Complex Exponentials

Complex exponentials are the shared language of phases, oscillations, plane waves, Fourier analysis, and energy-eigenstate time evolution. The expression $e^{i\theta}$ is not just shorthand for trigonometry; it is the natural multiplicative notation for phase.

This page focuses on scalar complex exponentials. For exponentials of matrices and operators, see Matrix Functions and Exponentials.

For the complex-analytic viewpoint in which $e^z$ is an entire function, see Analytic Functions.

Euler Formula

Euler’s formula is

$$e^{i\theta} = \cos\theta+i\sin\theta.$$

Thus $e^{i\theta}$ lies on the unit circle:

$$\left\lvert e^{i\theta}\right\rvert =1.$$

Complex conjugation reverses the phase:

$$\left(e^{i\theta}\right)^* = e^{-i\theta}.$$

Phases multiply by adding angles:

$$e^{i\theta}e^{i\phi} = e^{i(\theta+\phi)}.$$

This is why complex exponentials are more compact than sine and cosine when phases are shifted, multiplied, or superposed.

Oscillations

The real and imaginary parts of $e^{i\omega t}$ are ordinary oscillations:

$$e^{i\omega t} = \cos(\omega t)+i\sin(\omega t).$$

The derivative is especially simple:

$$\frac{d}{dt}e^{i\omega t} = i\omega e^{i\omega t}.$$

This eigenfunction property is the algebraic reason complex exponentials simplify linear differential equations with constant coefficients.

Real oscillations can be recovered by taking real or imaginary parts. For example,

$$\cos(\omega t) = \frac12 \left( e^{i\omega t}+e^{-i\omega t} \right).$$

The complex notation is usually a calculation device; the physical observable may be real, or it may be a complex quantum amplitude whose modulus squared gives a probability.

Plane Waves

A one-dimensional plane wave has the form

$$e^{i(kx-\omega t)}.$$

In quantum mechanics, one often writes the same phase as

$$e^{i(px-Et)/\hbar},$$

with

$$p=\hbar k, \qquad E=\hbar\omega.$$

The exponent must be dimensionless. Momentum times position and energy times time both have units of action, so division by $\hbar$ makes the phase meaningful.

The spatial factor satisfies

$$\frac{d}{dx}e^{ikx} = ik e^{ikx},$$

so

$$-i\hbar\frac{d}{dx}e^{ikx} = \hbar k\,e^{ikx}.$$

This is why a plane wave is a momentum eigenfunction with momentum $p=\hbar k$. On the full real line, however, a plane wave is not square-normalizable; it is a generalized eigenfunction used through delta normalization, boxes, or wave packets.

Fourier Modes

Fourier analysis expands functions in complex exponential modes. On a periodic interval of length $L$, the normalized modes

$$e_n(x) = \frac{1}{\sqrt L} e^{2\pi i n x/L}, \qquad n\in\mathbb Z,$$

are orthonormal:

$$\int_0^L e_m(x)^*e_n(x)\,dx = \delta_{mn}.$$

On the real line, the analogous basis is continuous and must be handled with distributions. The site convention for position and momentum uses phases $e^{\pm ipx/\hbar}$; see Fourier Transform and Fourier Transform Conventions.

Energy-Eigenstate Time Factors

For a time-independent Hamiltonian, an energy eigenstate evolves by

$$\lvert E;t\rangle = e^{-iEt/\hbar} \lvert E\rangle.$$

The phase has unit magnitude:

$$\left\lvert e^{-iEt/\hbar}\right\rvert =1.$$

Thus a single energy eigenstate has time-independent probabilities for time-independent measurements, even though its vector representative changes by a phase.

In a superposition,

$$\lvert\psi(t)\rangle = c_1e^{-iE_1t/\hbar}\lvert E_1\rangle + c_2e^{-iE_2t/\hbar}\lvert E_2\rangle,$$

the relative phase

$$e^{-i(E_2-E_1)t/\hbar}$$

can affect interference, expectation values, and transition amplitudes. A global phase is usually unobservable; relative phases are not.

Decay Versus Phase

Not every exponential is a phase. If the exponent has a real part,

$$e^{(-\kappa+ik)x} = e^{-\kappa x}e^{ikx}, \qquad \kappa>0,$$

then the factor has both decay and oscillation. Bound-state tails and classically forbidden regions often use real decaying exponentials such as $e^{-\kappa x}$, while allowed-region waves use oscillatory phases such as $e^{ikx}$.

Keeping this distinction clear prevents confusing probability-conserving phase evolution with exponential growth or decay.

Common Mistakes

• Writing $e^{ipx}$ when the intended dimensionless phase is $e^{ipx/\hbar}$.
• Treating a global phase as observable while forgetting that relative phases can be observable.
• Confusing a unit-magnitude phase $e^{i\theta}$ with a decaying exponential $e^{-\kappa x}$.
• Forgetting the sign convention in Fourier transforms.
• Normalizing a full-line plane wave as if it were an ordinary $L^2$ wavefunction.
• Assuming $e^{A+B}=e^Ae^B$ for operators; that requires commutation and belongs to matrix/operator exponentials.

Cross-Links

• Complex Numbers
• Analytic Functions
• Fourier Series
• Fourier Transform
• Momentum Representation
• Fourier Transform Conventions
• Matrix Functions and Exponentials
• Stationary States
• Time-Evolution Operator
• Superposition and Relative Phase

References

• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
• G. B. Folland, Fourier Analysis and Its Applications, American Mathematical Society, 1992.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. Show that $\left\lvert e^{i\theta}\right\rvert=1$.

Solution

Using Euler’s formula,

$$e^{i\theta} = \cos\theta+i\sin\theta.$$

Therefore

$$\left\lvert e^{i\theta}\right\rvert^2 = \cos^2\theta+\sin^2\theta =1.$$

Since the magnitude is nonnegative, $\left\lvert e^{i\theta}\right\rvert=1$.

2. Verify that $e^{ikx}$ is an eigenfunction of the momentum operator $-i\hbar d/dx$.

Solution

Compute

$$-i\hbar\frac{d}{dx}e^{ikx} = -i\hbar(ik)e^{ikx} = \hbar k e^{ikx}.$$

Thus the eigenvalue is $p=\hbar k$.

3. Why does a single energy eigenstate have time-independent probability density even though it has the factor $e^{-iEt/\hbar}$?

Solution

The phase has unit magnitude:

$$\left\lvert e^{-iEt/\hbar}\right\rvert^2 =1.$$

Therefore multiplying a wavefunction by this phase does not change $\lvert\psi(x,t)\rvert^2$. Superpositions of different energies can still show time-dependent interference because their relative phases change.

4. A table writes a wave as $e^{i(kx+\omega t)}$ instead of $e^{i(kx-\omega t)}$. What should you check before using it?

Solution

Check the sign convention for time dependence, Fourier transforms, and the definitions of positive frequency and outgoing or incoming waves. Different sign conventions can be consistent, but mixing them changes phases, propagation direction assignments, and Green-function prescriptions.