# 1.99: Visualizing the Difference Between a Superposition and a Mixture

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

The superposition principle, as Feynman said, is at the heart of quantum mechanics. While its mathematical expression is simple, its true meaning is difficult to grasp. For example, given a linear superposition (not normalized) of two states,

$|\Psi\rangle=|\phi_{1}\rangle+\left|\phi_{2}\right\rangle \nonumber$

one might assume that it represents a mixture of $$\phi_{1}$$ and $$\phi_{2}$$. In other words, half of the quons  are in state $$\phi_{1}$$ and half in $$\phi_{2}$$. However, the correct quantum mechanical interpretation of this equation is that the system represented by $$\Psi$$ is simultaneously in the states $$\phi_{1}$$ and $$\phi_{2}$$, properly weighted.

A mixture, half $$\phi_{1}$$ and half $$\phi_{2}$$, or any other ratio, cannot be represented by a wavefunction. It requires a density operator, which is a more general quantum mechanical construct that can be used to represent both pure states (superpositions) and mixtures, as shown below.

$\hat{\rho}_{\max }=|\Psi\rangle\langle\Psi|\qquad \hat{\rho}_{\min d}=\sum p_{i}| \Psi_{i}\rangle\langle\Psi_{i}| \nonumber$

In the equation on the right, pi is the fraction of the mixture in the state $$\Psi_{i}$$.

To illustrate how these equations distinguish between a mixture and a superposition, we will consider a superposition and a mixture of equally weighted gaussian functions representing one-dimensional wave packets. The normalization constants are omitted in the interest of mathematical clarity. The gaussians are centered at x = $$\pm$$ 4.

$\phi_{1}(x) :=\exp \left[-(x+4)^{2}\right] \qquad \phi_{2}(x) :=\exp \left[-(x-4)^{2}\right] \nonumber$ To visualize how the density operator discriminates between a superposition and a mixture, we calculate its matrix elements in coordinate space for the 50-50 superposition and mixture of $$\phi_{1}$$ and $$\phi_{2}$$. The superposition is considered first.

$\Psi(x) :=\phi_{1}(x)+\phi_{2}(x) \nonumber$

The matrix elements of this pure state are calculated as follows.

$\rho_{\text {pure}}=\left\langle x|\hat{\rho}_{\text {pure}}| x^{\prime}\right\rangle=\langle x | \Psi\rangle\left\langle\Psi | x^{\prime}\right\rangle \nonumber$

Looking at the right side we see that the matrix elements are the product of the probability amplitudes of a quon in state $$\Psi$$ being at x and xʹ. Next we display the density matrix graphically.

$\operatorname{DensityMatrixPure}\left(x, x^{\prime}\right) :=\Psi(x) \cdot \Psi\left(x^{\prime}\right) \nonumber$

$x_{0}=8 \qquad N :=80 \qquad i :=0 \ldots N \\ \mathrm{x}_{\mathrm{i}} :=-\mathrm{x}_{0}+\frac{2 \cdot \mathrm{x}_{0} \cdot \mathrm{i}}{\mathrm{N}} \qquad \mathrm{j} :=0 \ldots \mathrm{N} \qquad \mathrm{x}_{\mathrm{j}}^{\prime} :=-\mathrm{x}_{0}+\frac{2 \cdot \mathrm{x}_{0} \cdot \mathrm{j}}{\mathrm{N}} \nonumber$

$\operatorname{DensityMatrixPure}_{\mathrm{i},\mathrm{j}} : = \operatorname{DensityMatrixPure}\left(x, x^{\prime}\right) \nonumber$ The presence of off-diagonal elements in this density matrix is the signature of a quantum mechanical superposition. For example, from the quantum mechanical perspective bi-location is possible.

Now we turn our attention to the density matrix of a mixture of gaussian states.

$\rho_{\operatorname{mix}}=\left\langle x\left|\hat{\rho}_{\operatorname{mix}}\right| x^{\prime}\right\rangle=\sum_{i} p_{i}\left\langle x | \phi_{i}\right\rangle\left\langle\phi_{i} | x^{\prime}\right\rangle=\frac{1}{2}\left\langle x | \phi_{1}\right\rangle\left\langle\phi_{1} | x^{\prime}\right\rangle+\frac{1}{2}\left\langle x | \phi_{2}\right\rangle\left\langle\phi_{2} | x^{\prime}\right\rangle \nonumber$

$\operatorname{DensityMatrixMix}(\mathrm{x}, \mathrm{x'}) :=\frac{\phi_{1}(\mathrm{x}) \cdot \phi_{1}(\mathrm{x'})+\phi_{2}(\mathrm{x}) \cdot \phi_{2}\left(\mathrm{x'}^{\prime}\right)}{2} \nonumber$

$\operatorname{DensityMatrixMix}_{\mathrm{i},\mathrm{j}} : = \operatorname{DensityMatrixMix}\left(x, x^{\prime}\right) \nonumber$ The obvious difference between a superposition and a mixture is the absence of off-diagonal elements, $$\phi_{1}(\mathrm{x}) \cdot \phi_{2}\left(\mathrm{x}^{\prime}\right)+\phi_{2}(\mathrm{x}) \cdot \phi_{1}\left(\mathrm{x}^{\prime}\right)$$, in the mixed state. This indicates the mixture is in a definite but unknown state; it is an example of classical ignorance.

An equivalent way to describe the difference between a superposition and a mixture, is to say that to calculate the probability of measurement outcomes for a superpostion you add the probability amplitudes and square the sum. For a mixture you square the individual probability amplitudes and sum the squares.

1. Nick Herbert (Quantum Reality, page 64) suggested ʺquonʺ be used to stand for a generic quantum object. ʺA quon is any entity, no matter how immense, that exhibits both wave and particle aspects in the peculiar quantum manner.

This page titled 1.99: Visualizing the Difference Between a Superposition and a Mixture is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.