Superposition of Wavefunctions: Meaning and Examples

A superposition of wavefunctions is a new quantum state formed by adding possible states with complex coefficients. If $\psi_1$ and $\psi_2$ are wavefunctions for the same physical system, then

$$\psi = c_1\psi_1+c_2\psi_2$$

is also a possible state, as long as it is normalized and obeys the relevant boundary conditions. The coefficients $c_1$ and $c_2$ are probability amplitudes, not ordinary probabilities. Their squared magnitudes determine measurement probabilities only in an appropriate orthonormal basis.

Why addition is allowed

The time-dependent Schrödinger equation is linear. For a fixed Hamiltonian $\hat H$,

$$i\hbar\frac{\partial \psi}{\partial t}=\hat H\psi.$$

If $\psi_1$ and $\psi_2$ both solve this equation, then their linear combination does too:

$$\hat H(c_1\psi_1+c_2\psi_2)=c_1\hat H\psi_1+c_2\hat H\psi_2.$$

This mathematical property is the basis of quantum superposition. It does not mean that two classical waves or two particles have necessarily been physically placed on top of one another. It means that the state of one quantum system can contain several possible components at once.

For position-space wavefunctions, the combination is pointwise:

$$\psi(x,t)=c_1\psi_1(x,t)+c_2\psi_2(x,t).$$

The associated position probability density is

$$P(x,t)=|\psi(x,t)|^2.$$

Because the modulus is taken after adding the amplitudes, this generally is not equal to $|c_1\psi_1|^2+|c_2\psi_2|^2$. Expanding it gives

$$\begin{aligned} |\psi|^2 &=|c_1\psi_1+c_2\psi_2|^2\\ &=|c_1|^2|\psi_1|^2+|c_2|^2|\psi_2|^2\\ &\quad+c_1^*c_2\psi_1^*\psi_2+c_2^*c_1\psi_2^*\psi_1. \end{aligned}$$

The last two terms are interference terms. They can increase or decrease the probability in different regions.

Normalization

A wavefunction is normalized when the total probability is one:

$$\int_{-\infty}^{\infty}|\psi(x,t)|^2\,dx=1.$$

Suppose $\psi_1$ and $\psi_2$ are individually normalized and orthogonal, meaning

$$\int\psi_1^*(x)\psi_2(x)\,dx=0.$$

Then the superposition

$$\psi=c_1\psi_1+c_2\psi_2$$

is normalized precisely when

$$|c_1|^2+|c_2|^2=1.$$

For example,

$$\psi=\frac{1}{\sqrt{2}}\psi_1+\frac{i}{\sqrt{2}}\psi_2$$

has equal probabilities of being found in the $\psi_1$ and $\psi_2$ basis states: each probability is $1/2$. The factor $i$ does not change those two probabilities, but it can affect interference in another measurement basis.

If the states are not orthogonal, the simple condition $|c_1|^2+|c_2|^2=1$ is not sufficient. The norm is

$$\langle\psi|\psi\rangle=|c_1|^2+|c_2|^2+c_1^*c_2\langle\psi_1|\psi_2\rangle+c_2^*c_1\langle\psi_2|\psi_1\rangle.$$

In that case, choose an overall normalization factor so that this expression equals one, provided the unnormalized combination is nonzero.

Worked example: a two-level system

Consider an atom with two orthonormal energy eigenstates, $|g\rangle$ and $|e\rangle$. A possible state is

$$|\psi\rangle=\frac{1}{\sqrt{3}}|g\rangle+\sqrt{\frac{2}{3}}|e\rangle.$$

A measurement of energy gives

$$P(g)=\left|\frac{1}{\sqrt{3}}\right|^2=\frac{1}{3}, \qquad P(e)=\left|\sqrt{\frac{2}{3}}\right|^2=\frac{2}{3}.$$

Before the measurement, the system is not assigned one of these two energies with an ordinary hidden probability in this description. It is described by the combined state. After an ideal energy measurement, the state is projected into either $|g\rangle$ or $|e\rangle$, with the probabilities above.

If the energy eigenvalues are $E_g$ and $E_e$, time evolution adds phases:

$$|\psi(t)\rangle=\frac{1}{\sqrt{3}}e^{-iE_gt/\hbar}|g\rangle+\sqrt{\frac{2}{3}}e^{-iE_et/\hbar}|e\rangle.$$

The individual phase factors do not change energy probabilities. However, their relative phase can matter when measuring another observable, such as a transition probability or spin component.

Worked example: two spatial alternatives

Let two localized wave packets be $\psi_L(x)$ and $\psi_R(x)$, representing a particle traveling through a left and a right slit. For equal-amplitude, indistinguishable paths, and assuming the two packets are orthogonal or that any small overlap can be neglected, the state can be approximated by

$$\psi(x)=\frac{1}{\sqrt{2}}\bigl(\psi_L(x)+\psi_R(x)\bigr),$$

The detection probability is

$$|\psi(x)|^2=\frac{1}{2}|\psi_L(x)|^2+\frac{1}{2}|\psi_R(x)|^2+\operatorname{Re}\bigl(\psi_L^*(x)\psi_R(x)\bigr).$$

The final term produces bright and dark interference fringes. If an interaction records which slit the particle used, the alternatives become entangled with that record. The interference term is then reduced or disappears from the particle’s observed pattern. This is why superposition is connected to indistinguishable alternatives and interference, not simply to two visible objects occupying the same place.

Superposition and energy eigenstates

If $\psi_1$ and $\psi_2$ are energy eigenstates with different energies, their superposition is generally not an energy eigenstate. It has a definite time-dependent state but no single definite energy. Its expected energy is

$$\langle H\rangle=\langle\psi|\hat H|\psi\rangle.$$

For orthonormal eigenstates and a normalized state $|\psi\rangle=c_1|E_1\rangle+c_2|E_2\rangle$,

$$\langle H\rangle=|c_1|^2E_1+|c_2|^2E_2.$$

This expectation value is an average over many identically prepared measurements; it does not mean that one measurement returns an intermediate energy. An energy measurement returns one of the allowed eigenvalues, with the corresponding probabilities.

Common mistakes

• Adding probabilities instead of amplitudes: Add $\psi_1$ and $\psi_2$ first, then calculate $|\psi|^2$.
• Ignoring normalization: Arbitrary coefficients do not automatically produce a total probability of one.
• Treating coefficients as probabilities: $c_j$ is an amplitude; $|c_j|^2$ is a probability only when the states form an orthonormal measurement basis.
• Assuming every pair of functions is allowed: The functions must belong to the same state space and satisfy the system’s physical conditions.
• Calling a superposition a mixture: A coherent superposition has relative phases and can interfere. A statistical mixture has classical uncertainty and is represented by a density operator rather than one wavefunction.
• Assuming measurement reveals all components: A measurement projects onto the eigenstates of the chosen observable; it does not generally reveal the entire wavefunction in one trial.