hbox{and}}\quad [\mathbf {p} _{j},T]=0.$

and

$[\mathbf {r} _{j},T]=$

That is, for a stable molecule with time-independent wave function 2⟨T ⟩ = −⟨V ⟩.